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Random crumpling of paper disrupts the plane transforming it towards a compact sphere-like objects. Creasing straight lines gives order through predetermined sequencing towards specific reformations of the plane. This exploration looks at combining a straight-line creased grid with randomly crumpled creases.

The regularity of a self-distributive structurally ordered grid exhibits symmetry in circle-pattern not found elsewhere. It is independent from individual shapes or quality of material. http://wholemovement.com/blog/item/729-order-without-boundary-ii-geocoding

Crumpling appears without order. Yet there is a sense of organization in relationship to how the crumpling is done. The self-organization of a circle-patterned grid and random crumpling seem to compliment each other. In exploring this I have in some cases used printed images to add another layer of information to the reforming process.

Combining the circle-patterned grid and crumpling has parallels in nature as well as how we live our lives. Wrinkles record habits, a consistent tracking about what we have done, where we have been as we interact through space over time. These folds enliven the surface with unique expressions responsive to internal and external forces. A crumbled “mess” when opened suggests some kind of inherent order with a great deal of textural interest. Ordered folding is geometric in style and it lacks the uniqueness found in spontaneous crumpling.

Crumpled wad of paper Unwadded crumpled paper

To carefully unfold and then reform the creases of a crumpled surface reveals interesting variations with little possibility of exactly refolding the original wad ball.

Above left.) Paper folded to the circle-pattern hexagon grid, a 3-6 symmetry. The vertical line is the arbitrary-placed first fold. The two diagonal creases are the next two folds. These are the same proportional folds of division used in folding the circle. The rest of the creased grid lines are informed by the first three equally space creases.

Above right.) The same paper after it has been crumpled. Once the primary creases are established any random crumpling has little effect over the consistency and structural nature of reorganizing the folded grid.

Above left.) Paper reformed using the grid to reconfigure a five-fold symmetry.

Above right.) Same paper reformed to a square symmetry.

Below left.) This shows a rectangular shaped paper folded into a tetrahedron net; same as it would be in a circle. (The net creases are traced to make them easy to see.) These nine creases for the tetrahedron net are primarily used in the following demonstrations.

Below right.) I have tried to leave the center triangle in the net uncrumpled. Usually we cut off the “extra” paper leaving an isolated polygon. It is difficult to crumple one part of the paper without effecting the entire surface. Every crease is interconnected to all others through the paper itself.

Below.) Two reformations of the tetrahedron net above.

Below left.) The net is reformed to a tetrahedron.

Below right.) The same tetrahedron is formed before crumpling. It has the stylized geometric form. The tetrahedron pattern is identical in both.

Below left.) A regular tetrahedron with the excess paper more tightly crumpled.

Below right.) Three tetrahedra joined forming an open plane tetrahedron cavity.

Above left.) A fourth tetrahedron is added to the open face of the three on upper right, making a “solid” closed stellated tetrahedron. Each tetrahedron unit is consistent to the same size rectangle, roughly positioned to the same location on the paper, using the same measure to insure congruency of scale.

Above right.) Four tetrahedra rearranged to form a two-frequency tetrahedron revealing the open octahedron.

Above.) Four more tetrahedra added to the open triangle planes forming a stellated octahedron, or cube patterns. These added tetrahedra are without crumpling. There is consistent regularity of pattern not obvious in the random look of the form. Patterns are consistent; forms are a continually changing variable.

Below left.) Tracing the lines allows another way to see what is going on. Two papers differently creased are joined. One rectangle has been folded to the circle-pattern hexagon grid and reconfigured to pentagon symmetry. It has been placed on top of the other that has been crumbled.

Above right.) One paper has been both crumbled and folded to the circle-patterned hexagon grid and reconfigured to the pentagon.

Above.) Four pieces of copy paper folded to circle-pattern grid, partially crumpled and joined. The circle removed before folding is separately folded and crumbled; where folding the empty circle reveals the grid in the surrounding rectangle paper. This references last month’s blog where we saw the circle paper folds the grid *in* and the empty circle folds the grid *out*.

Again creases have been traced with a marker making them more pronounced. Glass is placed on the flat surface on top with the reformations below hanging out in 3-dimensions.

Tracing a folded pattern of creases is easy and straightforward, they are all straight lines; it is more difficult when tracing the crumpled folds. The angle of light changes our perception of where the crease and fold start and stop. Without shadow the crease appears in one place and direction; changing the light shifts perceived orientation. As with all tracings, the lines are a generalization of spatial relationships of movement. Tracing the residue of movement does not give an accurate picture of what has been in-formed. There is delight in the movement and spatial changes as well as coming to rest with it. I value are the unexpected surprises that happen along the way, they are hidden in the image. The limitation in folding any formula shields us from the unexpected, hids the experience of discovering the moment.

Below.) Four views of a single arrangement. Two diverse pictures have been joined combining crumpling and the hexagon grid into a single arrangement. The beauty is in experiencing the spatial relationships of images as they form an object.

Some of the 2-D images used are from past folded circles models photographed and combined with other imagery that now become material for exploring folding and crumpling. Both 2-D and 3-D are degrees of abstraction removed from the reality of expression.

You can find more about the images used in a previous blog http://wholemovement.com/blog/item/129-exploring-images-of-folding-circles

Below.) Two views of integrating three photo prints each folded to the circle-pattern grid and reformed. This is another way to explore the spatial relationships implied in 2-D images.

Below.) Two sides and one front view of a 3-D object by folding, crumpling, and joining three individual pictures. A transforming takes place from folding circles into 3-D objects, translating the objects to 2-D images, then folding those images back into into objects using the same folded creases used in the original circle folded models. This brings various stages of development together in a single object.

Above.) An early model using multiple circles folded to the tetrahedron net, where before reforming each circle, they have been crumpled to give an interesting surface and tactile appeal. Random crumpling softens the paper, yet seems to have little effect when reforming the structural net.

Below.) Four rectangular pieces of paper were first crumpled then folded in a four-frequency diameter circle-pattern hexagon grid. The units have been reformed to a variation on a truncated tetrahedron. Joined on their end points they form an octahedron with four open and four closed triangle planes. Given the crumpling and shape of paper it does not look like an octahedron, yet there are eight triangle planes in a regular octahedron arrangement.

Below.) Four more rectangles are folded to tetrahedra without crumpling. They are attached to the inverted triangle faces of the model above. There appears possibility for sustainable “new growth” when there is regularity of pattern. The open triangles are potential for continued growth.

Below.) A glossy cover stock 2ft square was folded to a randomly placed circle-pattern hexagon (3-6 grid.) Folding in 1/6 of the grid leaves a 5-10 symmetry formed to a pentagon. The paper outside the pentagon was then crumpled. Following are two variations in reforming the grid keeping to the pentagon symmetry.

Above left.) The pentagon is opened to show the hexagon star of the 3-6 symmetry grid.

Above right.) The grid is folded in reformed to a 4-8 symmetry. With each reformation the crumpled rectangle is changed.

Below.) Three circles and a scrap of crumpled paper combined to make something that looks like it might be biologically functional; particularly after adding a few twisty ties.

Below.) Four stages of opening the crumpled material around a solid tetrahedron made from four pieces of paper. The center triangle in each net is left flat and joined to form a solid regular tetrahedron with the rest of the paper around it being crumpled.

Below.) This series of six images show the tetrahedron from above with each of the four rectangular papers sequentially opened and flattened to show where the triangle is placed on that paper. These images lack the wonderful spatial quality displayed in these changes.

Tetrahedron with rectangles compressed. 1st rectangle opened flat.

2nd rectangle opened flat. 3rd rectangle opened flat.

4th rectangle opened flat. All rectangles opened somewhat equally.

Below.) Two separate images and one paper circle folded to the circle-pattern grid and combined. One image is crumpled, the other partially crumpled and reformed, with the third forming an icosahedron. Here 2-D images of 3-D objects are folded in 3-D, again using the same patterned grid used for the objects initial folded shown in the images. Each expression is layered into the next into what is now the 2-D images you see. Once this transforming process reached the virtual world it becomes a translations of zeros and ones. This is where this all started, by folding circles and straight line creases.

Two views give you an idea of the dimensionality of what otherwise would look flat.

The growing interest in origami expands the possibilities to further explore folding paper. So, when you wad up paper, unwad it and look at the beauty of what just randomly happened that you were going to throw away. If we looked at everything for what is beautiful before it becomes a throw-a-way, then maybe there would be more beauty and less garbage in the world. We have yet to realize the synergistic balance between regular ordering forward and the seemingly random spontaneous leaps that occur.

In believing the circle is image, we limit the imagination.

Draw a circle anywhere on a piece of paper.

Cut the circle from the paper.

How many circles are there?

Compare properties of both circles.

They are the same circle.

By removing the circle from the paper the other circle is left. One boundary separates the physical circle from the non-physical other circle. We might say one is positive, the other negative. One is where the other is not. The inverse of one is the other. The compliment is so intimate that without separation we miss the dual nature.

Fold both circles in half.

Notice the difference and similarities in each circle.

One circle contains the creases. With the other circle the paper references the unseen crease within the circle boundary. Having folded in half what is not there informs alignment with what is there.

Fold both half-folded circles twice more in ratio of 1:2 http://wholemovement.com/how-to-fold-circles.

Below: Having folded the half-folded circles into thirds, open them to find three diameters in each. Three chords are visible in the one removed; they are not visible in the other circle.

Three diameters, six radii define six areas and seven points. In the other circle the unseen diameters extended outward showing six line segments defining six areas and twelve points. The proportional folding is the same for both circles. Removing the image from the rectangle paper the other self-referencing, self-organizing circle is now a hole defined by context, yet functions in the same way as the folded paper. The creases not seen in the circle extend through the rectangle. The invisible is held with-in the visible, and for that reason often goes unrecognized. The unseen informs through space that which can be seen.

Fold circles to a higher frequency of the same pattern of division, see blog for instructions; http://wholemovement.com/blog/item/97-unity-origami

When folded to a higher frequency each circle reveals six diameters, the organization being in the first three. The second set of three diameters have a different proportioned division functioning as bisecting diameters to the first three.

Below left) Division of the first three diameters show four equal sections revealing a hexagon star and hexagon. With one the hexagon and star are on the inside boundary of the circle, with the other circle the star points and hexagon are creased into the rectangle on the outside of the circle boundary; one goes in the outer goes out.

Above right: When replacing the removed circle back into the other there are two ways to align the six diameters. One aligns the star points so both are in the same orientation; the other (pictured above) is where the star point diameters are in alignment with the in-between diameters of the other, a thirty-degree shift in position. The latter is the complete alignment of the grid showing two levels of grid division using all six diameters.

The creases in the paper have been drawn over for better visibility.

Below: Continuing a higher frequency folding in the rectangle (creasing lines parallel to lines already there) the equilateral triangle grid becomes obvious reflecting the same size grid we see in the folded circle. Coloring the same size and orientation of triangles makes the pattern clear. The removed circle shows more information; each triangle of the hexagon division is bisected in three directions rotating the grid thirty degrees to a smaller scale. This can be described as a fractal like “interference pattern” (upper right.)

Below: The removed circle has been reformed to a tetrahedron. This suggests the other circle will also reform into a tetrahedron. The form will look very different because of the difference in perimeter but the circle-pattern of arrangement will be the same. There are more equilateral triangles in the rectangle, which increases possibilities in multiples tetrahedra on different scales.

Below: The other circle can change symmetries from six diameters (hexagon,) to a five fold symmetry (pentagon,) to four (square,) and the three (triangle.) This is possible with the whole circle, the circle hole, and is demonstrable with any random shaped paper folded to the same circle-pattern. http://wholemovement.com/blog/item/729-order-without-boundary-ii-geocoding

Below: Fold creases into the rectangle reflecting what is already in the removed circle. Color the alternate areas of the second level division revealing another level of design using the creases.

Above: By replacing the removed circle into the rectangle the difference in scale becomes apparent. With consistency to the same alternate coloring of the folded grid the fractal scaling separates the inside and outside of the circle boundary.

Above: Both circles independently have been reformed to a square-based pyramid (half octahedron.) The pyramid from the rectangle is not complete until the removed circle is replaced, as if they had not been separated. The frequency difference is again apparent with the dual reformations aligned to the same symmetry showing consistency in pattern and difference in design.

Above: Left shows both forms folded into a "solid tetrahedron." On the right shows the other circle reformed to a tetrahedron pattern of four points in space. The circle-pattern is indicated by the arrangement of points (any four points in space is a tetrahedron pattern.) There are many possibilities reforming the tetrahedron using this grid-creased rectangle.

Below: Here are a few of many possible reconfigurations using the grid in the rectangle folded to the circle hole. There is no proportional relationship between inside circle and outside rectangle because of the initial arbitrary placement of the image. The diversity of forms suggests rich design possibilities in a proportionally aligned relationship of circle to perimeter. You might call this an organized, controlled crumpling of paper.

Circle-pattern is inherent to all shapes; all shapes are inherent in the circle-pattern. Were this not so we could not construct what we do with compass and straight edge, or remove the image from the plane and fold the circle into what is possible. By separating the circle from the rectangle part, itself a truncated circle, we see a great variety in different forms through consistency in pattern. The part and whole are so intimately bound that without one there is not the other, even in the appearance of separation.

Symmetry apparently has little to do with specific shapes and is more about proportional divisions of unity. Symmetry bound to pattern is therefore inherent to all shapes and forms. Transformation from one symmetry to another happens because they hold circle-pattern whole in common.

Below) This diagram relates sphere-to-circle compression with removing the circle image from the 2-D plane. Both forms carry spherical unity that can only be realized by moving from the illusion of 2-D to experience the dynamics by folding the circle. In this ways the intimate balance of dual compliments can be directly experienced.

Think about black holes as a function generating movement seen only in a spatial context. We are now theorizing about “white holes,” the opposite to black holes. Is not this what we have been folding; a black hole and a white whole? Sophisticated technological tools have expanded our ability to observe and measure what otherwise can not be seen, similarly the other circle stretches the idea of geometry and structural ordering and rearranging of systems observed to those that we do not see but experience the effects. This allows us to see how little we know about the unseen and unobservable aspects of physical reality. Space perceived as empty once again is shown to be occupied with higher-level phenomena that is beyond detection from lower levels of perception. The folds in the other circle hole are generated by the circle pattern not from the rectangle. There is not enough information within the shape of the rectangle to direct the circle-pattern movement in creasing.

The circumference informs both circle going in and going out. In this dual form there are two visible and two invisible circle planes, two curved planes between top and bottom separating the circle planes; one on the inside and the other on the outside of the paper, they both have the same volume. There are two circle edges where the three planes join. There are five congruent circle parts between two circles.

The dual circle, seen and unseen, reveal the same circle-pattern of organization; one folding into itself and the other folding out from itself. Like wise the concentric nature of the circle goes both endlessly in and out. The circle-pattern is unbounded by context allowing for countless expressions aligned through the formed and unformed concentric nature in unity on all scales.

Below: Two holes of different sizes, arbitrarily placed in a rectangle paper. Keeping the divisional creases in parallel, assures alignment between the two circles. Shown are a few of many reformations possible.

Below left: Drawing a non-concentric circle in a circle shape and removing it gives us three circles, two are congruent. To the right shows nine folds in the larger circle and three creases in the removed circle.

Below: A couple of reconfigurations.

Below: Four arbitrary placed circles cut from four larger circles as before, each with three folded diameters. Three diameters have been folded into the removed four circles. The other circle folding three diameters shows the four larger circles with chords that are not diameters. This makes the larger four circles when joined to a spherical vector equilibrium arrangement irregular and mismatched on the perimeter. The other circles form a regular and symmetrical bubble inside the irregularities of the four larger circles. When the removed circles are joined the same way there is the regular spherical form of the vector equilibrium that is identical to the other unseen bubble. (ref. blog http://wholemovement.com/how-to-fold-circles ) Both models are held together with bobby pins.

Below: The same four circles from above have been disassembled and rearranged to form a tetrahedron using the four removed circles as hubs to join the ends of the larger circles. The flexibility of the folded struts and removed circles accommodates a variety of angles and the irregularity of the off-centered effect of the arbitrary placement of the other circles. They are all tetrahedra in pattern, yet very different in form. With the model in lower left the circle planes are pushed in, the others show planes opened outward.

With this, I leave you to explore something that is not seen to find something that is.

Folding circles is a mind activity that takes place in the hands. This keeps my hands busy and my mind occupied with curiosity. Discovery is a recognition through mind attentive to what the hands are doing.

My mind periodically goes back to folding open forms and my hands play with what is familiar, having been done before. This way my eyes can see what the mind has missed. Again I looked at the open octahedron and explored joining multiples in various combinations. I chose the vector equilibrium with an open center because it reflects the open center of the octahedron units.

One paper plate folded to form an open octahedron unit.

Four open octahedron joined in a triangle arrangement.

Four sets of triangles joined in a tetrahedron pattern form a vector equilibrium system. Bobby pins are used to hold it together. It is open through the center in four directions and forms sixteen secondary open spaces not directly connected to the center, and has concave rhomboids on the six square faces. The arrangement of four tetrahedra in the same orientation sharing a common point forms the vector equilibrium with a defined center. Traditionally the vector equilibrium is called the cube octahedron and understood to be a solid figure. This open forming of the vector equilibrium using octahedra catches my interest.

I had some ¼” plywood veneer triangles that were angled to form octahedra left from another project. It seemed an interesting idea to form the above model in wood to see what difference in materials might make. This would keep my hands busy and my mind on the look out. It took 96 triangles to form 16 octahedron in 4 triangle sets to reconstruction the vector equilibrium arrangement.

Plywood pieces glued in open octahedra then glued and tape together.

The edges have been rounded and sanded. Wood dowels are added extending the triangle planes giving texture and visual interest.

The wood has been stained and the first round of painting applied.

Finished painting with gold leaf accents.

Following is a word-construction about processing the object with origin in the circle.

Cranium vector equilibrium with open center,

patterned system contained within its own inner cavities.

Unity without form

dividing personal and non-personal in relationship.

Structural pattern of three brings directives

for divergent sub-systems,

complements balanced in symmetry,

increasing outer boundary, organized

through inner connections.

From afar unity is relative; sort of, kind of like, but maybe not,

or possibly so, nothing seems, absolute, you think?

Construction fails unity

touched by unformed realization

of endless channels in distribution,

converging back to realization of never being less.

Between the doing of hand and the knowing of mind is the seeing of eye.

Looking at the octahedron I found something missed, not for lack of attention, only that my focus prevented seeing all that was there in the first place.

Observing six points of connection in the closest packing of four spheres tells us the octahedron is a spherical relationship. Both the octahedron and tetrahedron can be separated from spherical order as individualized polyhedra. Placing four tetrahedra point to point (spherical pattern) we find the same spatial octahedron relationship. The pattern is four, structure is three.

A single tetrahedron can reveal the octahedron by opening one vertex, allowing the properties of the tetrahedron to transform to those of an octahedron; 6 points showing 12 edge relationships of which 9 are formed, and 8 triangle planes, 4 surfaces and 4 open planes. Opening a tetrahedron doubles the number of triangle planes while increasing the number of vertex points half as much. if you count the inside surfaces then there are twelve planes.

Above) A tetrahedron opened to show the octahedron patterned relationship of 6 points.

Below) Two open tetrahedron joined edge-to-edge completing the "solid" enclosed octahedron.

Below right) Octahedron opened flat to show a net of eight small triangles made from two open tetrahedra. The two large triangles are short two more open tetrahedra to make a tetrahedron pattern.

Above left) Completing the tetrahedron pattern (three triangles around one make four) by using one of the large triangles as center and adding two more joining on half of the edge lengths as the first two. This pattern of of four makes a net of sixteen smaller triangles (4x4=16) from which is formed the regular twenty-sides icosahedron; 16 surfaces and 4 open faces.

Above) The sequence from tetrahedron net, to octahedron, to icosahedron net is a geometric progression of 4, 8, 16 triangles. This is reflected by the center triangles of 1, 2, 4, the number of circles used, reflecting spherical order. There is consistency in this development of the first three primary polyhedra that we do not see in traditional construction of individual figures.

There are a number of ways the octahedron net can be reformed, all are worth exploring. That these are all found within the circle assures they are transformational.

Below) Of the 6 edges around the middle of the octahedron, three adjacent edges are joined leaving three edges unattached. This opened figure can be reconfigured in three primary ways; the octahedron, a tetrahelix, and a bi-pentacap.

Below, a, b ,c) Three possible stable reformations from three open edges of the octahedron.

a. The octahedron shows three intersecting squares

b. A tetrahelix of three tetrahedra, one is the relationship between two. It can have a right or left handed twisting depending on whether the net is right or left hand.

c. The bi-pentacap with 10 triangle planes shows each pentagon has four triangle surfaces and one open face showing both inside and outside. The two pentacaps are joined by an open pentagon plane.

Here are other ways the octahedron net can be reformed without overlapping surfaces. The tetrahedron pattern has minimum two surfaces and two open faces that is consistent with the first fold of the circle in half.

d. Octahedron/tetrahedron combination. e. 2 open, 2 closed tetrahedra in a tetrahelix.

f. 5 tetrahedra in combination. g. One octahedron and two tetrahedra

h. Bi-pentacap showing 6 tetrahedra

i. An open square face without tetrahedra. This is the one configuration I missed and for the moment find interesting. It has 8 triangles surfaces and 2 open triangle faces; 10 triangle planes, the same number as the bi-pentacap (c. above.)

The open square face has no stability and when squeezed closed the unit will reform making 3 tetrahedra in a helix form (pic a. & e. above.) Squeezing opposite points towards each diagonal forms the edges of the helix, where one spins to a right and the other to the left. This also reflects the chirality in the net itself that can be right or left handed (See illus. of nets where joining tetrahedra is half way to the right or to the left.)

What happens when joining two sets together on the open square faces?

Above left) When joined on the squares a distorted icosahedron with twenty equilateral triangles is generated; 16 triangle surfaces with 4 open triangle planes. The properties are the same as the regular icosahedron except angles change the number of planes meeting at a vertex.

Above right) A regular icosahedron.

Below left) Two views showing two ways to put the square faces together. One changes the icosahedron pattern. They both have 20 equilateral triangle faces. The *left* shows the top and bottom surfaces are perpendicular to each other with the open faces on the same plane. On *right* shows the top and bottom surfaces are parallel in direction, with the open faces in a tetrahedral pattern. The open faces are consistent to the regular icosahedron. The tetrahedron arrangement of intervals are not found in the one on the left,

Above right) Another view of the same two pictured on the left. The red and black lines show two intersecting tetrahedra and the vertical green lines show the relationship of eight points of an elongated cube. The top and bottom look like rhomboids, they are not since the two triangles are not on the same plane. The two intersecting tetrahedron that form the cube are distorted yet all the triangles are congruent.

The regular icosahedron is not chiral. This distorted icosahedron shows two individual variations from the same arrangement by attaching on the square faces. There are other possible combinations of transformation that comes with open space that cannot happen with closed static forms.

Below) Joining open triangular planes in three possible orientations, where units are angled in different directions. Other variations are possible when joining both open and closed surfaces.

Below left) Three icosahedra joined on open faces so there is an open flow through the helix system.

Below right) Two more icosahedra were joined to the helix system on the left using a parallel top/bottom unit in the center. Four units are attached to the center unit on four open faces in a tetrahedron arrangement. Consistency in orientation and handedness from the first fold through all steps is important.

Above) Two views of 4 icosahedra joined on triangle faces in a tetrahedron pattern. Other combinations are possible. When the form becomes increasingly complex and individualized deviations occur that can stop any further generation.

Below) Four distorted icosahedra, consistent in orientation and chirality, are systematically joined through open faces in a tetrahelix pattern. Very different than the helix above.

Being consistent with each step in handedness and orientation while joining units is the easiest way to keep track of the variables. Consistency in pattern is the difference between generating order and falling into disorder. Pattern is consistent; the forms pattern takes often become confusing and chaotic when inconsistencies are introduced. I have taken many models apart looking for why they eventually break down and jam up, only to find I lost consistency in the process. Consistency in pattern will generate many unexpected surprises in a variety of forms.

Two circles reconfigured in two regular tetrahedra, joined to form the octahedron can than transform into a number of configurations showing combinations of 3, 4, 5, and 6 tetrahedra while at the same time generate 3, 4, and 5 fold symmetries. The two folded circles always remain two circles, regardless of the reformations. How do we explain a three-fold increase over the two tetrahedra we started with just by moving two circles? The first fold in the circle from which all folds are generated, is a pattern of movement revealing dual tetrahedra. Numbers keep track of things, they do not explain structural generation or the transforming process of spatial organization generating what often is unexpected.

Exploring without having an identifiable objective opens areas not predictable. As soon as I tie into what I think is going on, that becomes the objective of my exploration and I miss what else is there. It is a struggle to accommodate changes in our lives, yet transformational change is so easy and fluid through the movement of a paper circle as it reforms and reorganizes in multiples. Unencumbered geometry organizes systems that can be easily demonstrated by folding and joining circles. There is much to learn from folding the circle that has been missed by drawing pictures of them, folding polygons, and constructing static models; all which can be extremely engaging, although limited in-formation.

Over the years I have disregarded the impressed inner circle that gives form to paper plates. This deformation of the circle is not a property of the 3-D circle and is always flattened during creasing. Yet paper plates do come with this circle pressed into them, and concentricity is inherent in circles.

Jose Albers accordion pleating concentric circles has always intrigued me. There are been a few interesting developments coming out of Albers folding exercise. *Erik Demaine and Martin Demaine* at MIT have pushed it more towards art by cutting the circle, removing the center to give greater movement to the ribbons of folds as they are joined in complex curving systems of equilibrium. It is not uncommon to see paper engineers scoring curved lines and cutting paper moving away from the straight edges of traditional paper folding. Cutting the circle reveals many intriguing directions but the beauty of the spherical hyperbolic reconfiguration curving to itself is lost. It seems Albers point was the unexpected nature of the uncut circle plane by reforming through folding concentric circles that reveals a balanced movement around the center circle. http://wholemovement.com/blog/item/119-in-out-hyberbolic-surface.

In exploring this I went back to the great circle divisions of the spherical vector equilibrium, octahedron, and the icosidodecahedron, wondering how circumference sectors would change using concentric circles.

Below left) Four circles with three folded diameters systematically joined on edges form the spherical vector equilibrium where each circle shares a center point. This forms eight open tetrahedra. The impresses concentric circle shows a smaller scale spherical VE nested within the larger.

Below right) Four circles with three folded diameters plus three more creases to form an inscribed equilateral triangle that when joined has no center. The folded center on left moved to four places on the circumferences forming a single enclosed tetrahedron. Here we see both spherical and polyhedral forms of tetrahedra, reflecting the first fold of the circle in half forming both spherical and tetrahedral patterns of arrangement in the movement.

Below) Using the figure above on right, one folded tetrahedron http://wholemovement.com/how-to-fold-circles is joined to the center of each face of the large tetrahedron forming another tetrahedron of equal size. The circle sectors are flattened to the edges of the added tetrahedra forming vesicas showing the six faces of the cube resulting from two intersecting tetrahedra.

Below) Being consistent with all circumferences folded to the outside now allows for opening the cubic arrangement to spherical form revealing unusual divisions. On the right is a variation where one stellated tetrahedron is folded to a higher frequency grid enable to generate smaller triangles.

This is interesting but not where I want to go.

Starting again with one circle I fold the concentric inner circle that comes with the paper plate.

Above left) Creasing the inner circle shows the hyperbolic nature of concentric circles as the circle is folded and curves into itself. Right angle tension is created in the inner ring; a bobbed pin holds two opposite sides together.

Above right) Two circles have been folded (on left) and opened enough for one to fit into the other forming a tetrahedron arrangement where the outer circle of one joins the inner circle of the other at four points. Were the two circles joined on the outer circumference the two circles would lie flat one on the other.

This again substantiates the primacy of the tetrahedron but yields little formally.

Not to complicate things I decided to add one fold of the circle in half with the one concentric circle.

Above) This is one example of reforming the circle to one folded straight line and one folded circle line. Each half outside of the folded circle is folded in opposite directions from the concentric crease.

Below) Two views of one possibility in joining three of the units above.

Below) Adding more units further compounds the complexity of curving surfaces.Two views of six units joined in an octahedron pattern.

Below) Adding four more of the same units the three axis of the octahedron become more apparent.

Below) Four units attached in a tetrahedron pattern showing different arrangements of the right angle relationship between opposite edges.

Below) Using the model in the last picture above right, two are joined forming a dual tetrahedron pattern. One tetrahedron intersects the other showing six rhombic relationships in a distorted cube arrangement.

Below) Using another reformation of one straight and one inner circle, and joining multiples units reveals a variety of open systems in various polyhedral patterns. Two views of six circles arranged in a tetrahedron relationship bring out the octahedron relationship inherent in the tetrahedron pattern.

Below) Multiple concentric circles have been added increasing the level of complexity; again two views of each.

Below) The difference between the concentric circle folded by Albers and folding a diameter in the circle is the congruent concave and convex as each semicircle fits the other. When diameter fold is opened tension develops along the center line.

Below) On the left are two circles folded in half with four accordion pleats. In these units the circumference has been partially separated by folding them in opposite directions along the outside ring allowing space between them. On the right is a seven ring circle.

Above) Two examples of concentric circles reformed into spirals.

Below) A system of four circles joined showing different views.

Below) Two circles are folded to the same off-centered division to the creases inner circle. Joining the straight edges forms two intersecting circles where the vesica piscis becomes spatial.

The folds inherent to circle pattern are also found in all irregular parts of the circle. http://wholemovement.com/blog/item/130-order-without-boundary

This means concentric circles can be folded at any location on any shape of foldable material and there will always be a unique reconfiguration in relationship between placement and individual boundary configuration of the paper as it conforms to the circle effect on the plane surface.

Below) An example of concentric folding using an irregular rectangular piece of paper showing both sides standing in different positions.

I continue to delight in the exercise Jose Albers gave his students by exploring the beautiful forms that are produced in concentric circle unity, and the possibilities to combine with other ways in reforming circles.

To change a paradigm is to change one's frame of beliefs about the world to another. Is it possible in a world of infinite parts to think about an inclusive Whole? Is this even desirable in a world seemingly bounded by infinite boundaries?

The circle is fundamental in math as a 2-D image concept. Can we believe the circle is more than this image we draw, more than a defined circumference? Can we equally accept the circle as a 3-D form that comprehensively demonstrates through folding the idea of spherical unity that is inclusively whole?

Traditionally the idea of the circle is demonstrated by cutting a sphere in half. By compressing a sphere a 3-D circle is generated without destroying the wholeness of the only form that represents absolute unity. When the sphere is compressed the volume of the circle remains equal to the sphere. The surface boundary of the sphere changes properties. The sphere/circle is whole; nothing is taken away or added, it is transformed through movement from a spherical form to a circular form through compression. The circle/sphere functions as both Whole and discrete part simultaneously.

Euclidean construction of a circle is based on the definition of a point. Using a compass to draw a line that is considered as a set of points on an imaginary plane at a given distance from a single point, we draw and thus define a circle, calling the center point 'origin'. The small circle point contains even smaller equally concentric circles in the same way circles get progressively larger through opening the compass. There is no fixed outside or inside boundary, only unseen circles in alignment. The radius measures the openness of a compass used to draw a circle. It does not fully measure a circle, although we generalize and use it that way in the abstract.

The property of the image shows one continuous curved line defining a given area; we imagin individualized points. The properties of the 3-D circle show five distinct circles with volume. Think of a disc, such as a coin. Circles are dynamic, they move, can be moved; and through a consistent and systematic sequence of folds will self generate proportional relationships that are not possible with other shapes, yet contain all shapes.

If there is uncertainty about the difference between a 2-D and a 3-D circle then do the following:

Draw a circle, use a compass or trace around a circular object.

Cut the circle from the paper you drew it on.

Observe the difference between the image you drew, the hole that is left in the paper, and the circle in your hand.

Each of these must be understood for what they are in able to understand the interdependent nature of one to the other. There is nothing to suggest throwing out traditional understanding about the circle; on the contrary we are moved to enlarge what we believe by embracing the full nature of the object the picture represents.

This shift of perspective can be understood by looking at the word ‘geometry.’ Geometry means earth measure, measuring things of the earth. Geo refers to the earth. The earth is spherical and the sphere is *whole*. Metry means measure, keeping track through *movement* in space. We can now understand geometry comprehensively as *wholemovement; *a self-referencing system of the whole. This better describes, giving demonstration of our presently evolving worldview in a universe among many where much of what we see and know now was a short time ago unimagined. We can no longer afford to hold a geocentric view about ourselves, anymore than we can sustain the belief that the sun and celestial bodies revolve around the earth. We are a small part of something much larger and far more complex that yet imagined.

This suggest maybe it is time to consider shifting from a commonly held parts-to-whole thinking to a Whole–to-parts perspective. To start with the Whole in the form of the circle/sphere and observe information as revealed through movement, the order and proportional arrangements, the appropriateness of interactive systems, is perhaps worth considering. Through observation of what is generated from the circle, through folding and joining reveals what is not possible using other shapes or forms. The circle is its own center; an alignment of inner and outer boundaries. The whole is origin to all seen and unseen parts, realized and unrealized, revealing ongoing potential. The origin, both center and outer boundary, is already whole and cannot be constructed or deconstructed.

Broadening our perspective does not deny any mathematical value. What temporary benefits progress will eventually drop away being replaced by greater knowledge that elevates value towards a greater realization of human potential as we begin to see the finer reality of further abstraction. We need a more inclusive and comprehensive way of thinking about our universe, our place in it, and how we negotiate the inequalities and fragmentation, the separation that are no longer sustainable given our present understanding of the interrelated and interdependent nature of parts and systems. There is a fear of the unknown that isolates and keeps us from reaching out on a cosmic scale. Fear keeps us from reaching outside of the circle we have draw around ourselves.

Comparing properties of the image and the circle/sphere compression shows differences between 2-D & 3-D. The 3-D circle shows five discernable circles; three circle planes and two circle edge lines that contain a volumetric location. It all starts with a spherical point. Tradition shows two points making a relationship formed by connecting with a line. Three non linear points form three lines forming the edges of a triangle area. Similar components in both 2 & 3-D are points, lines, and planes. Folding the circle by touching any two points will generate two more points at the end of the crease, an axial line of division perpendicular to the distance between points that shows six relationships. These four points can be form into six edges that define four solid triangle planes. There are two solid planes and two open planes defined by five edges you can see, one edge you cannot see. Minimum description of a tetrahedron is for points in space. Edge relationships and planes are inherent in this spatial movement of two points of the circle. The first fold of the circle around the creased line/diameter/axis, in both directions forms two tetrahedra, one the inverse of the other at the same time showing a spherical pattern of movement as origin.

For those that like pictures with their words, I have included some photos of an old folded circle piece that bares relationship to this discussion. These pictures, “The Transient of Venus” show different views of an idea about Venus as it moves in a slow spiral around the sun observed from earth where for a short time appearing as a dark spot moving across the sun. This object is made using seven paper plate circles, four are folded to an open tetrahedron and joined with string holding them together, and the fifth is left unfolded, the other two form Venus and its path. It is painted conforming to the folded equilateral triangular grid matrix, indicating the activity between the surface and interior of the sun as Venus passes giving a temporary linear alignment of three spheres in space.

This form becomes an expression of multiple parts and by adding string, the black square box, and paint gives design to the discrete parts of the folded matrix. Here we have made in “parts” a multiple expression of the whole. Parts and whole are folded one into the other.

The circle/sphere is the only form that demonstrates, with some degree, the idea of a comprehensive Whole. In that regard no other form can be observed that is principle to all symmetries, and all subsequent generations of parts. What is principle is what comes first, not what is believed at the time to be most important. The whole, even defined as “nothing,” for lack of boundary, binds all potential to the principles of first movement. Does our compass open wide enough and will it close small enough to construct all circles contained in the one that can not be drawn?

From paper plate circles to scrap paper to scrap cardboard the circle pattern does not change. Using different materials folded to the circle pattern shows the adaptability and strength of structural organization and the limitations of different materials. Pattern and material form together reveal possibilities not possible in separation.

Below) are six views of a tetrahedron model using two pieces of scrap cardboard. Each piece is folded and scored as show in previous blogs (Order Without Boundary and Order Without Boundary II: Geocoding ). Only the folds necessary for the tetrahedron net as it appears in the circle have been folded. The measure for both folding is the same, the shape of the cardboard and placement of folding are different. Given those two variables every time a piece is folded the same way it will have a different proportional reconfiguration. This individual tetrahedron system can not be duplicated unless the configuration of paper is exactly the same as the position for folding, then the variables become consistently the same for all pieces. Then there would be a formula, not a discovery.

The surface that extends beyond the folded tetrahedron often makes it difficult to see the form of the tetrahedron or to even identify the pattern. Because of the differences in the original flat shapes it might even look arbitrary. Each piece is folded to form half a tetrahedron, two triangles each, then joined at right angles to each other forming a full tetrahedron. There is an inclosed "solid" tetrahedron that is central to holding the system together. You might be able to spot it by finding the small equilateral triangles.

Below) are multiple views of a truncated tetrahedron using four pieces of scrap cardboard. There are more creases added to the pattern in order to form the hexagon unit, one of the properties of the truncated tetrahedron (the folded 4-frequency diameter grid.) Each piece is different in configuration and placement of folds, but the measure and folding process remains consistently the same.

It is difficult to discern any symmetry or underlying pattern of organization by looking the diversity of different views. Each scrap is a different shape where each creased grid is located differently on each piece breaking the symmetry of anything recognizable. With the exception of one view above showing a vertical line of symmetry all other views seem random. In both models above, the formed pattern is not obvious by looking at the entire system. The form is an expression of two variables, the configuration of surface used and placement of the folded grid. All aspects are subject to the consistent organization of the circle pattern and sub-pattern of tetrahedron and truncated tetrahedron. The possibilities of variations appear to be countless depending on combinations of the two variables with the two constants. This reflects what we see with any two points when touched together and creased that will always generate two more points at the intersection of the crease and perimeter.

There are no folded line segments; each crease goes all the way across from edge-to-edge; as observed in folding the circle; the complete pattern for any shape. This provides a consistency of pattern and proportion throughout the surface allowing for a certain amount of reconfiguration beyond second level patterns as they are formed. To fold a line segment without connection to the perimeter is leaving information out, thus creating missing information.

The strength of these models lies with the structural nature of pattern and the rigidity of the material. When made from paper they are more fragile and appear less strong because of more flexibility in the extended surface area. Every material will have its effect on how we perceive the pattern and organization of order that seems to be primary in a forming process.

Three primary circle patterned grids; 3-6, 4-8, and 5-10 can be folded using a scrap of paper. They are inherent in any flat plane surface and can be folded without regard to boundary configuration. Last month’s blog, “Order Without Boundary” showed in the first fold, a ratio of 1:2 (a division of one unit in two parts) providing only three options in a regular division of the plane. The circle is primary from which other shapes can be folded, demonstrable by anyone willing to fold a paper circle in half, observe what is generated and use that information to continue folding.

Any point on a plane can generate one of three specific division of symmetry. This suggest every point on a plane is a center point for dividing the plane symmetrically. The circle itself is the center where symmetry becomes the means of encoding directives for division starting with the diameter. The circle is the center and a local centered systems can happen anywhere on the surface. This supports the idea of center as origin regardless of location or configuration.

All regular and irregular flat plane perimeters are inherent in the circle for the same reason the circle can be drawn on any plane. One center to one circle traditionally uses the compass to draw the circle giving importance to the radius as measure. Movement out, when going back stops at or goes through the center point. When moving into the center the scale changes to find other endless concentric circles. The center is both inside and outside by nature of the concentric alignment of the circumference.

Traditionally polygons are constructed using the circle and straight edge. Folding the circle produces the straight edge and yields the same information plus more. The triangle, square and pentagon are all discernible in the hexagon, primary in the circle. This arrangement of three straight line divisions from a single arbitrary fold is inherent to all shapes. The information is encoded to the angles of division around any location.

Years ago when tracing 2 and 3-D geometry constructions back to the circle I did not see that it was not about the shape of the circle, but about proportional movement that happens where straight line relationships hold consistent to a circle pattern. Information is encoded in the relationships of creases directive to what will and will not work. Folding is sequential coding that generates progressively complex systems, it is not construction.

Following are a few pictures expanding on the three primary polygons and the straight-line grids that exhibit the nature of a circular pattern.

*Above left) * A piece of paper folded to show the hexagon star with the 6 equally spaced bisectors. The star points are all equidistant from the center point making it an enfolded circular pattern. The lines forming the two large triangles do not go edge-to-edge; they are partially formed creases. For one sequence of folding see last months blog: http://wholemovement.com/blog/item/130-order-without-boundary

*Above right)* Each of the six partial creases forming the triangles has been fully creased across the paper, similar to every fold in the circle being a chord. This fully completes the division of the hexagon, six triangles around the center, each bisected in three directions with all lines extending to the perimeter.

*Above left)* The partial folds forming the hexagon have been folded to papers edge thereby expanding the matrix.

*Above right)* One sixth of the hexagon is folded in making a pentagon star. (6-1=5) The pentagon pyramid can also be formed with both concave and convex functions.

*Above left)* Folding in two sixths (6-2=4) leaves a square-based pyramid, half of an octahedron.

*Above right)* Three sixths or one half of the hexagon folded in forms two tetrahedra, a three sided regular pyramid with four triangles. One is inside of the other where both have an inside and an outside making a full count of four tetrahedra.

*Above) * The small equilateral triangles in one direction are colored to show the regularity of the triangle grid. The information is there to extend the grid over the entire plane.

This 3-6 folded grid gives options for reconfiguring the paper to the five, four, and three-fold symmetries.

The folding process is the same for the 4-8, and the 5-10 symmetries. With each reduction of symmetry the possibilities for complexity increases. Symmetry can be seen as the inherent gatekeeper for directional development.

*Above left)* The 5-10 folded grid shows the pentagon defined by partial folds as seen with the hexagon.

*Above right)* The completed five-fold grid is traced in black to show all lines folded to edge of shape.

*Above)* The completed grid is colored to show alternating areas.

*Above left)* The pentagon grid is reformed to a square-based pyramid. (5-1=4)

*Above right) * 5-2=3 shows a triangle-based pyramid, or tetrahedron. These are the same polyhedra formed by the hexagon grid, but they are differently proportioned because of the different in symmetry.

*Above left)* The square is defined by partial creases as observed with the hexagon and pentagon shapes. This is the same folding process using different angle directives.

*Above right) *Folding continues to fill in and out the square grid, in much the same way as the hexagon and pentagon.

*Above)* The square grid is colored showing the alternating triangle grid that is basic to a square division.

*Above left) * The 4-8 grid folded showing the right angle tetrahedron inside of a larger truncated tetrahedron.

*Above right)* Accordion pleating the tetrahedron is a function of parallel creases in the gird.

*Below)* The above folded paper is reconfigured into a cube with only six squares to the outside and the rest of the paper is folded to the inside.

* Below left) * Four right angle tetrahedra are folded to the same measured square grid in different places on different shapes of scrap paper.

*Below right)* The long edges of the tetrahedra are joined edge-to-edge in a tetrahedron pattern forming a cube. The excess paper coming from six of twelve diagonals to the cube show where the tetrahedra have been joined.

The coded order is that a 3-6-12 folded grid can be reconfigured to the pentagon, the square and the triangle. The 5-10 is limited to the square and triangle. The 4-8 grid is limited to the triangle. This is not just a construction formula, but the inherent function of the folding process. The 3-6-12 folding being first, contains the geometry and mathematical relationships that are fundamental to further development and is important for math.

The place to start is with observations by the folder; how they fold and what is there that was not before the fold. Angles are the most obvious directive since that is primary starting from the first fold, a ratio of 1:2. The individual proportions determines the direction of symmetry.

*Above)* The differences in angle divisions from any point of symmetry on any plane is determined by the proportional decision of congruent angles. Here the divisions of angles is revealed in the 3-6-12 grid. There is much to add, subtract, multiply and divide once we can determine the unit angles, and consider each proportionally within the entire plane. Even without knowing arithmetic or numbers, the proportional relationships of units to units are easy to see when they are all in the same place sharing functions.

*Above)* The square grid shows a different angle division. The central 90° angle is shared by both the 3-6-12 and 4-8 grid.

*Above)* The pentagon arrangement shows different angles that have different proportional properties from the other two. It shows 90° angles, but not centered as with the two above. This makes the proportions and ratios unique to the pentagon. There are different symmetries dividing the space around each point of intersection depending where they are located in the grid.

The number pattern for the angle degrees for each symmetry is surprisingly consistent and reflected in numbers.

With the first fold there are three possible choices to continue generation of structural order. Direction is coded in angle units consistent to 9 as the last number of the first level of natural numbers. Folding 60° in 180 gets two division of 60° three times and again 30° that will reveal 90°. Folding 90° again in half get 45°(4+5=9). By folding the 180 (9) into 144° angle (9) we automatically get 36° (9) with half of 144 being 72 (9). There are no other options to the consistency of angle direction encoded to the circle. When we start with polygons we are limited by the specific angles of the shape.

In looking at the angles of each folding 9 is a common base seen in reducing the numbers to a single digit .

**For 3-6-12**

30° 3

60° 6

90° 9

120° 3

150° 6

180° 9

630° 36 9

**For 4-8 **

45° 9

90° 9

135° 9

180° 9

450° 39 9

**For 5-10**

36° 9

72° 9

108° 9

144° 9

180° 9

540° 45 9

It would follow that the summation of all these numbers reduced down to a single digit would be 9. They all started from a single 90° movement. Three squared, structure to itself, is the number 9, the complete sequence before transition to the next level out. There is a consistent development even in the numbers as they represent angle units.

There many math functions and spatial systems that comes from these three folded grids. The more math you know the greater depth of discovery, particular when it comes to guiding children towards becoming aware of their own observations. At the same time math is not necessary to explore the beauty and complexities inherent in reforming and joining folded scrapes of paper.

While this information is accessible using any scrap of paper it is most visible in folding the circle shape. Information generated in the circle is not possible with other shapes since only the circle has a circumference. Every fold is a chord; an acknowledgement of the complete unity/unit of shape. Information is coded into the circle and revealed through a consistent directive of sequential folding. It is important that each crease be a full chord, otherwise contextual information is missing. This is equally important using irregular paper.

The 3-6 and 4-8 share a 90°central angle. Starting with either the three or the four folding both end up with 12 divisions. (only the 3-6-12 is consistent to the ratio where the other two are not.)The 4-8 is inherent in 3-6-12 making the 3-6-12 accessible through the 4-8. This interconnection between symmetries expands our ability to see what otherwise is not possible when things are separated.

*Below left)* Folding the 4-8 into a 12 symmetry division, six diameters, reveals two hexagons one enfolded to the other. The difference is not so much visual as procedural at this point.

*Above right)* Folding the hexagon star shows the internal triangular grid. The creases are outlined to better see the grid.

*Below left)* Starting with the 4-8 folding in a square and incorporating the 3-6-12 directives reveals the hexagon in the square intentionally not in alignment with the perimeter, showing the hexagon center can be anywhere in the square. Following different directives of the same pattern reveals many possibilities for combining symmetries, freeing us from having to rely on constructing polygons.

*Above right)* The creases are lined in black to see one possible combination between the square and hexagon sharing 12 fold symmetry.

*Above)* Alternate areas have been colored in to show the pentagon star is inherent to the combing of the 3-6 and 4-8. There are a great variety of intersections and shapes for reforming the square where the center of the hexagon can appear at any location in the square area on any scale. First there are pattern directives, then forming of branches that generate information with which to design. Symmetry seems to be the constant where discernable or not.

*Above)* The same folding of the hexagon using the regularity of the square shape as directive for the folding. This is more inline with traditional symmetry of polygon alignment.

That first fold shows coded information is activated by proportional folding which determines different symmetries. I call this information geocoding for lack of something more descriptive. It happens without construction or formulas. We do not think about geometric shapes as being coding devises for structural development. The folded ratio of 1:2 is structural; two quantities without separation is a set of three. The number 3 is structure coded into the first fold and with consistent development assures structural regeneration. Each of 3, 4, and 5 are further coded to branch into different matrices. Three, four, and five add up to twelve; another form of 1 and 2 or set of 3. Being first this primary code is core to all folding that follows revealing the geometry observed in nature and imbedded in the abstracted symbols we used mathematically. Geometry is as much a coding devise for spatial development of complex systems as it is a tool for proving abstract mathematical concepts.

We do not think of basic shapes as coding devises that generate complex geometries because of a traditional position about 2-D construction with emphasis on product. Geocoded information is important to the development of technology just as it is in the geometry which cannot be separated from the instructive information inherent to shapes that make up the forms and systems of the physical and biological world.

While I get excited about the product potential in folding circles, it is the information that it generates and the consistency of development that holds my attention. So here again, as twenty-five years ago, I look at the circle and ask, what it is? Why does it reveal what other shapes can not, yet contains information for all other shapes? How does it easily generate much that we have learned to construct? Why don’t children fold circles? Why isn’t folding circles part of our curriculum? Have we missed some crucial aspect of the circle by only drawing pictures of it? Has Euclid’s influence limited us? Is it because the idea of the circle is a symbol for nothing, where zero prevents us from seeing anything there? Maybe because of how we have defined the circle? I guess it all has something to do with why we continue to ignore folding circles.

Continuing the exploration of folding the circle pattern without the circle shape (http://wholemovement.com/blog/item/121-circles-from-scrap ) brings up questions about boundary and the separation between the inner center and outward centered locations. Folding any irregular shape of paper in a sequence of folding proportional to that first fold generates the same three patterned grids found in folding the circle. They come from the first fold in any plane surface and appear to have little to do with the shape of paper used yet each reveals a centered system of organized symmetry.

Polygons are truncated subsystems of the circle limited to the symmetry from number of sides. Conversely polygons of any shape inherently carry the circle pattern and center that can be revealed through folding. The first fold is arbitrary and can be made anywhere (http://wholemovement.com/blog/item/92-center-off-center) The first fold divides one shape in two parts (a ratio of 1:2.) As seen with the circle there are three principled options for consistently folding this ratio; 3-6, 4-8 and 5-10. (http://wholemovement.com/blog/itemlist/date/2010/9?catid=140 )

Once the first fold is made any of these primary symmetries can be initiated at any point along the first line that can be anywhere, thereby establishing a centered coordinate system through the development of a concentric pattern anywhere on the surface. It is through straight-line creases that we see proportional differences between the triangle/hexagon, triangle/pentagon and triangle/square.

FOLDING THE CIRCLE using the 3:6 ratio, a 4-frequency diameter hexagon star is formed. The ratio is extended to a 3:6:12 grid division of the circle. (Folds are lined in black for visibility.)

For folding instructions see http://wholemovement.com/blog/itemlist/date/2011/2?catid=140

Let’s see what this looks like folding scrap paper with an arbitrary perimeter.

1a. b.

1a & b.) Randomly fold any size paper to a ratio of one piece in two parts.

2a. b.

2a & b.) With folded edge towards you place a finger arbitrarily along the folded edge and fold over until the folded angle looks the same as the resultant adjacent angle. This approximates folding into thirds. Press slightly indicating fold; do not crease. Turn over bringing opposite edges together (one over and one under the middle section.) Slide back and forth until edges on both sides are even, then give a strong crease.

3a. b.

3a & b.) Open to flat paper finding three equally spaced straight lines intersecting at a point resulting from the placement of finger on the first fold. Refold to triangular shape just folded.

4a. b.

4a & b.) Place the center point somewhere on the edge between the folded over part and the center point on one side forming a right triangle, give a strong crease. Open the fold and locate where that crease meets the opposite edge.

5a. b.

5a & b.) Fold the same on the other side, bringing the center point to the point where the last crease intersects the edge. A strong crease will form another right angle triangle opposite in direction. Open to see two intersecting right angle bisectors on each edge showing a symmetrical crossing centered to a partially formed equilateral triangle.

6a. b.

6a & b.) Fold in half, giving a good crease when the two edges are even and open to find three intersecting creases (reflection of #3a.)

7a. b.

7a & b.) Open to flat paper to observe the hexagon star with twelve radians. Refold back to the equilateral triangle. Then fold between the two creased edge points creasing the third side of the equilateral triangle.

8a. b.

8a & b.) Open paper flat to shows six equally spaced bisected triangles in a hexagon arrangement. Refold to single triangle with three bisectors. Between the two end points remove excess paper outside of the triangle or by cutting straight across to make an equilateral triangle, or as this example shows, tear an arc approximating a circle.

For another reference for this folding go to: http://wholemovement.com/blog/itemlist/date/2011/2?catid=140 )

The material outside the polyhedral boundary gets in the way of how we perceive order and construct from regular polygons. Is there another purpose for having “excess” material? What can be learned from material that seems extraneous to traditional construction with polygons?

9a-c.) Below, the three symmetries, (*a*.) 3-6-12, (*b*.) 4-8 and (*c*.) 5-10 have all been folded in the same sequence with an arbitrary first fold across the rectangular paper, each with a different proportioned ratio of 1:2. The difference in symmetry is with the angle of the second fold that determines the following relationships. Looking at one triangular sector there is a proportional scaling out from the local center to the limits of the paper.

9a. b.

c.

10.) a.) Below the 3-6-12 hexagon is folded. b.) show the hexagon reconfigured into the pentagon [6-1=5.] c.) shows the hexagon reformed to the square [6-2=4.] and then (d.) reformed to the triangle [6-3=3.]

10a. b.

c. d.

Last month I used images of some models to explore 2-D expressions of the 3-D objects I have folded. One of the 2-D images is printed here in black and white, then folded back to a 3-D object completing the transformation from 3-D to 2-D and back to 3-D.

11a. b.

11a. & b.) A black/white square image printed on a rectangular 8 ½" x 11" paper. It has been folded and reconfigured into a pentagon arrangement.

12a.

12a.) Here the paper is folded down to a 4-8 symmetry in a square arrangement that is accordion pleated.

b.) 5 square units are joined forming a cube with one open plane to view the inside cubic space.

b. c.

c.) All six sides form an enclosed cube. Each folded unit has a different orientation leaving an irregular outer boundary to the cube form. The printed image visually disrupts recognizing the cubic form.

13a & b.) Below, the same cubic configuration without the printed image shows an arbitrary and irregular boundary. With the “excess” material cut away the twelve edges and six stellated-truncated square planes of the cube are more easily seen.

a. b.

14a-d.) Below four rectangular pieces of paper have been printed and folded to a tetrahedron configuration and joined in a tetrahedron pattern showing the open octahedron. There are many ways the material beyond the tetrahedron can be manipulated, reformed, and modified to change the boundaries without changing the opened center or tetrahedron symmetry. (Hairpins are holding the tetrahedra together.)

14a. b.

c. d.

15a & b.) Below the “excess” material has been partially eliminated and then cut down to the edges leaving the two-frequency tetrahedron.

a. b.

How much do we lose when constructing a polyhedral object from polygons where the pattern context supporting that object has been eliminated? How far out do boundaries go beyond our perception and to what extent do they affect what we see? It is reasonable to construct and assemble polygons, even folding polygons/polyhedra using circles. What value is there in seeing what seems extraneous or excessive material beyond the ordering of forms we are familiar with? Why deal with material that seems to get in the way? Folding/unfolding is an important life-forming function and an important part of geometry with renewed interest in origami and paper folding.

Expanding the perimeter of any shape does not take away from the order that comes from any individually centered point within an infinite matrix of points on a given plane. Every point reveals a centeredness that gives access to the matrix that generates that point of intersection that at once is everywhere.

The pattern of concentric circles, PCC, seems to prevail throughout being revealed through construction methods, truncation of the circle, reforming by folding the circle, and by folding any shape to show forth a polygonal system of symmetry.

These are all static pictures that represent what takes place prior to and beyond the object we experience.

### Exploring Images Through Folding Circles

Written by Bradford Hansen-SmithA few months of no posting brings another direction in exploring the circle. Using images of folded circles that show the structural nature of geometry and combining them with NASA images of the observable universe allows me to image what would otherwise not be possible to see by bringing together vastly differing scales in one frame.

Over the years I have used photo images of my sculpture collaging them together and with other images giving greater breadth and depth to the ideas that were embodied in the original work. These current images are a continuation of that process.

*Above*: the helix is made into the image of a cosmic flower with the center spiral derived from the same helix image. The second image shows the helix to be a dynamic cosmic force experienceable through local consciousness.

*Below*: is a spiral model with a design printed on the circles. The image has been modified to show a cosmic hot spot that appears to move structurally out forming a galactic cloud. Possibly the cloud is from itself forming a cosmic tendril of local significants. In the lower left is another space-star circle model.

*Above*: this image also uses the space-star model seen imediately above.

*Above*: an image combining two different folded circle spheres gives visualization to the dispersal of regulated energy.

*Below*: another spiral is in process of gathering cosmic debris and at the same time distributing as it moves both in and out of itself. The tape holding the model together is a nice touch.

*Above*: this single model has taken two different directions in these fanciful treatments suggesting familar life forms.

*Below*: A 3-D folded circle wall piece showing a variety of spatial layers. In keeping with the layering of the image, Photoshop was the right tool. The second image departs from the textural layering of the first and is pretty straight forward.

* *

*Below*: an old model is used to show a coming together, a collision or just attraction, of an object to itself where the image of the cosmos has moved to the foreground. The model is from an early exploration of folded circles where bobby pins, vinyl tubing, and string had been attached.

*Below*: is a model with an open interior space that appears to be a floating cube. This image suggest one possibility where floating is somewhere between gravity and illusion.

*Above*: a radiant image derived from the picture of a spherical model.

*Below*: another image using models, both previously used, where the forms take on some semblance of human consciousness.

Below: Two different models spanning 15 years have come together in a computer version of a layered projection through a checkered grid, which in Photoshop indcates an empty or transparent layer.

bove:

Random crumpling of paper disrupts the plane transforming it towards a compact sphere-like objects. Creasing straight lines gives order through predetermined sequencing towards specific reformations of the plane. This exploration looks at combining a straight-line creased grid with randomly crumpled creases.

The regularity of a self-distributive structurally ordered grid exhibits symmetry in circle-pattern not found elsewhere. It is independent from individual shapes or quality of material. http://wholemovement.com/blog/item/729-order-without-boundary-ii-geocoding

Crumpling appears without order. Yet there is a sense of organization in relationship to how the crumpling is done. The self-organization of a circle-patterned grid and random crumpling seem to compliment each other. In exploring this I have in some cases used printed images to add another layer of information to the reforming process.

Combining the circle-patterned grid and crumpling has parallels in nature as well as how we live our lives. Wrinkles record habits, a consistent tracking about what we have done, where we have been as we interact through space over time. These folds enliven the surface with unique expressions responsive to internal and external forces. A crumbled “mess” when opened suggests some kind of inherent order with a great deal of textural interest. Ordered folding is geometric in style and it lacks the uniqueness found in spontaneous crumpling.

Crumpled wad of paper Unwadded crumpled paper

To carefully unfold and then reform the creases of a crumpled surface reveals interesting variations with little possibility of exactly refolding the original wad ball.

Above left.) Paper folded to the circle-pattern hexagon grid, a 3-6 symmetry. The vertical line is the arbitrary-placed first fold. The two diagonal creases are the next two folds. These are the same proportional folds of division used in folding the circle. The rest of the creased grid lines are informed by the first three equally space creases.

Above right.) The same paper after it has been crumpled. Once the primary creases are established any random crumpling has little effect over the consistency and structural nature of reorganizing the folded grid.

Above left.) Paper reformed using the grid to reconfigure a five-fold symmetry.

Above right.) Same paper reformed to a square symmetry.

Below left.) This shows a rectangular shaped paper folded into a tetrahedron net; same as it would be in a circle. (The net creases are traced to make them easy to see.) These nine creases for the tetrahedron net are primarily used in the following demonstrations.

Below right.) I have tried to leave the center triangle in the net uncrumpled. Usually we cut off the “extra” paper leaving an isolated polygon. It is difficult to crumple one part of the paper without effecting the entire surface. Every crease is interconnected to all others through the paper itself.

Below.) Two reformations of the tetrahedron net above.

Below left.) The net is reformed to a tetrahedron.

Below right.) The same tetrahedron is formed before crumpling. It has the stylized geometric form. The tetrahedron pattern is identical in both.

Below left.) A regular tetrahedron with the excess paper more tightly crumpled.

Below right.) Three tetrahedra joined forming an open plane tetrahedron cavity.

Above left.) A fourth tetrahedron is added to the open face of the three on upper right, making a “solid” closed stellated tetrahedron. Each tetrahedron unit is consistent to the same size rectangle, roughly positioned to the same location on the paper, using the same measure to insure congruency of scale.

Above right.) Four tetrahedra rearranged to form a two-frequency tetrahedron revealing the open octahedron.

Above.) Four more tetrahedra added to the open triangle planes forming a stellated octahedron, or cube patterns. These added tetrahedra are without crumpling. There is consistent regularity of pattern not obvious in the random look of the form. Patterns are consistent; forms are a continually changing variable.

Below left.) Tracing the lines allows another way to see what is going on. Two papers differently creased are joined. One rectangle has been folded to the circle-pattern hexagon grid and reconfigured to pentagon symmetry. It has been placed on top of the other that has been crumbled.

Above right.) One paper has been both crumbled and folded to the circle-patterned hexagon grid and reconfigured to the pentagon.

Above.) Four pieces of copy paper folded to circle-pattern grid, partially crumpled and joined. The circle removed before folding is separately folded and crumbled; where folding the empty circle reveals the grid in the surrounding rectangle paper. This references last month’s blog where we saw the circle paper folds the grid *in* and the empty circle folds the grid *out*.

Again creases have been traced with a marker making them more pronounced. Glass is placed on the flat surface on top with the reformations below hanging out in 3-dimensions.

Tracing a folded pattern of creases is easy and straightforward, they are all straight lines; it is more difficult when tracing the crumpled folds. The angle of light changes our perception of where the crease and fold start and stop. Without shadow the crease appears in one place and direction; changing the light shifts perceived orientation. As with all tracings, the lines are a generalization of spatial relationships of movement. Tracing the residue of movement does not give an accurate picture of what has been in-formed. There is delight in the movement and spatial changes as well as coming to rest with it. I value are the unexpected surprises that happen along the way, they are hidden in the image. The limitation in folding any formula shields us from the unexpected, hids the experience of discovering the moment.

Below.) Four views of a single arrangement. Two diverse pictures have been joined combining crumpling and the hexagon grid into a single arrangement. The beauty is in experiencing the spatial relationships of images as they form an object.

Some of the 2-D images used are from past folded circles models photographed and combined with other imagery that now become material for exploring folding and crumpling. Both 2-D and 3-D are degrees of abstraction removed from the reality of expression.

You can find more about the images used in a previous blog http://wholemovement.com/blog/item/129-exploring-images-of-folding-circles

Below.) Two views of integrating three photo prints each folded to the circle-pattern grid and reformed. This is another way to explore the spatial relationships implied in 2-D images.

Below.) Two sides and one front view of a 3-D object by folding, crumpling, and joining three individual pictures. A transforming takes place from folding circles into 3-D objects, translating the objects to 2-D images, then folding those images back into into objects using the same folded creases used in the original circle folded models. This brings various stages of development together in a single object.

Above.) An early model using multiple circles folded to the tetrahedron net, where before reforming each circle, they have been crumpled to give an interesting surface and tactile appeal. Random crumpling softens the paper, yet seems to have little effect when reforming the structural net.

Below.) Four rectangular pieces of paper were first crumpled then folded in a four-frequency diameter circle-pattern hexagon grid. The units have been reformed to a variation on a truncated tetrahedron. Joined on their end points they form an octahedron with four open and four closed triangle planes. Given the crumpling and shape of paper it does not look like an octahedron, yet there are eight triangle planes in a regular octahedron arrangement.

Below.) Four more rectangles are folded to tetrahedra without crumpling. They are attached to the inverted triangle faces of the model above. There appears possibility for sustainable “new growth” when there is regularity of pattern. The open triangles are potential for continued growth.

Below.) A glossy cover stock 2ft square was folded to a randomly placed circle-pattern hexagon (3-6 grid.) Folding in 1/6 of the grid leaves a 5-10 symmetry formed to a pentagon. The paper outside the pentagon was then crumpled. Following are two variations in reforming the grid keeping to the pentagon symmetry.

Above left.) The pentagon is opened to show the hexagon star of the 3-6 symmetry grid.

Above right.) The grid is folded in reformed to a 4-8 symmetry. With each reformation the crumpled rectangle is changed.

Below.) Three circles and a scrap of crumpled paper combined to make something that looks like it might be biologically functional; particularly after adding a few twisty ties.

Below.) Four stages of opening the crumpled material around a solid tetrahedron made from four pieces of paper. The center triangle in each net is left flat and joined to form a solid regular tetrahedron with the rest of the paper around it being crumpled.

Below.) This series of six images show the tetrahedron from above with each of the four rectangular papers sequentially opened and flattened to show where the triangle is placed on that paper. These images lack the wonderful spatial quality displayed in these changes.

Tetrahedron with rectangles compressed. 1st rectangle opened flat.

2nd rectangle opened flat. 3rd rectangle opened flat.

4th rectangle opened flat. All rectangles opened somewhat equally.

Below.) Two separate images and one paper circle folded to the circle-pattern grid and combined. One image is crumpled, the other partially crumpled and reformed, with the third forming an icosahedron. Here 2-D images of 3-D objects are folded in 3-D, again using the same patterned grid used for the objects initial folded shown in the images. Each expression is layered into the next into what is now the 2-D images you see. Once this transforming process reached the virtual world it becomes a translations of zeros and ones. This is where this all started, by folding circles and straight line creases.

Two views give you an idea of the dimensionality of what otherwise would look flat.

The growing interest in origami expands the possibilities to further explore folding paper. So, when you wad up paper, unwad it and look at the beauty of what just randomly happened that you were going to throw away. If we looked at everything for what is beautiful before it becomes a throw-a-way, then maybe there would be more beauty and less garbage in the world. We have yet to realize the synergistic balance between regular ordering forward and the seemingly random spontaneous leaps that occur.

In believing the circle is image, we limit the imagination.

Draw a circle anywhere on a piece of paper.

Cut the circle from the paper.

How many circles are there?

Compare properties of both circles.

They are the same circle.

By removing the circle from the paper the other circle is left. One boundary separates the physical circle from the non-physical other circle. We might say one is positive, the other negative. One is where the other is not. The inverse of one is the other. The compliment is so intimate that without separation we miss the dual nature.

Fold both circles in half.

Notice the difference and similarities in each circle.

One circle contains the creases. With the other circle the paper references the unseen crease within the circle boundary. Having folded in half what is not there informs alignment with what is there.

Fold both half-folded circles twice more in ratio of 1:2 http://wholemovement.com/how-to-fold-circles.

Below: Having folded the half-folded circles into thirds, open them to find three diameters in each. Three chords are visible in the one removed; they are not visible in the other circle.

Three diameters, six radii define six areas and seven points. In the other circle the unseen diameters extended outward showing six line segments defining six areas and twelve points. The proportional folding is the same for both circles. Removing the image from the rectangle paper the other self-referencing, self-organizing circle is now a hole defined by context, yet functions in the same way as the folded paper. The creases not seen in the circle extend through the rectangle. The invisible is held with-in the visible, and for that reason often goes unrecognized. The unseen informs through space that which can be seen.

Fold circles to a higher frequency of the same pattern of division, see blog for instructions; http://wholemovement.com/blog/item/97-unity-origami

When folded to a higher frequency each circle reveals six diameters, the organization being in the first three. The second set of three diameters have a different proportioned division functioning as bisecting diameters to the first three.

Below left) Division of the first three diameters show four equal sections revealing a hexagon star and hexagon. With one the hexagon and star are on the inside boundary of the circle, with the other circle the star points and hexagon are creased into the rectangle on the outside of the circle boundary; one goes in the outer goes out.

Above right: When replacing the removed circle back into the other there are two ways to align the six diameters. One aligns the star points so both are in the same orientation; the other (pictured above) is where the star point diameters are in alignment with the in-between diameters of the other, a thirty-degree shift in position. The latter is the complete alignment of the grid showing two levels of grid division using all six diameters.

The creases in the paper have been drawn over for better visibility.

Below: Continuing a higher frequency folding in the rectangle (creasing lines parallel to lines already there) the equilateral triangle grid becomes obvious reflecting the same size grid we see in the folded circle. Coloring the same size and orientation of triangles makes the pattern clear. The removed circle shows more information; each triangle of the hexagon division is bisected in three directions rotating the grid thirty degrees to a smaller scale. This can be described as a fractal like “interference pattern” (upper right.)

Below: The removed circle has been reformed to a tetrahedron. This suggests the other circle will also reform into a tetrahedron. The form will look very different because of the difference in perimeter but the circle-pattern of arrangement will be the same. There are more equilateral triangles in the rectangle, which increases possibilities in multiples tetrahedra on different scales.

Below: The other circle can change symmetries from six diameters (hexagon,) to a five fold symmetry (pentagon,) to four (square,) and the three (triangle.) This is possible with the whole circle, the circle hole, and is demonstrable with any random shaped paper folded to the same circle-pattern. http://wholemovement.com/blog/item/729-order-without-boundary-ii-geocoding

Below: Fold creases into the rectangle reflecting what is already in the removed circle. Color the alternate areas of the second level division revealing another level of design using the creases.

Above: By replacing the removed circle into the rectangle the difference in scale becomes apparent. With consistency to the same alternate coloring of the folded grid the fractal scaling separates the inside and outside of the circle boundary.

Above: Both circles independently have been reformed to a square-based pyramid (half octahedron.) The pyramid from the rectangle is not complete until the removed circle is replaced, as if they had not been separated. The frequency difference is again apparent with the dual reformations aligned to the same symmetry showing consistency in pattern and difference in design.

Above: Left shows both forms folded into a "solid tetrahedron." On the right shows the other circle reformed to a tetrahedron pattern of four points in space. The circle-pattern is indicated by the arrangement of points (any four points in space is a tetrahedron pattern.) There are many possibilities reforming the tetrahedron using this grid-creased rectangle.

Below: Here are a few of many possible reconfigurations using the grid in the rectangle folded to the circle hole. There is no proportional relationship between inside circle and outside rectangle because of the initial arbitrary placement of the image. The diversity of forms suggests rich design possibilities in a proportionally aligned relationship of circle to perimeter. You might call this an organized, controlled crumpling of paper.

Circle-pattern is inherent to all shapes; all shapes are inherent in the circle-pattern. Were this not so we could not construct what we do with compass and straight edge, or remove the image from the plane and fold the circle into what is possible. By separating the circle from the rectangle part, itself a truncated circle, we see a great variety in different forms through consistency in pattern. The part and whole are so intimately bound that without one there is not the other, even in the appearance of separation.

Symmetry apparently has little to do with specific shapes and is more about proportional divisions of unity. Symmetry bound to pattern is therefore inherent to all shapes and forms. Transformation from one symmetry to another happens because they hold circle-pattern whole in common.

Below) This diagram relates sphere-to-circle compression with removing the circle image from the 2-D plane. Both forms carry spherical unity that can only be realized by moving from the illusion of 2-D to experience the dynamics by folding the circle. In this ways the intimate balance of dual compliments can be directly experienced.

Think about black holes as a function generating movement seen only in a spatial context. We are now theorizing about “white holes,” the opposite to black holes. Is not this what we have been folding; a black hole and a white whole? Sophisticated technological tools have expanded our ability to observe and measure what otherwise can not be seen, similarly the other circle stretches the idea of geometry and structural ordering and rearranging of systems observed to those that we do not see but experience the effects. This allows us to see how little we know about the unseen and unobservable aspects of physical reality. Space perceived as empty once again is shown to be occupied with higher-level phenomena that is beyond detection from lower levels of perception. The folds in the other circle hole are generated by the circle pattern not from the rectangle. There is not enough information within the shape of the rectangle to direct the circle-pattern movement in creasing.

The circumference informs both circle going in and going out. In this dual form there are two visible and two invisible circle planes, two curved planes between top and bottom separating the circle planes; one on the inside and the other on the outside of the paper, they both have the same volume. There are two circle edges where the three planes join. There are five congruent circle parts between two circles.

The dual circle, seen and unseen, reveal the same circle-pattern of organization; one folding into itself and the other folding out from itself. Like wise the concentric nature of the circle goes both endlessly in and out. The circle-pattern is unbounded by context allowing for countless expressions aligned through the formed and unformed concentric nature in unity on all scales.

Below: Two holes of different sizes, arbitrarily placed in a rectangle paper. Keeping the divisional creases in parallel, assures alignment between the two circles. Shown are a few of many reformations possible.

Below left: Drawing a non-concentric circle in a circle shape and removing it gives us three circles, two are congruent. To the right shows nine folds in the larger circle and three creases in the removed circle.

Below: A couple of reconfigurations.

Below: Four arbitrary placed circles cut from four larger circles as before, each with three folded diameters. Three diameters have been folded into the removed four circles. The other circle folding three diameters shows the four larger circles with chords that are not diameters. This makes the larger four circles when joined to a spherical vector equilibrium arrangement irregular and mismatched on the perimeter. The other circles form a regular and symmetrical bubble inside the irregularities of the four larger circles. When the removed circles are joined the same way there is the regular spherical form of the vector equilibrium that is identical to the other unseen bubble. (ref. blog http://wholemovement.com/how-to-fold-circles ) Both models are held together with bobby pins.

Below: The same four circles from above have been disassembled and rearranged to form a tetrahedron using the four removed circles as hubs to join the ends of the larger circles. The flexibility of the folded struts and removed circles accommodates a variety of angles and the irregularity of the off-centered effect of the arbitrary placement of the other circles. They are all tetrahedra in pattern, yet very different in form. With the model in lower left the circle planes are pushed in, the others show planes opened outward.

With this, I leave you to explore something that is not seen to find something that is.

Folding circles is a mind activity that takes place in the hands. This keeps my hands busy and my mind occupied with curiosity. Discovery is a recognition through mind attentive to what the hands are doing.

My mind periodically goes back to folding open forms and my hands play with what is familiar, having been done before. This way my eyes can see what the mind has missed. Again I looked at the open octahedron and explored joining multiples in various combinations. I chose the vector equilibrium with an open center because it reflects the open center of the octahedron units.

One paper plate folded to form an open octahedron unit.

Four open octahedron joined in a triangle arrangement.

Four sets of triangles joined in a tetrahedron pattern form a vector equilibrium system. Bobby pins are used to hold it together. It is open through the center in four directions and forms sixteen secondary open spaces not directly connected to the center, and has concave rhomboids on the six square faces. The arrangement of four tetrahedra in the same orientation sharing a common point forms the vector equilibrium with a defined center. Traditionally the vector equilibrium is called the cube octahedron and understood to be a solid figure. This open forming of the vector equilibrium using octahedra catches my interest.

I had some ¼” plywood veneer triangles that were angled to form octahedra left from another project. It seemed an interesting idea to form the above model in wood to see what difference in materials might make. This would keep my hands busy and my mind on the look out. It took 96 triangles to form 16 octahedron in 4 triangle sets to reconstruction the vector equilibrium arrangement.

Plywood pieces glued in open octahedra then glued and tape together.

The edges have been rounded and sanded. Wood dowels are added extending the triangle planes giving texture and visual interest.

The wood has been stained and the first round of painting applied.

Finished painting with gold leaf accents.

Following is a word-construction about processing the object with origin in the circle.

Cranium vector equilibrium with open center,

patterned system contained within its own inner cavities.

Unity without form

dividing personal and non-personal in relationship.

Structural pattern of three brings directives

for divergent sub-systems,

complements balanced in symmetry,

increasing outer boundary, organized

through inner connections.

From afar unity is relative; sort of, kind of like, but maybe not,

or possibly so, nothing seems, absolute, you think?

Construction fails unity

touched by unformed realization

of endless channels in distribution,

converging back to realization of never being less.

Between the doing of hand and the knowing of mind is the seeing of eye.

Looking at the octahedron I found something missed, not for lack of attention, only that my focus prevented seeing all that was there in the first place.

Observing six points of connection in the closest packing of four spheres tells us the octahedron is a spherical relationship. Both the octahedron and tetrahedron can be separated from spherical order as individualized polyhedra. Placing four tetrahedra point to point (spherical pattern) we find the same spatial octahedron relationship. The pattern is four, structure is three.

A single tetrahedron can reveal the octahedron by opening one vertex, allowing the properties of the tetrahedron to transform to those of an octahedron; 6 points showing 12 edge relationships of which 9 are formed, and 8 triangle planes, 4 surfaces and 4 open planes. Opening a tetrahedron doubles the number of triangle planes while increasing the number of vertex points half as much. if you count the inside surfaces then there are twelve planes.

Above) A tetrahedron opened to show the octahedron patterned relationship of 6 points.

Below) Two open tetrahedron joined edge-to-edge completing the "solid" enclosed octahedron.

Below right) Octahedron opened flat to show a net of eight small triangles made from two open tetrahedra. The two large triangles are short two more open tetrahedra to make a tetrahedron pattern.

Above left) Completing the tetrahedron pattern (three triangles around one make four) by using one of the large triangles as center and adding two more joining on half of the edge lengths as the first two. This pattern of of four makes a net of sixteen smaller triangles (4x4=16) from which is formed the regular twenty-sides icosahedron; 16 surfaces and 4 open faces.

Above) The sequence from tetrahedron net, to octahedron, to icosahedron net is a geometric progression of 4, 8, 16 triangles. This is reflected by the center triangles of 1, 2, 4, the number of circles used, reflecting spherical order. There is consistency in this development of the first three primary polyhedra that we do not see in traditional construction of individual figures.

There are a number of ways the octahedron net can be reformed, all are worth exploring. That these are all found within the circle assures they are transformational.

Below) Of the 6 edges around the middle of the octahedron, three adjacent edges are joined leaving three edges unattached. This opened figure can be reconfigured in three primary ways; the octahedron, a tetrahelix, and a bi-pentacap.

Below, a, b ,c) Three possible stable reformations from three open edges of the octahedron.

a. The octahedron shows three intersecting squares

b. A tetrahelix of three tetrahedra, one is the relationship between two. It can have a right or left handed twisting depending on whether the net is right or left hand.

c. The bi-pentacap with 10 triangle planes shows each pentagon has four triangle surfaces and one open face showing both inside and outside. The two pentacaps are joined by an open pentagon plane.

Here are other ways the octahedron net can be reformed without overlapping surfaces. The tetrahedron pattern has minimum two surfaces and two open faces that is consistent with the first fold of the circle in half.

d. Octahedron/tetrahedron combination. e. 2 open, 2 closed tetrahedra in a tetrahelix.

f. 5 tetrahedra in combination. g. One octahedron and two tetrahedra

h. Bi-pentacap showing 6 tetrahedra

i. An open square face without tetrahedra. This is the one configuration I missed and for the moment find interesting. It has 8 triangles surfaces and 2 open triangle faces; 10 triangle planes, the same number as the bi-pentacap (c. above.)

The open square face has no stability and when squeezed closed the unit will reform making 3 tetrahedra in a helix form (pic a. & e. above.) Squeezing opposite points towards each diagonal forms the edges of the helix, where one spins to a right and the other to the left. This also reflects the chirality in the net itself that can be right or left handed (See illus. of nets where joining tetrahedra is half way to the right or to the left.)

What happens when joining two sets together on the open square faces?

Above left) When joined on the squares a distorted icosahedron with twenty equilateral triangles is generated; 16 triangle surfaces with 4 open triangle planes. The properties are the same as the regular icosahedron except angles change the number of planes meeting at a vertex.

Above right) A regular icosahedron.

Below left) Two views showing two ways to put the square faces together. One changes the icosahedron pattern. They both have 20 equilateral triangle faces. The *left* shows the top and bottom surfaces are perpendicular to each other with the open faces on the same plane. On *right* shows the top and bottom surfaces are parallel in direction, with the open faces in a tetrahedral pattern. The open faces are consistent to the regular icosahedron. The tetrahedron arrangement of intervals are not found in the one on the left,

Above right) Another view of the same two pictured on the left. The red and black lines show two intersecting tetrahedra and the vertical green lines show the relationship of eight points of an elongated cube. The top and bottom look like rhomboids, they are not since the two triangles are not on the same plane. The two intersecting tetrahedron that form the cube are distorted yet all the triangles are congruent.

The regular icosahedron is not chiral. This distorted icosahedron shows two individual variations from the same arrangement by attaching on the square faces. There are other possible combinations of transformation that comes with open space that cannot happen with closed static forms.

Below) Joining open triangular planes in three possible orientations, where units are angled in different directions. Other variations are possible when joining both open and closed surfaces.

Below left) Three icosahedra joined on open faces so there is an open flow through the helix system.

Below right) Two more icosahedra were joined to the helix system on the left using a parallel top/bottom unit in the center. Four units are attached to the center unit on four open faces in a tetrahedron arrangement. Consistency in orientation and handedness from the first fold through all steps is important.

Above) Two views of 4 icosahedra joined on triangle faces in a tetrahedron pattern. Other combinations are possible. When the form becomes increasingly complex and individualized deviations occur that can stop any further generation.

Below) Four distorted icosahedra, consistent in orientation and chirality, are systematically joined through open faces in a tetrahelix pattern. Very different than the helix above.

Being consistent with each step in handedness and orientation while joining units is the easiest way to keep track of the variables. Consistency in pattern is the difference between generating order and falling into disorder. Pattern is consistent; the forms pattern takes often become confusing and chaotic when inconsistencies are introduced. I have taken many models apart looking for why they eventually break down and jam up, only to find I lost consistency in the process. Consistency in pattern will generate many unexpected surprises in a variety of forms.

Two circles reconfigured in two regular tetrahedra, joined to form the octahedron can than transform into a number of configurations showing combinations of 3, 4, 5, and 6 tetrahedra while at the same time generate 3, 4, and 5 fold symmetries. The two folded circles always remain two circles, regardless of the reformations. How do we explain a three-fold increase over the two tetrahedra we started with just by moving two circles? The first fold in the circle from which all folds are generated, is a pattern of movement revealing dual tetrahedra. Numbers keep track of things, they do not explain structural generation or the transforming process of spatial organization generating what often is unexpected.

Exploring without having an identifiable objective opens areas not predictable. As soon as I tie into what I think is going on, that becomes the objective of my exploration and I miss what else is there. It is a struggle to accommodate changes in our lives, yet transformational change is so easy and fluid through the movement of a paper circle as it reforms and reorganizes in multiples. Unencumbered geometry organizes systems that can be easily demonstrated by folding and joining circles. There is much to learn from folding the circle that has been missed by drawing pictures of them, folding polygons, and constructing static models; all which can be extremely engaging, although limited in-formation.

Over the years I have disregarded the impressed inner circle that gives form to paper plates. This deformation of the circle is not a property of the 3-D circle and is always flattened during creasing. Yet paper plates do come with this circle pressed into them, and concentricity is inherent in circles.

Jose Albers accordion pleating concentric circles has always intrigued me. There are been a few interesting developments coming out of Albers folding exercise. *Erik Demaine and Martin Demaine* at MIT have pushed it more towards art by cutting the circle, removing the center to give greater movement to the ribbons of folds as they are joined in complex curving systems of equilibrium. It is not uncommon to see paper engineers scoring curved lines and cutting paper moving away from the straight edges of traditional paper folding. Cutting the circle reveals many intriguing directions but the beauty of the spherical hyperbolic reconfiguration curving to itself is lost. It seems Albers point was the unexpected nature of the uncut circle plane by reforming through folding concentric circles that reveals a balanced movement around the center circle. http://wholemovement.com/blog/item/119-in-out-hyberbolic-surface.

In exploring this I went back to the great circle divisions of the spherical vector equilibrium, octahedron, and the icosidodecahedron, wondering how circumference sectors would change using concentric circles.

Below left) Four circles with three folded diameters systematically joined on edges form the spherical vector equilibrium where each circle shares a center point. This forms eight open tetrahedra. The impresses concentric circle shows a smaller scale spherical VE nested within the larger.

Below right) Four circles with three folded diameters plus three more creases to form an inscribed equilateral triangle that when joined has no center. The folded center on left moved to four places on the circumferences forming a single enclosed tetrahedron. Here we see both spherical and polyhedral forms of tetrahedra, reflecting the first fold of the circle in half forming both spherical and tetrahedral patterns of arrangement in the movement.

Below) Using the figure above on right, one folded tetrahedron http://wholemovement.com/how-to-fold-circles is joined to the center of each face of the large tetrahedron forming another tetrahedron of equal size. The circle sectors are flattened to the edges of the added tetrahedra forming vesicas showing the six faces of the cube resulting from two intersecting tetrahedra.

Below) Being consistent with all circumferences folded to the outside now allows for opening the cubic arrangement to spherical form revealing unusual divisions. On the right is a variation where one stellated tetrahedron is folded to a higher frequency grid enable to generate smaller triangles.

This is interesting but not where I want to go.

Starting again with one circle I fold the concentric inner circle that comes with the paper plate.

Above left) Creasing the inner circle shows the hyperbolic nature of concentric circles as the circle is folded and curves into itself. Right angle tension is created in the inner ring; a bobbed pin holds two opposite sides together.

Above right) Two circles have been folded (on left) and opened enough for one to fit into the other forming a tetrahedron arrangement where the outer circle of one joins the inner circle of the other at four points. Were the two circles joined on the outer circumference the two circles would lie flat one on the other.

This again substantiates the primacy of the tetrahedron but yields little formally.

Not to complicate things I decided to add one fold of the circle in half with the one concentric circle.

Above) This is one example of reforming the circle to one folded straight line and one folded circle line. Each half outside of the folded circle is folded in opposite directions from the concentric crease.

Below) Two views of one possibility in joining three of the units above.

Below) Adding more units further compounds the complexity of curving surfaces.Two views of six units joined in an octahedron pattern.

Below) Adding four more of the same units the three axis of the octahedron become more apparent.

Below) Four units attached in a tetrahedron pattern showing different arrangements of the right angle relationship between opposite edges.

Below) Using the model in the last picture above right, two are joined forming a dual tetrahedron pattern. One tetrahedron intersects the other showing six rhombic relationships in a distorted cube arrangement.

Below) Using another reformation of one straight and one inner circle, and joining multiples units reveals a variety of open systems in various polyhedral patterns. Two views of six circles arranged in a tetrahedron relationship bring out the octahedron relationship inherent in the tetrahedron pattern.

Below) Multiple concentric circles have been added increasing the level of complexity; again two views of each.

Below) The difference between the concentric circle folded by Albers and folding a diameter in the circle is the congruent concave and convex as each semicircle fits the other. When diameter fold is opened tension develops along the center line.

Below) On the left are two circles folded in half with four accordion pleats. In these units the circumference has been partially separated by folding them in opposite directions along the outside ring allowing space between them. On the right is a seven ring circle.

Above) Two examples of concentric circles reformed into spirals.

Below) A system of four circles joined showing different views.

Below) Two circles are folded to the same off-centered division to the creases inner circle. Joining the straight edges forms two intersecting circles where the vesica piscis becomes spatial.

The folds inherent to circle pattern are also found in all irregular parts of the circle. http://wholemovement.com/blog/item/130-order-without-boundary

This means concentric circles can be folded at any location on any shape of foldable material and there will always be a unique reconfiguration in relationship between placement and individual boundary configuration of the paper as it conforms to the circle effect on the plane surface.

Below) An example of concentric folding using an irregular rectangular piece of paper showing both sides standing in different positions.

I continue to delight in the exercise Jose Albers gave his students by exploring the beautiful forms that are produced in concentric circle unity, and the possibilities to combine with other ways in reforming circles.

To change a paradigm is to change one's frame of beliefs about the world to another. Is it possible in a world of infinite parts to think about an inclusive Whole? Is this even desirable in a world seemingly bounded by infinite boundaries?

The circle is fundamental in math as a 2-D image concept. Can we believe the circle is more than this image we draw, more than a defined circumference? Can we equally accept the circle as a 3-D form that comprehensively demonstrates through folding the idea of spherical unity that is inclusively whole?

Traditionally the idea of the circle is demonstrated by cutting a sphere in half. By compressing a sphere a 3-D circle is generated without destroying the wholeness of the only form that represents absolute unity. When the sphere is compressed the volume of the circle remains equal to the sphere. The surface boundary of the sphere changes properties. The sphere/circle is whole; nothing is taken away or added, it is transformed through movement from a spherical form to a circular form through compression. The circle/sphere functions as both Whole and discrete part simultaneously.

Euclidean construction of a circle is based on the definition of a point. Using a compass to draw a line that is considered as a set of points on an imaginary plane at a given distance from a single point, we draw and thus define a circle, calling the center point 'origin'. The small circle point contains even smaller equally concentric circles in the same way circles get progressively larger through opening the compass. There is no fixed outside or inside boundary, only unseen circles in alignment. The radius measures the openness of a compass used to draw a circle. It does not fully measure a circle, although we generalize and use it that way in the abstract.

The property of the image shows one continuous curved line defining a given area; we imagin individualized points. The properties of the 3-D circle show five distinct circles with volume. Think of a disc, such as a coin. Circles are dynamic, they move, can be moved; and through a consistent and systematic sequence of folds will self generate proportional relationships that are not possible with other shapes, yet contain all shapes.

If there is uncertainty about the difference between a 2-D and a 3-D circle then do the following:

Draw a circle, use a compass or trace around a circular object.

Cut the circle from the paper you drew it on.

Observe the difference between the image you drew, the hole that is left in the paper, and the circle in your hand.

Each of these must be understood for what they are in able to understand the interdependent nature of one to the other. There is nothing to suggest throwing out traditional understanding about the circle; on the contrary we are moved to enlarge what we believe by embracing the full nature of the object the picture represents.

This shift of perspective can be understood by looking at the word ‘geometry.’ Geometry means earth measure, measuring things of the earth. Geo refers to the earth. The earth is spherical and the sphere is *whole*. Metry means measure, keeping track through *movement* in space. We can now understand geometry comprehensively as *wholemovement; *a self-referencing system of the whole. This better describes, giving demonstration of our presently evolving worldview in a universe among many where much of what we see and know now was a short time ago unimagined. We can no longer afford to hold a geocentric view about ourselves, anymore than we can sustain the belief that the sun and celestial bodies revolve around the earth. We are a small part of something much larger and far more complex that yet imagined.

This suggest maybe it is time to consider shifting from a commonly held parts-to-whole thinking to a Whole–to-parts perspective. To start with the Whole in the form of the circle/sphere and observe information as revealed through movement, the order and proportional arrangements, the appropriateness of interactive systems, is perhaps worth considering. Through observation of what is generated from the circle, through folding and joining reveals what is not possible using other shapes or forms. The circle is its own center; an alignment of inner and outer boundaries. The whole is origin to all seen and unseen parts, realized and unrealized, revealing ongoing potential. The origin, both center and outer boundary, is already whole and cannot be constructed or deconstructed.

Broadening our perspective does not deny any mathematical value. What temporary benefits progress will eventually drop away being replaced by greater knowledge that elevates value towards a greater realization of human potential as we begin to see the finer reality of further abstraction. We need a more inclusive and comprehensive way of thinking about our universe, our place in it, and how we negotiate the inequalities and fragmentation, the separation that are no longer sustainable given our present understanding of the interrelated and interdependent nature of parts and systems. There is a fear of the unknown that isolates and keeps us from reaching out on a cosmic scale. Fear keeps us from reaching outside of the circle we have draw around ourselves.

Comparing properties of the image and the circle/sphere compression shows differences between 2-D & 3-D. The 3-D circle shows five discernable circles; three circle planes and two circle edge lines that contain a volumetric location. It all starts with a spherical point. Tradition shows two points making a relationship formed by connecting with a line. Three non linear points form three lines forming the edges of a triangle area. Similar components in both 2 & 3-D are points, lines, and planes. Folding the circle by touching any two points will generate two more points at the end of the crease, an axial line of division perpendicular to the distance between points that shows six relationships. These four points can be form into six edges that define four solid triangle planes. There are two solid planes and two open planes defined by five edges you can see, one edge you cannot see. Minimum description of a tetrahedron is for points in space. Edge relationships and planes are inherent in this spatial movement of two points of the circle. The first fold of the circle around the creased line/diameter/axis, in both directions forms two tetrahedra, one the inverse of the other at the same time showing a spherical pattern of movement as origin.

For those that like pictures with their words, I have included some photos of an old folded circle piece that bares relationship to this discussion. These pictures, “The Transient of Venus” show different views of an idea about Venus as it moves in a slow spiral around the sun observed from earth where for a short time appearing as a dark spot moving across the sun. This object is made using seven paper plate circles, four are folded to an open tetrahedron and joined with string holding them together, and the fifth is left unfolded, the other two form Venus and its path. It is painted conforming to the folded equilateral triangular grid matrix, indicating the activity between the surface and interior of the sun as Venus passes giving a temporary linear alignment of three spheres in space.

This form becomes an expression of multiple parts and by adding string, the black square box, and paint gives design to the discrete parts of the folded matrix. Here we have made in “parts” a multiple expression of the whole. Parts and whole are folded one into the other.

The circle/sphere is the only form that demonstrates, with some degree, the idea of a comprehensive Whole. In that regard no other form can be observed that is principle to all symmetries, and all subsequent generations of parts. What is principle is what comes first, not what is believed at the time to be most important. The whole, even defined as “nothing,” for lack of boundary, binds all potential to the principles of first movement. Does our compass open wide enough and will it close small enough to construct all circles contained in the one that can not be drawn?

From paper plate circles to scrap paper to scrap cardboard the circle pattern does not change. Using different materials folded to the circle pattern shows the adaptability and strength of structural organization and the limitations of different materials. Pattern and material form together reveal possibilities not possible in separation.

Below) are six views of a tetrahedron model using two pieces of scrap cardboard. Each piece is folded and scored as show in previous blogs (Order Without Boundary and Order Without Boundary II: Geocoding ). Only the folds necessary for the tetrahedron net as it appears in the circle have been folded. The measure for both folding is the same, the shape of the cardboard and placement of folding are different. Given those two variables every time a piece is folded the same way it will have a different proportional reconfiguration. This individual tetrahedron system can not be duplicated unless the configuration of paper is exactly the same as the position for folding, then the variables become consistently the same for all pieces. Then there would be a formula, not a discovery.

The surface that extends beyond the folded tetrahedron often makes it difficult to see the form of the tetrahedron or to even identify the pattern. Because of the differences in the original flat shapes it might even look arbitrary. Each piece is folded to form half a tetrahedron, two triangles each, then joined at right angles to each other forming a full tetrahedron. There is an inclosed "solid" tetrahedron that is central to holding the system together. You might be able to spot it by finding the small equilateral triangles.

Below) are multiple views of a truncated tetrahedron using four pieces of scrap cardboard. There are more creases added to the pattern in order to form the hexagon unit, one of the properties of the truncated tetrahedron (the folded 4-frequency diameter grid.) Each piece is different in configuration and placement of folds, but the measure and folding process remains consistently the same.

It is difficult to discern any symmetry or underlying pattern of organization by looking the diversity of different views. Each scrap is a different shape where each creased grid is located differently on each piece breaking the symmetry of anything recognizable. With the exception of one view above showing a vertical line of symmetry all other views seem random. In both models above, the formed pattern is not obvious by looking at the entire system. The form is an expression of two variables, the configuration of surface used and placement of the folded grid. All aspects are subject to the consistent organization of the circle pattern and sub-pattern of tetrahedron and truncated tetrahedron. The possibilities of variations appear to be countless depending on combinations of the two variables with the two constants. This reflects what we see with any two points when touched together and creased that will always generate two more points at the intersection of the crease and perimeter.

There are no folded line segments; each crease goes all the way across from edge-to-edge; as observed in folding the circle; the complete pattern for any shape. This provides a consistency of pattern and proportion throughout the surface allowing for a certain amount of reconfiguration beyond second level patterns as they are formed. To fold a line segment without connection to the perimeter is leaving information out, thus creating missing information.

The strength of these models lies with the structural nature of pattern and the rigidity of the material. When made from paper they are more fragile and appear less strong because of more flexibility in the extended surface area. Every material will have its effect on how we perceive the pattern and organization of order that seems to be primary in a forming process.

Three primary circle patterned grids; 3-6, 4-8, and 5-10 can be folded using a scrap of paper. They are inherent in any flat plane surface and can be folded without regard to boundary configuration. Last month’s blog, “Order Without Boundary” showed in the first fold, a ratio of 1:2 (a division of one unit in two parts) providing only three options in a regular division of the plane. The circle is primary from which other shapes can be folded, demonstrable by anyone willing to fold a paper circle in half, observe what is generated and use that information to continue folding.

Any point on a plane can generate one of three specific division of symmetry. This suggest every point on a plane is a center point for dividing the plane symmetrically. The circle itself is the center where symmetry becomes the means of encoding directives for division starting with the diameter. The circle is the center and a local centered systems can happen anywhere on the surface. This supports the idea of center as origin regardless of location or configuration.

All regular and irregular flat plane perimeters are inherent in the circle for the same reason the circle can be drawn on any plane. One center to one circle traditionally uses the compass to draw the circle giving importance to the radius as measure. Movement out, when going back stops at or goes through the center point. When moving into the center the scale changes to find other endless concentric circles. The center is both inside and outside by nature of the concentric alignment of the circumference.

Traditionally polygons are constructed using the circle and straight edge. Folding the circle produces the straight edge and yields the same information plus more. The triangle, square and pentagon are all discernible in the hexagon, primary in the circle. This arrangement of three straight line divisions from a single arbitrary fold is inherent to all shapes. The information is encoded to the angles of division around any location.

Years ago when tracing 2 and 3-D geometry constructions back to the circle I did not see that it was not about the shape of the circle, but about proportional movement that happens where straight line relationships hold consistent to a circle pattern. Information is encoded in the relationships of creases directive to what will and will not work. Folding is sequential coding that generates progressively complex systems, it is not construction.

Following are a few pictures expanding on the three primary polygons and the straight-line grids that exhibit the nature of a circular pattern.

*Above left) * A piece of paper folded to show the hexagon star with the 6 equally spaced bisectors. The star points are all equidistant from the center point making it an enfolded circular pattern. The lines forming the two large triangles do not go edge-to-edge; they are partially formed creases. For one sequence of folding see last months blog: http://wholemovement.com/blog/item/130-order-without-boundary

*Above right)* Each of the six partial creases forming the triangles has been fully creased across the paper, similar to every fold in the circle being a chord. This fully completes the division of the hexagon, six triangles around the center, each bisected in three directions with all lines extending to the perimeter.

*Above left)* The partial folds forming the hexagon have been folded to papers edge thereby expanding the matrix.

*Above right)* One sixth of the hexagon is folded in making a pentagon star. (6-1=5) The pentagon pyramid can also be formed with both concave and convex functions.

*Above left)* Folding in two sixths (6-2=4) leaves a square-based pyramid, half of an octahedron.

*Above right)* Three sixths or one half of the hexagon folded in forms two tetrahedra, a three sided regular pyramid with four triangles. One is inside of the other where both have an inside and an outside making a full count of four tetrahedra.

*Above) * The small equilateral triangles in one direction are colored to show the regularity of the triangle grid. The information is there to extend the grid over the entire plane.

This 3-6 folded grid gives options for reconfiguring the paper to the five, four, and three-fold symmetries.

The folding process is the same for the 4-8, and the 5-10 symmetries. With each reduction of symmetry the possibilities for complexity increases. Symmetry can be seen as the inherent gatekeeper for directional development.

*Above left)* The 5-10 folded grid shows the pentagon defined by partial folds as seen with the hexagon.

*Above right)* The completed five-fold grid is traced in black to show all lines folded to edge of shape.

*Above)* The completed grid is colored to show alternating areas.

*Above left)* The pentagon grid is reformed to a square-based pyramid. (5-1=4)

*Above right) * 5-2=3 shows a triangle-based pyramid, or tetrahedron. These are the same polyhedra formed by the hexagon grid, but they are differently proportioned because of the different in symmetry.

*Above left)* The square is defined by partial creases as observed with the hexagon and pentagon shapes. This is the same folding process using different angle directives.

*Above right) *Folding continues to fill in and out the square grid, in much the same way as the hexagon and pentagon.

*Above)* The square grid is colored showing the alternating triangle grid that is basic to a square division.

*Above left) * The 4-8 grid folded showing the right angle tetrahedron inside of a larger truncated tetrahedron.

*Above right)* Accordion pleating the tetrahedron is a function of parallel creases in the gird.

*Below)* The above folded paper is reconfigured into a cube with only six squares to the outside and the rest of the paper is folded to the inside.

* Below left) * Four right angle tetrahedra are folded to the same measured square grid in different places on different shapes of scrap paper.

*Below right)* The long edges of the tetrahedra are joined edge-to-edge in a tetrahedron pattern forming a cube. The excess paper coming from six of twelve diagonals to the cube show where the tetrahedra have been joined.

The coded order is that a 3-6-12 folded grid can be reconfigured to the pentagon, the square and the triangle. The 5-10 is limited to the square and triangle. The 4-8 grid is limited to the triangle. This is not just a construction formula, but the inherent function of the folding process. The 3-6-12 folding being first, contains the geometry and mathematical relationships that are fundamental to further development and is important for math.

The place to start is with observations by the folder; how they fold and what is there that was not before the fold. Angles are the most obvious directive since that is primary starting from the first fold, a ratio of 1:2. The individual proportions determines the direction of symmetry.

*Above)* The differences in angle divisions from any point of symmetry on any plane is determined by the proportional decision of congruent angles. Here the divisions of angles is revealed in the 3-6-12 grid. There is much to add, subtract, multiply and divide once we can determine the unit angles, and consider each proportionally within the entire plane. Even without knowing arithmetic or numbers, the proportional relationships of units to units are easy to see when they are all in the same place sharing functions.

*Above)* The square grid shows a different angle division. The central 90° angle is shared by both the 3-6-12 and 4-8 grid.

*Above)* The pentagon arrangement shows different angles that have different proportional properties from the other two. It shows 90° angles, but not centered as with the two above. This makes the proportions and ratios unique to the pentagon. There are different symmetries dividing the space around each point of intersection depending where they are located in the grid.

The number pattern for the angle degrees for each symmetry is surprisingly consistent and reflected in numbers.

With the first fold there are three possible choices to continue generation of structural order. Direction is coded in angle units consistent to 9 as the last number of the first level of natural numbers. Folding 60° in 180 gets two division of 60° three times and again 30° that will reveal 90°. Folding 90° again in half get 45°(4+5=9). By folding the 180 (9) into 144° angle (9) we automatically get 36° (9) with half of 144 being 72 (9). There are no other options to the consistency of angle direction encoded to the circle. When we start with polygons we are limited by the specific angles of the shape.

In looking at the angles of each folding 9 is a common base seen in reducing the numbers to a single digit .

**For 3-6-12**

30° 3

60° 6

90° 9

120° 3

150° 6

180° 9

630° 36 9

**For 4-8 **

45° 9

90° 9

135° 9

180° 9

450° 39 9

**For 5-10**

36° 9

72° 9

108° 9

144° 9

180° 9

540° 45 9

It would follow that the summation of all these numbers reduced down to a single digit would be 9. They all started from a single 90° movement. Three squared, structure to itself, is the number 9, the complete sequence before transition to the next level out. There is a consistent development even in the numbers as they represent angle units.

There many math functions and spatial systems that comes from these three folded grids. The more math you know the greater depth of discovery, particular when it comes to guiding children towards becoming aware of their own observations. At the same time math is not necessary to explore the beauty and complexities inherent in reforming and joining folded scrapes of paper.

While this information is accessible using any scrap of paper it is most visible in folding the circle shape. Information generated in the circle is not possible with other shapes since only the circle has a circumference. Every fold is a chord; an acknowledgement of the complete unity/unit of shape. Information is coded into the circle and revealed through a consistent directive of sequential folding. It is important that each crease be a full chord, otherwise contextual information is missing. This is equally important using irregular paper.

The 3-6 and 4-8 share a 90°central angle. Starting with either the three or the four folding both end up with 12 divisions. (only the 3-6-12 is consistent to the ratio where the other two are not.)The 4-8 is inherent in 3-6-12 making the 3-6-12 accessible through the 4-8. This interconnection between symmetries expands our ability to see what otherwise is not possible when things are separated.

*Below left)* Folding the 4-8 into a 12 symmetry division, six diameters, reveals two hexagons one enfolded to the other. The difference is not so much visual as procedural at this point.

*Above right)* Folding the hexagon star shows the internal triangular grid. The creases are outlined to better see the grid.

*Below left)* Starting with the 4-8 folding in a square and incorporating the 3-6-12 directives reveals the hexagon in the square intentionally not in alignment with the perimeter, showing the hexagon center can be anywhere in the square. Following different directives of the same pattern reveals many possibilities for combining symmetries, freeing us from having to rely on constructing polygons.

*Above right)* The creases are lined in black to see one possible combination between the square and hexagon sharing 12 fold symmetry.

*Above)* Alternate areas have been colored in to show the pentagon star is inherent to the combing of the 3-6 and 4-8. There are a great variety of intersections and shapes for reforming the square where the center of the hexagon can appear at any location in the square area on any scale. First there are pattern directives, then forming of branches that generate information with which to design. Symmetry seems to be the constant where discernable or not.

*Above)* The same folding of the hexagon using the regularity of the square shape as directive for the folding. This is more inline with traditional symmetry of polygon alignment.

That first fold shows coded information is activated by proportional folding which determines different symmetries. I call this information geocoding for lack of something more descriptive. It happens without construction or formulas. We do not think about geometric shapes as being coding devises for structural development. The folded ratio of 1:2 is structural; two quantities without separation is a set of three. The number 3 is structure coded into the first fold and with consistent development assures structural regeneration. Each of 3, 4, and 5 are further coded to branch into different matrices. Three, four, and five add up to twelve; another form of 1 and 2 or set of 3. Being first this primary code is core to all folding that follows revealing the geometry observed in nature and imbedded in the abstracted symbols we used mathematically. Geometry is as much a coding devise for spatial development of complex systems as it is a tool for proving abstract mathematical concepts.

We do not think of basic shapes as coding devises that generate complex geometries because of a traditional position about 2-D construction with emphasis on product. Geocoded information is important to the development of technology just as it is in the geometry which cannot be separated from the instructive information inherent to shapes that make up the forms and systems of the physical and biological world.

While I get excited about the product potential in folding circles, it is the information that it generates and the consistency of development that holds my attention. So here again, as twenty-five years ago, I look at the circle and ask, what it is? Why does it reveal what other shapes can not, yet contains information for all other shapes? How does it easily generate much that we have learned to construct? Why don’t children fold circles? Why isn’t folding circles part of our curriculum? Have we missed some crucial aspect of the circle by only drawing pictures of it? Has Euclid’s influence limited us? Is it because the idea of the circle is a symbol for nothing, where zero prevents us from seeing anything there? Maybe because of how we have defined the circle? I guess it all has something to do with why we continue to ignore folding circles.

Continuing the exploration of folding the circle pattern without the circle shape (http://wholemovement.com/blog/item/121-circles-from-scrap ) brings up questions about boundary and the separation between the inner center and outward centered locations. Folding any irregular shape of paper in a sequence of folding proportional to that first fold generates the same three patterned grids found in folding the circle. They come from the first fold in any plane surface and appear to have little to do with the shape of paper used yet each reveals a centered system of organized symmetry.

Polygons are truncated subsystems of the circle limited to the symmetry from number of sides. Conversely polygons of any shape inherently carry the circle pattern and center that can be revealed through folding. The first fold is arbitrary and can be made anywhere (http://wholemovement.com/blog/item/92-center-off-center) The first fold divides one shape in two parts (a ratio of 1:2.) As seen with the circle there are three principled options for consistently folding this ratio; 3-6, 4-8 and 5-10. (http://wholemovement.com/blog/itemlist/date/2010/9?catid=140 )

Once the first fold is made any of these primary symmetries can be initiated at any point along the first line that can be anywhere, thereby establishing a centered coordinate system through the development of a concentric pattern anywhere on the surface. It is through straight-line creases that we see proportional differences between the triangle/hexagon, triangle/pentagon and triangle/square.

FOLDING THE CIRCLE using the 3:6 ratio, a 4-frequency diameter hexagon star is formed. The ratio is extended to a 3:6:12 grid division of the circle. (Folds are lined in black for visibility.)

For folding instructions see http://wholemovement.com/blog/itemlist/date/2011/2?catid=140

Let’s see what this looks like folding scrap paper with an arbitrary perimeter.

1a. b.

1a & b.) Randomly fold any size paper to a ratio of one piece in two parts.

2a. b.

2a & b.) With folded edge towards you place a finger arbitrarily along the folded edge and fold over until the folded angle looks the same as the resultant adjacent angle. This approximates folding into thirds. Press slightly indicating fold; do not crease. Turn over bringing opposite edges together (one over and one under the middle section.) Slide back and forth until edges on both sides are even, then give a strong crease.

3a. b.

3a & b.) Open to flat paper finding three equally spaced straight lines intersecting at a point resulting from the placement of finger on the first fold. Refold to triangular shape just folded.

4a. b.

4a & b.) Place the center point somewhere on the edge between the folded over part and the center point on one side forming a right triangle, give a strong crease. Open the fold and locate where that crease meets the opposite edge.

5a. b.

5a & b.) Fold the same on the other side, bringing the center point to the point where the last crease intersects the edge. A strong crease will form another right angle triangle opposite in direction. Open to see two intersecting right angle bisectors on each edge showing a symmetrical crossing centered to a partially formed equilateral triangle.

6a. b.

6a & b.) Fold in half, giving a good crease when the two edges are even and open to find three intersecting creases (reflection of #3a.)

7a. b.

7a & b.) Open to flat paper to observe the hexagon star with twelve radians. Refold back to the equilateral triangle. Then fold between the two creased edge points creasing the third side of the equilateral triangle.

8a. b.

8a & b.) Open paper flat to shows six equally spaced bisected triangles in a hexagon arrangement. Refold to single triangle with three bisectors. Between the two end points remove excess paper outside of the triangle or by cutting straight across to make an equilateral triangle, or as this example shows, tear an arc approximating a circle.

For another reference for this folding go to: http://wholemovement.com/blog/itemlist/date/2011/2?catid=140 )

The material outside the polyhedral boundary gets in the way of how we perceive order and construct from regular polygons. Is there another purpose for having “excess” material? What can be learned from material that seems extraneous to traditional construction with polygons?

9a-c.) Below, the three symmetries, (*a*.) 3-6-12, (*b*.) 4-8 and (*c*.) 5-10 have all been folded in the same sequence with an arbitrary first fold across the rectangular paper, each with a different proportioned ratio of 1:2. The difference in symmetry is with the angle of the second fold that determines the following relationships. Looking at one triangular sector there is a proportional scaling out from the local center to the limits of the paper.

9a. b.

c.

10.) a.) Below the 3-6-12 hexagon is folded. b.) show the hexagon reconfigured into the pentagon [6-1=5.] c.) shows the hexagon reformed to the square [6-2=4.] and then (d.) reformed to the triangle [6-3=3.]

10a. b.

c. d.

Last month I used images of some models to explore 2-D expressions of the 3-D objects I have folded. One of the 2-D images is printed here in black and white, then folded back to a 3-D object completing the transformation from 3-D to 2-D and back to 3-D.

11a. b.

11a. & b.) A black/white square image printed on a rectangular 8 ½" x 11" paper. It has been folded and reconfigured into a pentagon arrangement.

12a.

12a.) Here the paper is folded down to a 4-8 symmetry in a square arrangement that is accordion pleated.

b.) 5 square units are joined forming a cube with one open plane to view the inside cubic space.

b. c.

c.) All six sides form an enclosed cube. Each folded unit has a different orientation leaving an irregular outer boundary to the cube form. The printed image visually disrupts recognizing the cubic form.

13a & b.) Below, the same cubic configuration without the printed image shows an arbitrary and irregular boundary. With the “excess” material cut away the twelve edges and six stellated-truncated square planes of the cube are more easily seen.

a. b.

14a-d.) Below four rectangular pieces of paper have been printed and folded to a tetrahedron configuration and joined in a tetrahedron pattern showing the open octahedron. There are many ways the material beyond the tetrahedron can be manipulated, reformed, and modified to change the boundaries without changing the opened center or tetrahedron symmetry. (Hairpins are holding the tetrahedra together.)

14a. b.

c. d.

15a & b.) Below the “excess” material has been partially eliminated and then cut down to the edges leaving the two-frequency tetrahedron.

a. b.

How much do we lose when constructing a polyhedral object from polygons where the pattern context supporting that object has been eliminated? How far out do boundaries go beyond our perception and to what extent do they affect what we see? It is reasonable to construct and assemble polygons, even folding polygons/polyhedra using circles. What value is there in seeing what seems extraneous or excessive material beyond the ordering of forms we are familiar with? Why deal with material that seems to get in the way? Folding/unfolding is an important life-forming function and an important part of geometry with renewed interest in origami and paper folding.

Expanding the perimeter of any shape does not take away from the order that comes from any individually centered point within an infinite matrix of points on a given plane. Every point reveals a centeredness that gives access to the matrix that generates that point of intersection that at once is everywhere.

The pattern of concentric circles, PCC, seems to prevail throughout being revealed through construction methods, truncation of the circle, reforming by folding the circle, and by folding any shape to show forth a polygonal system of symmetry.

These are all static pictures that represent what takes place prior to and beyond the object we experience.

### Exploring Images Through Folding Circles

Written by Bradford Hansen-SmithA few months of no posting brings another direction in exploring the circle. Using images of folded circles that show the structural nature of geometry and combining them with NASA images of the observable universe allows me to image what would otherwise not be possible to see by bringing together vastly differing scales in one frame.

Over the years I have used photo images of my sculpture collaging them together and with other images giving greater breadth and depth to the ideas that were embodied in the original work. These current images are a continuation of that process.

*Above*: the helix is made into the image of a cosmic flower with the center spiral derived from the same helix image. The second image shows the helix to be a dynamic cosmic force experienceable through local consciousness.

*Below*: is a spiral model with a design printed on the circles. The image has been modified to show a cosmic hot spot that appears to move structurally out forming a galactic cloud. Possibly the cloud is from itself forming a cosmic tendril of local significants. In the lower left is another space-star circle model.

*Above*: this image also uses the space-star model seen imediately above.

*Above*: an image combining two different folded circle spheres gives visualization to the dispersal of regulated energy.

*Below*: another spiral is in process of gathering cosmic debris and at the same time distributing as it moves both in and out of itself. The tape holding the model together is a nice touch.

*Above*: this single model has taken two different directions in these fanciful treatments suggesting familar life forms.

*Below*: A 3-D folded circle wall piece showing a variety of spatial layers. In keeping with the layering of the image, Photoshop was the right tool. The second image departs from the textural layering of the first and is pretty straight forward.

* *

*Below*: an old model is used to show a coming together, a collision or just attraction, of an object to itself where the image of the cosmos has moved to the foreground. The model is from an early exploration of folded circles where bobby pins, vinyl tubing, and string had been attached.

*Below*: is a model with an open interior space that appears to be a floating cube. This image suggest one possibility where floating is somewhere between gravity and illusion.

*Above*: a radiant image derived from the picture of a spherical model.

*Below*: another image using models, both previously used, where the forms take on some semblance of human consciousness.

Below: Two different models spanning 15 years have come together in a computer version of a layered projection through a checkered grid, which in Photoshop indcates an empty or transparent layer.

bove:

Random crumpling of paper disrupts the plane transforming it towards a compact sphere-like objects. Creasing straight lines gives order through predetermined sequencing towards specific reformations of the plane. This exploration looks at combining a straight-line creased grid with randomly crumpled creases.

The regularity of a self-distributive structurally ordered grid exhibits symmetry in circle-pattern not found elsewhere. It is independent from individual shapes or quality of material. http://wholemovement.com/blog/item/729-order-without-boundary-ii-geocoding

Crumpling appears without order. Yet there is a sense of organization in relationship to how the crumpling is done. The self-organization of a circle-patterned grid and random crumpling seem to compliment each other. In exploring this I have in some cases used printed images to add another layer of information to the reforming process.

Combining the circle-patterned grid and crumpling has parallels in nature as well as how we live our lives. Wrinkles record habits, a consistent tracking about what we have done, where we have been as we interact through space over time. These folds enliven the surface with unique expressions responsive to internal and external forces. A crumbled “mess” when opened suggests some kind of inherent order with a great deal of textural interest. Ordered folding is geometric in style and it lacks the uniqueness found in spontaneous crumpling.

Crumpled wad of paper Unwadded crumpled paper

To carefully unfold and then reform the creases of a crumpled surface reveals interesting variations with little possibility of exactly refolding the original wad ball.

Above left.) Paper folded to the circle-pattern hexagon grid, a 3-6 symmetry. The vertical line is the arbitrary-placed first fold. The two diagonal creases are the next two folds. These are the same proportional folds of division used in folding the circle. The rest of the creased grid lines are informed by the first three equally space creases.

Above right.) The same paper after it has been crumpled. Once the primary creases are established any random crumpling has little effect over the consistency and structural nature of reorganizing the folded grid.

Above left.) Paper reformed using the grid to reconfigure a five-fold symmetry.

Above right.) Same paper reformed to a square symmetry.

Below left.) This shows a rectangular shaped paper folded into a tetrahedron net; same as it would be in a circle. (The net creases are traced to make them easy to see.) These nine creases for the tetrahedron net are primarily used in the following demonstrations.

Below right.) I have tried to leave the center triangle in the net uncrumpled. Usually we cut off the “extra” paper leaving an isolated polygon. It is difficult to crumple one part of the paper without effecting the entire surface. Every crease is interconnected to all others through the paper itself.

Below.) Two reformations of the tetrahedron net above.

Below left.) The net is reformed to a tetrahedron.

Below right.) The same tetrahedron is formed before crumpling. It has the stylized geometric form. The tetrahedron pattern is identical in both.

Below left.) A regular tetrahedron with the excess paper more tightly crumpled.

Below right.) Three tetrahedra joined forming an open plane tetrahedron cavity.

Above left.) A fourth tetrahedron is added to the open face of the three on upper right, making a “solid” closed stellated tetrahedron. Each tetrahedron unit is consistent to the same size rectangle, roughly positioned to the same location on the paper, using the same measure to insure congruency of scale.

Above right.) Four tetrahedra rearranged to form a two-frequency tetrahedron revealing the open octahedron.

Above.) Four more tetrahedra added to the open triangle planes forming a stellated octahedron, or cube patterns. These added tetrahedra are without crumpling. There is consistent regularity of pattern not obvious in the random look of the form. Patterns are consistent; forms are a continually changing variable.

Below left.) Tracing the lines allows another way to see what is going on. Two papers differently creased are joined. One rectangle has been folded to the circle-pattern hexagon grid and reconfigured to pentagon symmetry. It has been placed on top of the other that has been crumbled.

Above right.) One paper has been both crumbled and folded to the circle-patterned hexagon grid and reconfigured to the pentagon.

Above.) Four pieces of copy paper folded to circle-pattern grid, partially crumpled and joined. The circle removed before folding is separately folded and crumbled; where folding the empty circle reveals the grid in the surrounding rectangle paper. This references last month’s blog where we saw the circle paper folds the grid *in* and the empty circle folds the grid *out*.

Again creases have been traced with a marker making them more pronounced. Glass is placed on the flat surface on top with the reformations below hanging out in 3-dimensions.

Tracing a folded pattern of creases is easy and straightforward, they are all straight lines; it is more difficult when tracing the crumpled folds. The angle of light changes our perception of where the crease and fold start and stop. Without shadow the crease appears in one place and direction; changing the light shifts perceived orientation. As with all tracings, the lines are a generalization of spatial relationships of movement. Tracing the residue of movement does not give an accurate picture of what has been in-formed. There is delight in the movement and spatial changes as well as coming to rest with it. I value are the unexpected surprises that happen along the way, they are hidden in the image. The limitation in folding any formula shields us from the unexpected, hids the experience of discovering the moment.

Below.) Four views of a single arrangement. Two diverse pictures have been joined combining crumpling and the hexagon grid into a single arrangement. The beauty is in experiencing the spatial relationships of images as they form an object.

Some of the 2-D images used are from past folded circles models photographed and combined with other imagery that now become material for exploring folding and crumpling. Both 2-D and 3-D are degrees of abstraction removed from the reality of expression.

You can find more about the images used in a previous blog http://wholemovement.com/blog/item/129-exploring-images-of-folding-circles

Below.) Two views of integrating three photo prints each folded to the circle-pattern grid and reformed. This is another way to explore the spatial relationships implied in 2-D images.

Below.) Two sides and one front view of a 3-D object by folding, crumpling, and joining three individual pictures. A transforming takes place from folding circles into 3-D objects, translating the objects to 2-D images, then folding those images back into into objects using the same folded creases used in the original circle folded models. This brings various stages of development together in a single object.

Above.) An early model using multiple circles folded to the tetrahedron net, where before reforming each circle, they have been crumpled to give an interesting surface and tactile appeal. Random crumpling softens the paper, yet seems to have little effect when reforming the structural net.

Below.) Four rectangular pieces of paper were first crumpled then folded in a four-frequency diameter circle-pattern hexagon grid. The units have been reformed to a variation on a truncated tetrahedron. Joined on their end points they form an octahedron with four open and four closed triangle planes. Given the crumpling and shape of paper it does not look like an octahedron, yet there are eight triangle planes in a regular octahedron arrangement.

Below.) Four more rectangles are folded to tetrahedra without crumpling. They are attached to the inverted triangle faces of the model above. There appears possibility for sustainable “new growth” when there is regularity of pattern. The open triangles are potential for continued growth.

Below.) A glossy cover stock 2ft square was folded to a randomly placed circle-pattern hexagon (3-6 grid.) Folding in 1/6 of the grid leaves a 5-10 symmetry formed to a pentagon. The paper outside the pentagon was then crumpled. Following are two variations in reforming the grid keeping to the pentagon symmetry.

Above left.) The pentagon is opened to show the hexagon star of the 3-6 symmetry grid.

Above right.) The grid is folded in reformed to a 4-8 symmetry. With each reformation the crumpled rectangle is changed.

Below.) Three circles and a scrap of crumpled paper combined to make something that looks like it might be biologically functional; particularly after adding a few twisty ties.

Below.) Four stages of opening the crumpled material around a solid tetrahedron made from four pieces of paper. The center triangle in each net is left flat and joined to form a solid regular tetrahedron with the rest of the paper around it being crumpled.

Below.) This series of six images show the tetrahedron from above with each of the four rectangular papers sequentially opened and flattened to show where the triangle is placed on that paper. These images lack the wonderful spatial quality displayed in these changes.

Tetrahedron with rectangles compressed. 1st rectangle opened flat.

2nd rectangle opened flat. 3rd rectangle opened flat.

4th rectangle opened flat. All rectangles opened somewhat equally.

Below.) Two separate images and one paper circle folded to the circle-pattern grid and combined. One image is crumpled, the other partially crumpled and reformed, with the third forming an icosahedron. Here 2-D images of 3-D objects are folded in 3-D, again using the same patterned grid used for the objects initial folded shown in the images. Each expression is layered into the next into what is now the 2-D images you see. Once this transforming process reached the virtual world it becomes a translations of zeros and ones. This is where this all started, by folding circles and straight line creases.

Two views give you an idea of the dimensionality of what otherwise would look flat.

The growing interest in origami expands the possibilities to further explore folding paper. So, when you wad up paper, unwad it and look at the beauty of what just randomly happened that you were going to throw away. If we looked at everything for what is beautiful before it becomes a throw-a-way, then maybe there would be more beauty and less garbage in the world. We have yet to realize the synergistic balance between regular ordering forward and the seemingly random spontaneous leaps that occur.

In believing the circle is image, we limit the imagination.

Draw a circle anywhere on a piece of paper.

Cut the circle from the paper.

How many circles are there?

Compare properties of both circles.

They are the same circle.

By removing the circle from the paper the other circle is left. One boundary separates the physical circle from the non-physical other circle. We might say one is positive, the other negative. One is where the other is not. The inverse of one is the other. The compliment is so intimate that without separation we miss the dual nature.

Fold both circles in half.

Notice the difference and similarities in each circle.

One circle contains the creases. With the other circle the paper references the unseen crease within the circle boundary. Having folded in half what is not there informs alignment with what is there.

Fold both half-folded circles twice more in ratio of 1:2 http://wholemovement.com/how-to-fold-circles.

Below: Having folded the half-folded circles into thirds, open them to find three diameters in each. Three chords are visible in the one removed; they are not visible in the other circle.

Three diameters, six radii define six areas and seven points. In the other circle the unseen diameters extended outward showing six line segments defining six areas and twelve points. The proportional folding is the same for both circles. Removing the image from the rectangle paper the other self-referencing, self-organizing circle is now a hole defined by context, yet functions in the same way as the folded paper. The creases not seen in the circle extend through the rectangle. The invisible is held with-in the visible, and for that reason often goes unrecognized. The unseen informs through space that which can be seen.

Fold circles to a higher frequency of the same pattern of division, see blog for instructions; http://wholemovement.com/blog/item/97-unity-origami

When folded to a higher frequency each circle reveals six diameters, the organization being in the first three. The second set of three diameters have a different proportioned division functioning as bisecting diameters to the first three.

Below left) Division of the first three diameters show four equal sections revealing a hexagon star and hexagon. With one the hexagon and star are on the inside boundary of the circle, with the other circle the star points and hexagon are creased into the rectangle on the outside of the circle boundary; one goes in the outer goes out.

Above right: When replacing the removed circle back into the other there are two ways to align the six diameters. One aligns the star points so both are in the same orientation; the other (pictured above) is where the star point diameters are in alignment with the in-between diameters of the other, a thirty-degree shift in position. The latter is the complete alignment of the grid showing two levels of grid division using all six diameters.

The creases in the paper have been drawn over for better visibility.

Below: Continuing a higher frequency folding in the rectangle (creasing lines parallel to lines already there) the equilateral triangle grid becomes obvious reflecting the same size grid we see in the folded circle. Coloring the same size and orientation of triangles makes the pattern clear. The removed circle shows more information; each triangle of the hexagon division is bisected in three directions rotating the grid thirty degrees to a smaller scale. This can be described as a fractal like “interference pattern” (upper right.)

Below: The removed circle has been reformed to a tetrahedron. This suggests the other circle will also reform into a tetrahedron. The form will look very different because of the difference in perimeter but the circle-pattern of arrangement will be the same. There are more equilateral triangles in the rectangle, which increases possibilities in multiples tetrahedra on different scales.

Below: The other circle can change symmetries from six diameters (hexagon,) to a five fold symmetry (pentagon,) to four (square,) and the three (triangle.) This is possible with the whole circle, the circle hole, and is demonstrable with any random shaped paper folded to the same circle-pattern. http://wholemovement.com/blog/item/729-order-without-boundary-ii-geocoding

Below: Fold creases into the rectangle reflecting what is already in the removed circle. Color the alternate areas of the second level division revealing another level of design using the creases.

Above: By replacing the removed circle into the rectangle the difference in scale becomes apparent. With consistency to the same alternate coloring of the folded grid the fractal scaling separates the inside and outside of the circle boundary.

Above: Both circles independently have been reformed to a square-based pyramid (half octahedron.) The pyramid from the rectangle is not complete until the removed circle is replaced, as if they had not been separated. The frequency difference is again apparent with the dual reformations aligned to the same symmetry showing consistency in pattern and difference in design.

Above: Left shows both forms folded into a "solid tetrahedron." On the right shows the other circle reformed to a tetrahedron pattern of four points in space. The circle-pattern is indicated by the arrangement of points (any four points in space is a tetrahedron pattern.) There are many possibilities reforming the tetrahedron using this grid-creased rectangle.

Below: Here are a few of many possible reconfigurations using the grid in the rectangle folded to the circle hole. There is no proportional relationship between inside circle and outside rectangle because of the initial arbitrary placement of the image. The diversity of forms suggests rich design possibilities in a proportionally aligned relationship of circle to perimeter. You might call this an organized, controlled crumpling of paper.

Circle-pattern is inherent to all shapes; all shapes are inherent in the circle-pattern. Were this not so we could not construct what we do with compass and straight edge, or remove the image from the plane and fold the circle into what is possible. By separating the circle from the rectangle part, itself a truncated circle, we see a great variety in different forms through consistency in pattern. The part and whole are so intimately bound that without one there is not the other, even in the appearance of separation.

Symmetry apparently has little to do with specific shapes and is more about proportional divisions of unity. Symmetry bound to pattern is therefore inherent to all shapes and forms. Transformation from one symmetry to another happens because they hold circle-pattern whole in common.

Below) This diagram relates sphere-to-circle compression with removing the circle image from the 2-D plane. Both forms carry spherical unity that can only be realized by moving from the illusion of 2-D to experience the dynamics by folding the circle. In this ways the intimate balance of dual compliments can be directly experienced.

Think about black holes as a function generating movement seen only in a spatial context. We are now theorizing about “white holes,” the opposite to black holes. Is not this what we have been folding; a black hole and a white whole? Sophisticated technological tools have expanded our ability to observe and measure what otherwise can not be seen, similarly the other circle stretches the idea of geometry and structural ordering and rearranging of systems observed to those that we do not see but experience the effects. This allows us to see how little we know about the unseen and unobservable aspects of physical reality. Space perceived as empty once again is shown to be occupied with higher-level phenomena that is beyond detection from lower levels of perception. The folds in the other circle hole are generated by the circle pattern not from the rectangle. There is not enough information within the shape of the rectangle to direct the circle-pattern movement in creasing.

The circumference informs both circle going in and going out. In this dual form there are two visible and two invisible circle planes, two curved planes between top and bottom separating the circle planes; one on the inside and the other on the outside of the paper, they both have the same volume. There are two circle edges where the three planes join. There are five congruent circle parts between two circles.

The dual circle, seen and unseen, reveal the same circle-pattern of organization; one folding into itself and the other folding out from itself. Like wise the concentric nature of the circle goes both endlessly in and out. The circle-pattern is unbounded by context allowing for countless expressions aligned through the formed and unformed concentric nature in unity on all scales.

Below: Two holes of different sizes, arbitrarily placed in a rectangle paper. Keeping the divisional creases in parallel, assures alignment between the two circles. Shown are a few of many reformations possible.

Below left: Drawing a non-concentric circle in a circle shape and removing it gives us three circles, two are congruent. To the right shows nine folds in the larger circle and three creases in the removed circle.

Below: A couple of reconfigurations.

Below: Four arbitrary placed circles cut from four larger circles as before, each with three folded diameters. Three diameters have been folded into the removed four circles. The other circle folding three diameters shows the four larger circles with chords that are not diameters. This makes the larger four circles when joined to a spherical vector equilibrium arrangement irregular and mismatched on the perimeter. The other circles form a regular and symmetrical bubble inside the irregularities of the four larger circles. When the removed circles are joined the same way there is the regular spherical form of the vector equilibrium that is identical to the other unseen bubble. (ref. blog http://wholemovement.com/how-to-fold-circles ) Both models are held together with bobby pins.

Below: The same four circles from above have been disassembled and rearranged to form a tetrahedron using the four removed circles as hubs to join the ends of the larger circles. The flexibility of the folded struts and removed circles accommodates a variety of angles and the irregularity of the off-centered effect of the arbitrary placement of the other circles. They are all tetrahedra in pattern, yet very different in form. With the model in lower left the circle planes are pushed in, the others show planes opened outward.

With this, I leave you to explore something that is not seen to find something that is.

Folding circles is a mind activity that takes place in the hands. This keeps my hands busy and my mind occupied with curiosity. Discovery is a recognition through mind attentive to what the hands are doing.

My mind periodically goes back to folding open forms and my hands play with what is familiar, having been done before. This way my eyes can see what the mind has missed. Again I looked at the open octahedron and explored joining multiples in various combinations. I chose the vector equilibrium with an open center because it reflects the open center of the octahedron units.

One paper plate folded to form an open octahedron unit.

Four open octahedron joined in a triangle arrangement.

Four sets of triangles joined in a tetrahedron pattern form a vector equilibrium system. Bobby pins are used to hold it together. It is open through the center in four directions and forms sixteen secondary open spaces not directly connected to the center, and has concave rhomboids on the six square faces. The arrangement of four tetrahedra in the same orientation sharing a common point forms the vector equilibrium with a defined center. Traditionally the vector equilibrium is called the cube octahedron and understood to be a solid figure. This open forming of the vector equilibrium using octahedra catches my interest.

I had some ¼” plywood veneer triangles that were angled to form octahedra left from another project. It seemed an interesting idea to form the above model in wood to see what difference in materials might make. This would keep my hands busy and my mind on the look out. It took 96 triangles to form 16 octahedron in 4 triangle sets to reconstruction the vector equilibrium arrangement.

Plywood pieces glued in open octahedra then glued and tape together.

The edges have been rounded and sanded. Wood dowels are added extending the triangle planes giving texture and visual interest.

The wood has been stained and the first round of painting applied.

Finished painting with gold leaf accents.

Following is a word-construction about processing the object with origin in the circle.

Cranium vector equilibrium with open center,

patterned system contained within its own inner cavities.

Unity without form

dividing personal and non-personal in relationship.

Structural pattern of three brings directives

for divergent sub-systems,

complements balanced in symmetry,

increasing outer boundary, organized

through inner connections.

From afar unity is relative; sort of, kind of like, but maybe not,

or possibly so, nothing seems, absolute, you think?

Construction fails unity

touched by unformed realization

of endless channels in distribution,

converging back to realization of never being less.

Between the doing of hand and the knowing of mind is the seeing of eye.

Looking at the octahedron I found something missed, not for lack of attention, only that my focus prevented seeing all that was there in the first place.

Observing six points of connection in the closest packing of four spheres tells us the octahedron is a spherical relationship. Both the octahedron and tetrahedron can be separated from spherical order as individualized polyhedra. Placing four tetrahedra point to point (spherical pattern) we find the same spatial octahedron relationship. The pattern is four, structure is three.

A single tetrahedron can reveal the octahedron by opening one vertex, allowing the properties of the tetrahedron to transform to those of an octahedron; 6 points showing 12 edge relationships of which 9 are formed, and 8 triangle planes, 4 surfaces and 4 open planes. Opening a tetrahedron doubles the number of triangle planes while increasing the number of vertex points half as much. if you count the inside surfaces then there are twelve planes.

Above) A tetrahedron opened to show the octahedron patterned relationship of 6 points.

Below) Two open tetrahedron joined edge-to-edge completing the "solid" enclosed octahedron.

Below right) Octahedron opened flat to show a net of eight small triangles made from two open tetrahedra. The two large triangles are short two more open tetrahedra to make a tetrahedron pattern.

Above left) Completing the tetrahedron pattern (three triangles around one make four) by using one of the large triangles as center and adding two more joining on half of the edge lengths as the first two. This pattern of of four makes a net of sixteen smaller triangles (4x4=16) from which is formed the regular twenty-sides icosahedron; 16 surfaces and 4 open faces.

Above) The sequence from tetrahedron net, to octahedron, to icosahedron net is a geometric progression of 4, 8, 16 triangles. This is reflected by the center triangles of 1, 2, 4, the number of circles used, reflecting spherical order. There is consistency in this development of the first three primary polyhedra that we do not see in traditional construction of individual figures.

There are a number of ways the octahedron net can be reformed, all are worth exploring. That these are all found within the circle assures they are transformational.

Below) Of the 6 edges around the middle of the octahedron, three adjacent edges are joined leaving three edges unattached. This opened figure can be reconfigured in three primary ways; the octahedron, a tetrahelix, and a bi-pentacap.

Below, a, b ,c) Three possible stable reformations from three open edges of the octahedron.

a. The octahedron shows three intersecting squares

b. A tetrahelix of three tetrahedra, one is the relationship between two. It can have a right or left handed twisting depending on whether the net is right or left hand.

c. The bi-pentacap with 10 triangle planes shows each pentagon has four triangle surfaces and one open face showing both inside and outside. The two pentacaps are joined by an open pentagon plane.

Here are other ways the octahedron net can be reformed without overlapping surfaces. The tetrahedron pattern has minimum two surfaces and two open faces that is consistent with the first fold of the circle in half.

d. Octahedron/tetrahedron combination. e. 2 open, 2 closed tetrahedra in a tetrahelix.

f. 5 tetrahedra in combination. g. One octahedron and two tetrahedra

h. Bi-pentacap showing 6 tetrahedra

i. An open square face without tetrahedra. This is the one configuration I missed and for the moment find interesting. It has 8 triangles surfaces and 2 open triangle faces; 10 triangle planes, the same number as the bi-pentacap (c. above.)

The open square face has no stability and when squeezed closed the unit will reform making 3 tetrahedra in a helix form (pic a. & e. above.) Squeezing opposite points towards each diagonal forms the edges of the helix, where one spins to a right and the other to the left. This also reflects the chirality in the net itself that can be right or left handed (See illus. of nets where joining tetrahedra is half way to the right or to the left.)

What happens when joining two sets together on the open square faces?

Above left) When joined on the squares a distorted icosahedron with twenty equilateral triangles is generated; 16 triangle surfaces with 4 open triangle planes. The properties are the same as the regular icosahedron except angles change the number of planes meeting at a vertex.

Above right) A regular icosahedron.

Below left) Two views showing two ways to put the square faces together. One changes the icosahedron pattern. They both have 20 equilateral triangle faces. The *left* shows the top and bottom surfaces are perpendicular to each other with the open faces on the same plane. On *right* shows the top and bottom surfaces are parallel in direction, with the open faces in a tetrahedral pattern. The open faces are consistent to the regular icosahedron. The tetrahedron arrangement of intervals are not found in the one on the left,

Above right) Another view of the same two pictured on the left. The red and black lines show two intersecting tetrahedra and the vertical green lines show the relationship of eight points of an elongated cube. The top and bottom look like rhomboids, they are not since the two triangles are not on the same plane. The two intersecting tetrahedron that form the cube are distorted yet all the triangles are congruent.

The regular icosahedron is not chiral. This distorted icosahedron shows two individual variations from the same arrangement by attaching on the square faces. There are other possible combinations of transformation that comes with open space that cannot happen with closed static forms.

Below) Joining open triangular planes in three possible orientations, where units are angled in different directions. Other variations are possible when joining both open and closed surfaces.

Below left) Three icosahedra joined on open faces so there is an open flow through the helix system.

Below right) Two more icosahedra were joined to the helix system on the left using a parallel top/bottom unit in the center. Four units are attached to the center unit on four open faces in a tetrahedron arrangement. Consistency in orientation and handedness from the first fold through all steps is important.

Above) Two views of 4 icosahedra joined on triangle faces in a tetrahedron pattern. Other combinations are possible. When the form becomes increasingly complex and individualized deviations occur that can stop any further generation.

Below) Four distorted icosahedra, consistent in orientation and chirality, are systematically joined through open faces in a tetrahelix pattern. Very different than the helix above.

Being consistent with each step in handedness and orientation while joining units is the easiest way to keep track of the variables. Consistency in pattern is the difference between generating order and falling into disorder. Pattern is consistent; the forms pattern takes often become confusing and chaotic when inconsistencies are introduced. I have taken many models apart looking for why they eventually break down and jam up, only to find I lost consistency in the process. Consistency in pattern will generate many unexpected surprises in a variety of forms.

Two circles reconfigured in two regular tetrahedra, joined to form the octahedron can than transform into a number of configurations showing combinations of 3, 4, 5, and 6 tetrahedra while at the same time generate 3, 4, and 5 fold symmetries. The two folded circles always remain two circles, regardless of the reformations. How do we explain a three-fold increase over the two tetrahedra we started with just by moving two circles? The first fold in the circle from which all folds are generated, is a pattern of movement revealing dual tetrahedra. Numbers keep track of things, they do not explain structural generation or the transforming process of spatial organization generating what often is unexpected.

Exploring without having an identifiable objective opens areas not predictable. As soon as I tie into what I think is going on, that becomes the objective of my exploration and I miss what else is there. It is a struggle to accommodate changes in our lives, yet transformational change is so easy and fluid through the movement of a paper circle as it reforms and reorganizes in multiples. Unencumbered geometry organizes systems that can be easily demonstrated by folding and joining circles. There is much to learn from folding the circle that has been missed by drawing pictures of them, folding polygons, and constructing static models; all which can be extremely engaging, although limited in-formation.

Over the years I have disregarded the impressed inner circle that gives form to paper plates. This deformation of the circle is not a property of the 3-D circle and is always flattened during creasing. Yet paper plates do come with this circle pressed into them, and concentricity is inherent in circles.

Jose Albers accordion pleating concentric circles has always intrigued me. There are been a few interesting developments coming out of Albers folding exercise. *Erik Demaine and Martin Demaine* at MIT have pushed it more towards art by cutting the circle, removing the center to give greater movement to the ribbons of folds as they are joined in complex curving systems of equilibrium. It is not uncommon to see paper engineers scoring curved lines and cutting paper moving away from the straight edges of traditional paper folding. Cutting the circle reveals many intriguing directions but the beauty of the spherical hyperbolic reconfiguration curving to itself is lost. It seems Albers point was the unexpected nature of the uncut circle plane by reforming through folding concentric circles that reveals a balanced movement around the center circle. http://wholemovement.com/blog/item/119-in-out-hyberbolic-surface.

In exploring this I went back to the great circle divisions of the spherical vector equilibrium, octahedron, and the icosidodecahedron, wondering how circumference sectors would change using concentric circles.

Below left) Four circles with three folded diameters systematically joined on edges form the spherical vector equilibrium where each circle shares a center point. This forms eight open tetrahedra. The impresses concentric circle shows a smaller scale spherical VE nested within the larger.

Below right) Four circles with three folded diameters plus three more creases to form an inscribed equilateral triangle that when joined has no center. The folded center on left moved to four places on the circumferences forming a single enclosed tetrahedron. Here we see both spherical and polyhedral forms of tetrahedra, reflecting the first fold of the circle in half forming both spherical and tetrahedral patterns of arrangement in the movement.

Below) Using the figure above on right, one folded tetrahedron http://wholemovement.com/how-to-fold-circles is joined to the center of each face of the large tetrahedron forming another tetrahedron of equal size. The circle sectors are flattened to the edges of the added tetrahedra forming vesicas showing the six faces of the cube resulting from two intersecting tetrahedra.

Below) Being consistent with all circumferences folded to the outside now allows for opening the cubic arrangement to spherical form revealing unusual divisions. On the right is a variation where one stellated tetrahedron is folded to a higher frequency grid enable to generate smaller triangles.

This is interesting but not where I want to go.

Starting again with one circle I fold the concentric inner circle that comes with the paper plate.

Above left) Creasing the inner circle shows the hyperbolic nature of concentric circles as the circle is folded and curves into itself. Right angle tension is created in the inner ring; a bobbed pin holds two opposite sides together.

Above right) Two circles have been folded (on left) and opened enough for one to fit into the other forming a tetrahedron arrangement where the outer circle of one joins the inner circle of the other at four points. Were the two circles joined on the outer circumference the two circles would lie flat one on the other.

This again substantiates the primacy of the tetrahedron but yields little formally.

Not to complicate things I decided to add one fold of the circle in half with the one concentric circle.

Above) This is one example of reforming the circle to one folded straight line and one folded circle line. Each half outside of the folded circle is folded in opposite directions from the concentric crease.

Below) Two views of one possibility in joining three of the units above.

Below) Adding more units further compounds the complexity of curving surfaces.Two views of six units joined in an octahedron pattern.

Below) Adding four more of the same units the three axis of the octahedron become more apparent.

Below) Four units attached in a tetrahedron pattern showing different arrangements of the right angle relationship between opposite edges.

Below) Using the model in the last picture above right, two are joined forming a dual tetrahedron pattern. One tetrahedron intersects the other showing six rhombic relationships in a distorted cube arrangement.

Below) Using another reformation of one straight and one inner circle, and joining multiples units reveals a variety of open systems in various polyhedral patterns. Two views of six circles arranged in a tetrahedron relationship bring out the octahedron relationship inherent in the tetrahedron pattern.

Below) Multiple concentric circles have been added increasing the level of complexity; again two views of each.

Below) The difference between the concentric circle folded by Albers and folding a diameter in the circle is the congruent concave and convex as each semicircle fits the other. When diameter fold is opened tension develops along the center line.

Below) On the left are two circles folded in half with four accordion pleats. In these units the circumference has been partially separated by folding them in opposite directions along the outside ring allowing space between them. On the right is a seven ring circle.

Above) Two examples of concentric circles reformed into spirals.

Below) A system of four circles joined showing different views.

Below) Two circles are folded to the same off-centered division to the creases inner circle. Joining the straight edges forms two intersecting circles where the vesica piscis becomes spatial.

The folds inherent to circle pattern are also found in all irregular parts of the circle. http://wholemovement.com/blog/item/130-order-without-boundary

This means concentric circles can be folded at any location on any shape of foldable material and there will always be a unique reconfiguration in relationship between placement and individual boundary configuration of the paper as it conforms to the circle effect on the plane surface.

Below) An example of concentric folding using an irregular rectangular piece of paper showing both sides standing in different positions.

I continue to delight in the exercise Jose Albers gave his students by exploring the beautiful forms that are produced in concentric circle unity, and the possibilities to combine with other ways in reforming circles.

To change a paradigm is to change one's frame of beliefs about the world to another. Is it possible in a world of infinite parts to think about an inclusive Whole? Is this even desirable in a world seemingly bounded by infinite boundaries?

The circle is fundamental in math as a 2-D image concept. Can we believe the circle is more than this image we draw, more than a defined circumference? Can we equally accept the circle as a 3-D form that comprehensively demonstrates through folding the idea of spherical unity that is inclusively whole?

Traditionally the idea of the circle is demonstrated by cutting a sphere in half. By compressing a sphere a 3-D circle is generated without destroying the wholeness of the only form that represents absolute unity. When the sphere is compressed the volume of the circle remains equal to the sphere. The surface boundary of the sphere changes properties. The sphere/circle is whole; nothing is taken away or added, it is transformed through movement from a spherical form to a circular form through compression. The circle/sphere functions as both Whole and discrete part simultaneously.

Euclidean construction of a circle is based on the definition of a point. Using a compass to draw a line that is considered as a set of points on an imaginary plane at a given distance from a single point, we draw and thus define a circle, calling the center point 'origin'. The small circle point contains even smaller equally concentric circles in the same way circles get progressively larger through opening the compass. There is no fixed outside or inside boundary, only unseen circles in alignment. The radius measures the openness of a compass used to draw a circle. It does not fully measure a circle, although we generalize and use it that way in the abstract.

The property of the image shows one continuous curved line defining a given area; we imagin individualized points. The properties of the 3-D circle show five distinct circles with volume. Think of a disc, such as a coin. Circles are dynamic, they move, can be moved; and through a consistent and systematic sequence of folds will self generate proportional relationships that are not possible with other shapes, yet contain all shapes.

If there is uncertainty about the difference between a 2-D and a 3-D circle then do the following:

Draw a circle, use a compass or trace around a circular object.

Cut the circle from the paper you drew it on.

Observe the difference between the image you drew, the hole that is left in the paper, and the circle in your hand.

Each of these must be understood for what they are in able to understand the interdependent nature of one to the other. There is nothing to suggest throwing out traditional understanding about the circle; on the contrary we are moved to enlarge what we believe by embracing the full nature of the object the picture represents.

This shift of perspective can be understood by looking at the word ‘geometry.’ Geometry means earth measure, measuring things of the earth. Geo refers to the earth. The earth is spherical and the sphere is *whole*. Metry means measure, keeping track through *movement* in space. We can now understand geometry comprehensively as *wholemovement; *a self-referencing system of the whole. This better describes, giving demonstration of our presently evolving worldview in a universe among many where much of what we see and know now was a short time ago unimagined. We can no longer afford to hold a geocentric view about ourselves, anymore than we can sustain the belief that the sun and celestial bodies revolve around the earth. We are a small part of something much larger and far more complex that yet imagined.

This suggest maybe it is time to consider shifting from a commonly held parts-to-whole thinking to a Whole–to-parts perspective. To start with the Whole in the form of the circle/sphere and observe information as revealed through movement, the order and proportional arrangements, the appropriateness of interactive systems, is perhaps worth considering. Through observation of what is generated from the circle, through folding and joining reveals what is not possible using other shapes or forms. The circle is its own center; an alignment of inner and outer boundaries. The whole is origin to all seen and unseen parts, realized and unrealized, revealing ongoing potential. The origin, both center and outer boundary, is already whole and cannot be constructed or deconstructed.

Broadening our perspective does not deny any mathematical value. What temporary benefits progress will eventually drop away being replaced by greater knowledge that elevates value towards a greater realization of human potential as we begin to see the finer reality of further abstraction. We need a more inclusive and comprehensive way of thinking about our universe, our place in it, and how we negotiate the inequalities and fragmentation, the separation that are no longer sustainable given our present understanding of the interrelated and interdependent nature of parts and systems. There is a fear of the unknown that isolates and keeps us from reaching out on a cosmic scale. Fear keeps us from reaching outside of the circle we have draw around ourselves.

Comparing properties of the image and the circle/sphere compression shows differences between 2-D & 3-D. The 3-D circle shows five discernable circles; three circle planes and two circle edge lines that contain a volumetric location. It all starts with a spherical point. Tradition shows two points making a relationship formed by connecting with a line. Three non linear points form three lines forming the edges of a triangle area. Similar components in both 2 & 3-D are points, lines, and planes. Folding the circle by touching any two points will generate two more points at the end of the crease, an axial line of division perpendicular to the distance between points that shows six relationships. These four points can be form into six edges that define four solid triangle planes. There are two solid planes and two open planes defined by five edges you can see, one edge you cannot see. Minimum description of a tetrahedron is for points in space. Edge relationships and planes are inherent in this spatial movement of two points of the circle. The first fold of the circle around the creased line/diameter/axis, in both directions forms two tetrahedra, one the inverse of the other at the same time showing a spherical pattern of movement as origin.

For those that like pictures with their words, I have included some photos of an old folded circle piece that bares relationship to this discussion. These pictures, “The Transient of Venus” show different views of an idea about Venus as it moves in a slow spiral around the sun observed from earth where for a short time appearing as a dark spot moving across the sun. This object is made using seven paper plate circles, four are folded to an open tetrahedron and joined with string holding them together, and the fifth is left unfolded, the other two form Venus and its path. It is painted conforming to the folded equilateral triangular grid matrix, indicating the activity between the surface and interior of the sun as Venus passes giving a temporary linear alignment of three spheres in space.

This form becomes an expression of multiple parts and by adding string, the black square box, and paint gives design to the discrete parts of the folded matrix. Here we have made in “parts” a multiple expression of the whole. Parts and whole are folded one into the other.

The circle/sphere is the only form that demonstrates, with some degree, the idea of a comprehensive Whole. In that regard no other form can be observed that is principle to all symmetries, and all subsequent generations of parts. What is principle is what comes first, not what is believed at the time to be most important. The whole, even defined as “nothing,” for lack of boundary, binds all potential to the principles of first movement. Does our compass open wide enough and will it close small enough to construct all circles contained in the one that can not be drawn?

From paper plate circles to scrap paper to scrap cardboard the circle pattern does not change. Using different materials folded to the circle pattern shows the adaptability and strength of structural organization and the limitations of different materials. Pattern and material form together reveal possibilities not possible in separation.

Below) are six views of a tetrahedron model using two pieces of scrap cardboard. Each piece is folded and scored as show in previous blogs (Order Without Boundary and Order Without Boundary II: Geocoding ). Only the folds necessary for the tetrahedron net as it appears in the circle have been folded. The measure for both folding is the same, the shape of the cardboard and placement of folding are different. Given those two variables every time a piece is folded the same way it will have a different proportional reconfiguration. This individual tetrahedron system can not be duplicated unless the configuration of paper is exactly the same as the position for folding, then the variables become consistently the same for all pieces. Then there would be a formula, not a discovery.

The surface that extends beyond the folded tetrahedron often makes it difficult to see the form of the tetrahedron or to even identify the pattern. Because of the differences in the original flat shapes it might even look arbitrary. Each piece is folded to form half a tetrahedron, two triangles each, then joined at right angles to each other forming a full tetrahedron. There is an inclosed "solid" tetrahedron that is central to holding the system together. You might be able to spot it by finding the small equilateral triangles.

Below) are multiple views of a truncated tetrahedron using four pieces of scrap cardboard. There are more creases added to the pattern in order to form the hexagon unit, one of the properties of the truncated tetrahedron (the folded 4-frequency diameter grid.) Each piece is different in configuration and placement of folds, but the measure and folding process remains consistently the same.

It is difficult to discern any symmetry or underlying pattern of organization by looking the diversity of different views. Each scrap is a different shape where each creased grid is located differently on each piece breaking the symmetry of anything recognizable. With the exception of one view above showing a vertical line of symmetry all other views seem random. In both models above, the formed pattern is not obvious by looking at the entire system. The form is an expression of two variables, the configuration of surface used and placement of the folded grid. All aspects are subject to the consistent organization of the circle pattern and sub-pattern of tetrahedron and truncated tetrahedron. The possibilities of variations appear to be countless depending on combinations of the two variables with the two constants. This reflects what we see with any two points when touched together and creased that will always generate two more points at the intersection of the crease and perimeter.

There are no folded line segments; each crease goes all the way across from edge-to-edge; as observed in folding the circle; the complete pattern for any shape. This provides a consistency of pattern and proportion throughout the surface allowing for a certain amount of reconfiguration beyond second level patterns as they are formed. To fold a line segment without connection to the perimeter is leaving information out, thus creating missing information.

The strength of these models lies with the structural nature of pattern and the rigidity of the material. When made from paper they are more fragile and appear less strong because of more flexibility in the extended surface area. Every material will have its effect on how we perceive the pattern and organization of order that seems to be primary in a forming process.

Three primary circle patterned grids; 3-6, 4-8, and 5-10 can be folded using a scrap of paper. They are inherent in any flat plane surface and can be folded without regard to boundary configuration. Last month’s blog, “Order Without Boundary” showed in the first fold, a ratio of 1:2 (a division of one unit in two parts) providing only three options in a regular division of the plane. The circle is primary from which other shapes can be folded, demonstrable by anyone willing to fold a paper circle in half, observe what is generated and use that information to continue folding.

Any point on a plane can generate one of three specific division of symmetry. This suggest every point on a plane is a center point for dividing the plane symmetrically. The circle itself is the center where symmetry becomes the means of encoding directives for division starting with the diameter. The circle is the center and a local centered systems can happen anywhere on the surface. This supports the idea of center as origin regardless of location or configuration.

All regular and irregular flat plane perimeters are inherent in the circle for the same reason the circle can be drawn on any plane. One center to one circle traditionally uses the compass to draw the circle giving importance to the radius as measure. Movement out, when going back stops at or goes through the center point. When moving into the center the scale changes to find other endless concentric circles. The center is both inside and outside by nature of the concentric alignment of the circumference.

Traditionally polygons are constructed using the circle and straight edge. Folding the circle produces the straight edge and yields the same information plus more. The triangle, square and pentagon are all discernible in the hexagon, primary in the circle. This arrangement of three straight line divisions from a single arbitrary fold is inherent to all shapes. The information is encoded to the angles of division around any location.

Years ago when tracing 2 and 3-D geometry constructions back to the circle I did not see that it was not about the shape of the circle, but about proportional movement that happens where straight line relationships hold consistent to a circle pattern. Information is encoded in the relationships of creases directive to what will and will not work. Folding is sequential coding that generates progressively complex systems, it is not construction.

Following are a few pictures expanding on the three primary polygons and the straight-line grids that exhibit the nature of a circular pattern.

*Above left) * A piece of paper folded to show the hexagon star with the 6 equally spaced bisectors. The star points are all equidistant from the center point making it an enfolded circular pattern. The lines forming the two large triangles do not go edge-to-edge; they are partially formed creases. For one sequence of folding see last months blog: http://wholemovement.com/blog/item/130-order-without-boundary

*Above right)* Each of the six partial creases forming the triangles has been fully creased across the paper, similar to every fold in the circle being a chord. This fully completes the division of the hexagon, six triangles around the center, each bisected in three directions with all lines extending to the perimeter.

*Above left)* The partial folds forming the hexagon have been folded to papers edge thereby expanding the matrix.

*Above right)* One sixth of the hexagon is folded in making a pentagon star. (6-1=5) The pentagon pyramid can also be formed with both concave and convex functions.

*Above left)* Folding in two sixths (6-2=4) leaves a square-based pyramid, half of an octahedron.

*Above right)* Three sixths or one half of the hexagon folded in forms two tetrahedra, a three sided regular pyramid with four triangles. One is inside of the other where both have an inside and an outside making a full count of four tetrahedra.

*Above) * The small equilateral triangles in one direction are colored to show the regularity of the triangle grid. The information is there to extend the grid over the entire plane.

This 3-6 folded grid gives options for reconfiguring the paper to the five, four, and three-fold symmetries.

The folding process is the same for the 4-8, and the 5-10 symmetries. With each reduction of symmetry the possibilities for complexity increases. Symmetry can be seen as the inherent gatekeeper for directional development.

*Above left)* The 5-10 folded grid shows the pentagon defined by partial folds as seen with the hexagon.

*Above right)* The completed five-fold grid is traced in black to show all lines folded to edge of shape.

*Above)* The completed grid is colored to show alternating areas.

*Above left)* The pentagon grid is reformed to a square-based pyramid. (5-1=4)

*Above right) * 5-2=3 shows a triangle-based pyramid, or tetrahedron. These are the same polyhedra formed by the hexagon grid, but they are differently proportioned because of the different in symmetry.

*Above left)* The square is defined by partial creases as observed with the hexagon and pentagon shapes. This is the same folding process using different angle directives.

*Above right) *Folding continues to fill in and out the square grid, in much the same way as the hexagon and pentagon.

*Above)* The square grid is colored showing the alternating triangle grid that is basic to a square division.

*Above left) * The 4-8 grid folded showing the right angle tetrahedron inside of a larger truncated tetrahedron.

*Above right)* Accordion pleating the tetrahedron is a function of parallel creases in the gird.

*Below)* The above folded paper is reconfigured into a cube with only six squares to the outside and the rest of the paper is folded to the inside.

* Below left) * Four right angle tetrahedra are folded to the same measured square grid in different places on different shapes of scrap paper.

*Below right)* The long edges of the tetrahedra are joined edge-to-edge in a tetrahedron pattern forming a cube. The excess paper coming from six of twelve diagonals to the cube show where the tetrahedra have been joined.

The coded order is that a 3-6-12 folded grid can be reconfigured to the pentagon, the square and the triangle. The 5-10 is limited to the square and triangle. The 4-8 grid is limited to the triangle. This is not just a construction formula, but the inherent function of the folding process. The 3-6-12 folding being first, contains the geometry and mathematical relationships that are fundamental to further development and is important for math.

The place to start is with observations by the folder; how they fold and what is there that was not before the fold. Angles are the most obvious directive since that is primary starting from the first fold, a ratio of 1:2. The individual proportions determines the direction of symmetry.

*Above)* The differences in angle divisions from any point of symmetry on any plane is determined by the proportional decision of congruent angles. Here the divisions of angles is revealed in the 3-6-12 grid. There is much to add, subtract, multiply and divide once we can determine the unit angles, and consider each proportionally within the entire plane. Even without knowing arithmetic or numbers, the proportional relationships of units to units are easy to see when they are all in the same place sharing functions.

*Above)* The square grid shows a different angle division. The central 90° angle is shared by both the 3-6-12 and 4-8 grid.

*Above)* The pentagon arrangement shows different angles that have different proportional properties from the other two. It shows 90° angles, but not centered as with the two above. This makes the proportions and ratios unique to the pentagon. There are different symmetries dividing the space around each point of intersection depending where they are located in the grid.

The number pattern for the angle degrees for each symmetry is surprisingly consistent and reflected in numbers.

With the first fold there are three possible choices to continue generation of structural order. Direction is coded in angle units consistent to 9 as the last number of the first level of natural numbers. Folding 60° in 180 gets two division of 60° three times and again 30° that will reveal 90°. Folding 90° again in half get 45°(4+5=9). By folding the 180 (9) into 144° angle (9) we automatically get 36° (9) with half of 144 being 72 (9). There are no other options to the consistency of angle direction encoded to the circle. When we start with polygons we are limited by the specific angles of the shape.

In looking at the angles of each folding 9 is a common base seen in reducing the numbers to a single digit .

**For 3-6-12**

30° 3

60° 6

90° 9

120° 3

150° 6

180° 9

630° 36 9

**For 4-8 **

45° 9

90° 9

135° 9

180° 9

450° 39 9

**For 5-10**

36° 9

72° 9

108° 9

144° 9

180° 9

540° 45 9

It would follow that the summation of all these numbers reduced down to a single digit would be 9. They all started from a single 90° movement. Three squared, structure to itself, is the number 9, the complete sequence before transition to the next level out. There is a consistent development even in the numbers as they represent angle units.

There many math functions and spatial systems that comes from these three folded grids. The more math you know the greater depth of discovery, particular when it comes to guiding children towards becoming aware of their own observations. At the same time math is not necessary to explore the beauty and complexities inherent in reforming and joining folded scrapes of paper.

While this information is accessible using any scrap of paper it is most visible in folding the circle shape. Information generated in the circle is not possible with other shapes since only the circle has a circumference. Every fold is a chord; an acknowledgement of the complete unity/unit of shape. Information is coded into the circle and revealed through a consistent directive of sequential folding. It is important that each crease be a full chord, otherwise contextual information is missing. This is equally important using irregular paper.

The 3-6 and 4-8 share a 90°central angle. Starting with either the three or the four folding both end up with 12 divisions. (only the 3-6-12 is consistent to the ratio where the other two are not.)The 4-8 is inherent in 3-6-12 making the 3-6-12 accessible through the 4-8. This interconnection between symmetries expands our ability to see what otherwise is not possible when things are separated.

*Below left)* Folding the 4-8 into a 12 symmetry division, six diameters, reveals two hexagons one enfolded to the other. The difference is not so much visual as procedural at this point.

*Above right)* Folding the hexagon star shows the internal triangular grid. The creases are outlined to better see the grid.

*Below left)* Starting with the 4-8 folding in a square and incorporating the 3-6-12 directives reveals the hexagon in the square intentionally not in alignment with the perimeter, showing the hexagon center can be anywhere in the square. Following different directives of the same pattern reveals many possibilities for combining symmetries, freeing us from having to rely on constructing polygons.

*Above right)* The creases are lined in black to see one possible combination between the square and hexagon sharing 12 fold symmetry.

*Above)* Alternate areas have been colored in to show the pentagon star is inherent to the combing of the 3-6 and 4-8. There are a great variety of intersections and shapes for reforming the square where the center of the hexagon can appear at any location in the square area on any scale. First there are pattern directives, then forming of branches that generate information with which to design. Symmetry seems to be the constant where discernable or not.

*Above)* The same folding of the hexagon using the regularity of the square shape as directive for the folding. This is more inline with traditional symmetry of polygon alignment.

That first fold shows coded information is activated by proportional folding which determines different symmetries. I call this information geocoding for lack of something more descriptive. It happens without construction or formulas. We do not think about geometric shapes as being coding devises for structural development. The folded ratio of 1:2 is structural; two quantities without separation is a set of three. The number 3 is structure coded into the first fold and with consistent development assures structural regeneration. Each of 3, 4, and 5 are further coded to branch into different matrices. Three, four, and five add up to twelve; another form of 1 and 2 or set of 3. Being first this primary code is core to all folding that follows revealing the geometry observed in nature and imbedded in the abstracted symbols we used mathematically. Geometry is as much a coding devise for spatial development of complex systems as it is a tool for proving abstract mathematical concepts.

We do not think of basic shapes as coding devises that generate complex geometries because of a traditional position about 2-D construction with emphasis on product. Geocoded information is important to the development of technology just as it is in the geometry which cannot be separated from the instructive information inherent to shapes that make up the forms and systems of the physical and biological world.

While I get excited about the product potential in folding circles, it is the information that it generates and the consistency of development that holds my attention. So here again, as twenty-five years ago, I look at the circle and ask, what it is? Why does it reveal what other shapes can not, yet contains information for all other shapes? How does it easily generate much that we have learned to construct? Why don’t children fold circles? Why isn’t folding circles part of our curriculum? Have we missed some crucial aspect of the circle by only drawing pictures of it? Has Euclid’s influence limited us? Is it because the idea of the circle is a symbol for nothing, where zero prevents us from seeing anything there? Maybe because of how we have defined the circle? I guess it all has something to do with why we continue to ignore folding circles.

Continuing the exploration of folding the circle pattern without the circle shape (http://wholemovement.com/blog/item/121-circles-from-scrap ) brings up questions about boundary and the separation between the inner center and outward centered locations. Folding any irregular shape of paper in a sequence of folding proportional to that first fold generates the same three patterned grids found in folding the circle. They come from the first fold in any plane surface and appear to have little to do with the shape of paper used yet each reveals a centered system of organized symmetry.

Polygons are truncated subsystems of the circle limited to the symmetry from number of sides. Conversely polygons of any shape inherently carry the circle pattern and center that can be revealed through folding. The first fold is arbitrary and can be made anywhere (http://wholemovement.com/blog/item/92-center-off-center) The first fold divides one shape in two parts (a ratio of 1:2.) As seen with the circle there are three principled options for consistently folding this ratio; 3-6, 4-8 and 5-10. (http://wholemovement.com/blog/itemlist/date/2010/9?catid=140 )

Once the first fold is made any of these primary symmetries can be initiated at any point along the first line that can be anywhere, thereby establishing a centered coordinate system through the development of a concentric pattern anywhere on the surface. It is through straight-line creases that we see proportional differences between the triangle/hexagon, triangle/pentagon and triangle/square.

FOLDING THE CIRCLE using the 3:6 ratio, a 4-frequency diameter hexagon star is formed. The ratio is extended to a 3:6:12 grid division of the circle. (Folds are lined in black for visibility.)

For folding instructions see http://wholemovement.com/blog/itemlist/date/2011/2?catid=140

Let’s see what this looks like folding scrap paper with an arbitrary perimeter.

1a. b.

1a & b.) Randomly fold any size paper to a ratio of one piece in two parts.

2a. b.

2a & b.) With folded edge towards you place a finger arbitrarily along the folded edge and fold over until the folded angle looks the same as the resultant adjacent angle. This approximates folding into thirds. Press slightly indicating fold; do not crease. Turn over bringing opposite edges together (one over and one under the middle section.) Slide back and forth until edges on both sides are even, then give a strong crease.

3a. b.

3a & b.) Open to flat paper finding three equally spaced straight lines intersecting at a point resulting from the placement of finger on the first fold. Refold to triangular shape just folded.

4a. b.

4a & b.) Place the center point somewhere on the edge between the folded over part and the center point on one side forming a right triangle, give a strong crease. Open the fold and locate where that crease meets the opposite edge.

5a. b.

5a & b.) Fold the same on the other side, bringing the center point to the point where the last crease intersects the edge. A strong crease will form another right angle triangle opposite in direction. Open to see two intersecting right angle bisectors on each edge showing a symmetrical crossing centered to a partially formed equilateral triangle.

6a. b.

6a & b.) Fold in half, giving a good crease when the two edges are even and open to find three intersecting creases (reflection of #3a.)

7a. b.

7a & b.) Open to flat paper to observe the hexagon star with twelve radians. Refold back to the equilateral triangle. Then fold between the two creased edge points creasing the third side of the equilateral triangle.

8a. b.

8a & b.) Open paper flat to shows six equally spaced bisected triangles in a hexagon arrangement. Refold to single triangle with three bisectors. Between the two end points remove excess paper outside of the triangle or by cutting straight across to make an equilateral triangle, or as this example shows, tear an arc approximating a circle.

For another reference for this folding go to: http://wholemovement.com/blog/itemlist/date/2011/2?catid=140 )

The material outside the polyhedral boundary gets in the way of how we perceive order and construct from regular polygons. Is there another purpose for having “excess” material? What can be learned from material that seems extraneous to traditional construction with polygons?

9a-c.) Below, the three symmetries, (*a*.) 3-6-12, (*b*.) 4-8 and (*c*.) 5-10 have all been folded in the same sequence with an arbitrary first fold across the rectangular paper, each with a different proportioned ratio of 1:2. The difference in symmetry is with the angle of the second fold that determines the following relationships. Looking at one triangular sector there is a proportional scaling out from the local center to the limits of the paper.

9a. b.

c.

10.) a.) Below the 3-6-12 hexagon is folded. b.) show the hexagon reconfigured into the pentagon [6-1=5.] c.) shows the hexagon reformed to the square [6-2=4.] and then (d.) reformed to the triangle [6-3=3.]

10a. b.

c. d.

Last month I used images of some models to explore 2-D expressions of the 3-D objects I have folded. One of the 2-D images is printed here in black and white, then folded back to a 3-D object completing the transformation from 3-D to 2-D and back to 3-D.

11a. b.

11a. & b.) A black/white square image printed on a rectangular 8 ½" x 11" paper. It has been folded and reconfigured into a pentagon arrangement.

12a.

12a.) Here the paper is folded down to a 4-8 symmetry in a square arrangement that is accordion pleated.

b.) 5 square units are joined forming a cube with one open plane to view the inside cubic space.

b. c.

c.) All six sides form an enclosed cube. Each folded unit has a different orientation leaving an irregular outer boundary to the cube form. The printed image visually disrupts recognizing the cubic form.

13a & b.) Below, the same cubic configuration without the printed image shows an arbitrary and irregular boundary. With the “excess” material cut away the twelve edges and six stellated-truncated square planes of the cube are more easily seen.

a. b.

14a-d.) Below four rectangular pieces of paper have been printed and folded to a tetrahedron configuration and joined in a tetrahedron pattern showing the open octahedron. There are many ways the material beyond the tetrahedron can be manipulated, reformed, and modified to change the boundaries without changing the opened center or tetrahedron symmetry. (Hairpins are holding the tetrahedra together.)

14a. b.

c. d.

15a & b.) Below the “excess” material has been partially eliminated and then cut down to the edges leaving the two-frequency tetrahedron.

a. b.

How much do we lose when constructing a polyhedral object from polygons where the pattern context supporting that object has been eliminated? How far out do boundaries go beyond our perception and to what extent do they affect what we see? It is reasonable to construct and assemble polygons, even folding polygons/polyhedra using circles. What value is there in seeing what seems extraneous or excessive material beyond the ordering of forms we are familiar with? Why deal with material that seems to get in the way? Folding/unfolding is an important life-forming function and an important part of geometry with renewed interest in origami and paper folding.

Expanding the perimeter of any shape does not take away from the order that comes from any individually centered point within an infinite matrix of points on a given plane. Every point reveals a centeredness that gives access to the matrix that generates that point of intersection that at once is everywhere.

The pattern of concentric circles, PCC, seems to prevail throughout being revealed through construction methods, truncation of the circle, reforming by folding the circle, and by folding any shape to show forth a polygonal system of symmetry.

These are all static pictures that represent what takes place prior to and beyond the object we experience.

### Exploring Images Through Folding Circles

Written by Bradford Hansen-SmithA few months of no posting brings another direction in exploring the circle. Using images of folded circles that show the structural nature of geometry and combining them with NASA images of the observable universe allows me to image what would otherwise not be possible to see by bringing together vastly differing scales in one frame.

Over the years I have used photo images of my sculpture collaging them together and with other images giving greater breadth and depth to the ideas that were embodied in the original work. These current images are a continuation of that process.

*Above*: the helix is made into the image of a cosmic flower with the center spiral derived from the same helix image. The second image shows the helix to be a dynamic cosmic force experienceable through local consciousness.

*Below*: is a spiral model with a design printed on the circles. The image has been modified to show a cosmic hot spot that appears to move structurally out forming a galactic cloud. Possibly the cloud is from itself forming a cosmic tendril of local significants. In the lower left is another space-star circle model.

*Above*: this image also uses the space-star model seen imediately above.

*Above*: an image combining two different folded circle spheres gives visualization to the dispersal of regulated energy.

*Below*: another spiral is in process of gathering cosmic debris and at the same time distributing as it moves both in and out of itself. The tape holding the model together is a nice touch.

*Above*: this single model has taken two different directions in these fanciful treatments suggesting familar life forms.

*Below*: A 3-D folded circle wall piece showing a variety of spatial layers. In keeping with the layering of the image, Photoshop was the right tool. The second image departs from the textural layering of the first and is pretty straight forward.

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*Below*: an old model is used to show a coming together, a collision or just attraction, of an object to itself where the image of the cosmos has moved to the foreground. The model is from an early exploration of folded circles where bobby pins, vinyl tubing, and string had been attached.

*Below*: is a model with an open interior space that appears to be a floating cube. This image suggest one possibility where floating is somewhere between gravity and illusion.

*Above*: a radiant image derived from the picture of a spherical model.

*Below*: another image using models, both previously used, where the forms take on some semblance of human consciousness.

Below: Two different models spanning 15 years have come together in a computer version of a layered projection through a checkered grid, which in Photoshop indcates an empty or transparent layer.

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