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## Bradford Hansen-Smith

### Unity Origami

Last month I posted a few pictures of folded circles on Wholemovement Facebook page. Some of you may have wondered how the complexity was achieved from paper plates circles. I thought it might be of interest to see the process of folding the triangle grid matrix that generates the forms used in assembling those and many other complex systems I have recently been exploring.

This process is similar to unit origami in that you are joining similar paper units to form complex systems. The similarity stops there. Because we are using circles and the circle demonstrates unity beyond all other shapes and forms, we then might call folding paper circles “unity” origami. This is not the unity of adding things together, it is a singular unity wherein all things are already together as potential. It is then only a matter of giving individual expression through reformation of the circle and various arrangements in combination. Squares, triangles, and all other reformations of the circle are never less than the Whole circle. They are all formed from the same folded grid, simply reformed differently in multiples and joined. Unity origami does not carry the limitations and restriction that come along with folding square units. The circle demonstrates spherical unity revealed through patterned movements of reformation that are inherent only with the circle.

Steps in folding an equilateral triangle grid matrix.

First fold the circle in half by touching any two points on the circumference together, a simply process of self-aligning of the circle.

Fold one end point of the half folded circle to a position half way between the two edges as you change the length by moving along the circumference arc. Do not try to measure; use your eyes to see the proportions. It may be easier to look for equal angles keeping the circumference even.

Do not crease yet.

** **

Fold unfolded part under and line up the end points; one fold over and one fold under like a “Z”. When the points are touching the edges will be even. After turning over and checking both sides to see that everything is even, t**hen crease**. (This is the hardest part of the entire process, it requires using your eyes coordinating with your fingers.)

Open to the circle; there are three diameters dividing the circle in six equal sections, seven points. There are three choices to continue folding point to point. We will take one; the other two will also lead to the same complete 8-frequency grid.

Fold end point of one diameter to the opposite end point and crease. Do that the same with all three diameters; end to opposite end. This generates three more diameters that now divides the circle into twelve equal sections.

Fold one end point to the center point and crease.

Open the circle. The radius of the folded diameter is now divided in half and with the fold between radii of two diameters forms an isosceles triangle. Fold one end point from the first fold (the isosceles triangle) to the center similar to before and crease.

Continue around the circle touching every end point of each isosceles triangle to the center and creasing (this is every other end point to the center.)

When you are back to where you started there will be a folded hexagon star of two intersecting inscribed triangles and three bisecting diameters. The three diameters form six star points with three bisecting diameters half way between each star point.

Each diameter now has two new points of intersection not there before. Each star point diameter is divided into four equal sections. I call this a 4-frequency diameter grid. The hexagon diameters are divided equally, the bisecting diameters are unequal in division. Does this look like the circle you just folded?

Start with the one diameter; two end points, the center point and two new points that divide the diameter into four equal segments.

Fold the end point to the furthest new point on the same diameter and crease.

Fold the same end point to the closest new point on the same diameter and crease.

There are now three parallel folds that divide one radius into four equal sections. Do this to each star point diameter (six times.) If you go from one to the other in sequence you will know when you are finished coming to the first crease.

You have folded a grid of three diameters where each is divided into eight equal sections.

This I call an 8-frequency diameter grid. The three bisecting diameters are now part of the grid of three sets of seven parallel lines each. It is like an octave in music. All the notes you need are there to form endless arrangements between any combination using endless possibilities of intervals. The other two choices with the first three diameters that we did not take are now folded within this 8-frequency grid.

This octave can be further divided by following the same process of folding the same six end points to the new points of intersection on each individual diameter dividing each section again in half. It goes from the first fold 1, 2, 4, 8, 16, 32 and so on, until you reach the size limitation of the circle. If you want to take it to a higher frequency start with a larger circle.

The following pictures show the development from the first, a 1-frequency grid to a 32-frequency grid. The higher the frequency the greater complexity can be generated from folding one circle. It is the 8-frequency octave that is fundamental to scaling in and out, all to the pattern of three.

The pattern does not change while the possibilities in reconfiguring a single circle build in complexity with the increase in frequency. Kids in workshops sometimes fold a 32-frequency grid after showing them only the 8-frequency. They did not know what to do with it when done because of the enormous amount of information, but they felt acommplished doing it. You will get the most out of a higher frequency grid if you have worked with increasing levels of frequencies first. Some students find it a challenge and engaging to just fold and explore the possibilities of reconfiguring a high frequency grid of creases.

You can see this process is straight forward always generating information to continue the folding. Each of these individual frequency levels have different directions to explore and will reveal very different reconfigurations of the circle and when you start combining them the possibilities become endless. This is not different that any other kind of frequency modulation except you are doing it with a circle.

The more you understand the tetrahedron as basic to all pattern formation and that the other four regular polyhedra are patterned arrangements for reconfiguring and joining the more you will get from various frequency gird levels. Go to the previous five "Center Off-Center" blogs for more information about some basic instruction for folding the tetrahedron, octahedron and the icosahedron.

There is another useful folding for all frequency levels. I call it an in-folded hexagon rather than an inscribed hexagon since we are folding it in; unless you figure we are drawing it out from the circle by creasing. The hexagon comes directly from the in-folded equilateral triangle.

Here is a 16-frequency folded grid circle. Fold the circumference in forming the equilateral triangle. Each side is folded under the previous side locking one into the other.

The equilateral triangle is folded in half forming a right triangle. Fold each diameter, the three perpendicular bisectors, one at a time.

Open the triangle and put the circumference folds on the outside making three small vesicas. This is a another way to see the proportional difference between the diameter length and distance around the circumference. This proportional folding demonstrates what we call pi.

Open to the circle and fold in the circumference informing a hexagon shape. This is a more familiar representation of pi. There are now many more possibilities for using the circumference in exploring all levels of reconfiguring the circle.

For your first time in folding the grid do not be over concerned if all lines are not exactly parallel, some will be slightly off, but with attention to touching points they will all be close enough. Remember when points are touching the lines will be where they need to be, and subsequent points will be in alignment.

This is not a difficult process and it will become clear as you begin fold by fold. I would like to hear about and see and what you come up with. Feel free to contact me if you have questions or need help with folding.

Enjoy the exploration.

### Center Off-Center #5

Folding the circle in half seems intuitive or at least a well conditioned first response. So let's fold it in less than half.

Fold an off-center crease (below top left.) Line up the long part of the circumference with itself so the angle that is made on the off-center fold and new edge looks divided in half (below top right.) Turn over and fold the unfolded part to line up to the edge just formed dividing the folded circle into thirds; even up the edges and crease (bottom left.) The dark lines in the opened circle are the resulting creases (below bottom right.)

My brother asked what would happen if the fold was aligned to the smaller fold of the circumference when folding into thirds instead of the larger outside edge as pictured above. So the next step was to align the same right hand point to the smaller circumference edge dividing the new angle in half. Then turn over and line up all the straight edges and crease, again dividing the off-center folded circle into thirds. The lines (below) show a symmetry of folding the same end point to both the larger and smaller parts of the circumference, just as if we folded form both ends exactly the same to the large section. One point of crossing is both a right and left handed fold just by turning the circle over and doing the same thing. Orientation is an important and curious factor.

There are two diameters crossing at the center of the circle that intersect at two places with the first off-center fold. This forms two off-center points and five chords with two triangles on the first folded line where one point of one triangle is to the center of the circle.

Using information from the folds we can make parallel creases by accordion folding in all three directions forming an equilateral triangle grid (below left.) Using the triangle pointed to the center for position we can then fill in all triangles of the same orientation to see better how this triangle grid lines up with the circle (below right.)

As we see the grid does not line up with the circle.

We have previously established that folding the circle is about alignment, not about the center. Here we have a triangular grid centered to the circle without alignment. We can see the consistent symmetry of the grid and that a point of intersection is to the center of the circle but there is no true relationship of the dividing grid to the circumference. The first fold was arbitrarily off-center and still is.

Below are two more arbitrary off-center folds showing the same equilateral triangular grid to a different scale, depending on where the crease is off-center. A grid developed from an off-center fold will never be aligned to the circle.

Alignment comes from the concentric nature of the circle not from the triangulation or regularity of a grid. For this reason no polygon or polyhedron can be whole and will never reveal as much information as the circle to itself. Adherence to alignment between the furthest out and furthest in boundary of the circle reveals an order that far exceeds all other relationships since the movement in both directions is for all practical purposes infinite. When we start out misaligned it is sometimes difficult to discern when the reference is less than a circle. The accuracy of alignment with that first move within the established boundary has everything to do with determining subsequent development.

Let's see how it works folding the 4-8 symmetry. The process of folding is the same, the proportions are different.

The 4-8 symmetry shows one diameter and a change in proportions of triangles (above left.) There is no long or short circumference in the division, the folding is the same from both ends of first fold. Again there is information to accordion fold the right angle triangle grid matrix. Another diameter can be located by lining points of intersection perpendicular to the center crease which will place the center of the circle, thought it is not formed to this grid level. As before the grid is out of alignment with the circle boundary.

Folding the off-center crease to a 5-10 symmetry shows five chords where two are diameters. Again folding on both sides of the off-center line (below top left) show again differently proportioned triangles. By folding the triangle grid we see a very different division of creases (below top right.) By coloring in the triangles of the same orientation shows an out of alignment to the circle boundary.

We have seen folding the circle in half reveals alignment (previous post; Sept 20, #2 and Oct 19, #3.) Folding the half folded circle into thirds consistently forms three equally spaced diameters. This happens with the same consistency but with different proportions that correspond to the 4-8 and 5-10 symmetry. Let's look at the grid from folding the circle in half and how that is different from what we have just seen with developing the grid from the off-center folds.

Fold in half and then fold into thirds showing three diameters. From this folded information we can fold the equilateral triangle grid similar to what we did previously, only in this case we are folding all possible combinations of touching points and then creasing. This reveals the triangle grid showing the enfolded hexagon star and three more diameters where there is exact alignment to the circle

The circle is equally divided into twelve sectors. There are three sets of three parallel creases and three diameters. Twelve lines in the grid reflect a pattern formation of three. This is not arbitrary, there is self-organizing and order that comes from alignment of the inner and outer boundary of the circle. This alignment is critical for the full functioning continuation of folding the circle (below bottom.)

This is not so much about the triangular matrix or symmetry as it is unity of the circle. There is an order of proportional organization, balance and symmetrical arrangement of finite parts that only occurs with alignment. Any fold out of alignment and off-center will always reveal a consistent grid of triangles from which the center of the circle can be located. But there is only one way to align the circle to gain full benefit from the inclusive nature and potential of the circle and that is to fold the circle in half.

This one fold aligns the circle, in a proportional ratio of 1:2 that is directive for all that follows. Sequential development reveals three possibilities of symmetries; 3:6, 4:8 and 5:10 (post Oct 19, #3.) The one Whole two parts ratio sets the structural pattern of three, a triunity that happens first with the compression of the sphere to a circle form, thus reflected in the first fold. Consistent developed from that first fold is a true expression of circle/sphere unity. This embraces all limited expressions from off-center folding and truncations into polygons.

The off-center folded grid, centered but not aligned, can be brought into alignment when you cut back, or move out to boundary towards the concentric nature of circles corresponding to the local center and the primary points of intersection of the grid. Even with alignment missing, there is always information to get realigned.

This in-the-hand demonstration seems to have direct implications of how we might think about our off-centeredness and misalignment and how we might bring principled organization and balanced to the symmetrical and infinitely concentric proportional nature of life.

This picture is not unlike the disorder of the planet we are living on. It is how I find my couch at the end of the year, with not even a place to sit down. This coming year I plan to find ways towards aligning my personal off-centeredness (a self-centered perspective) to a more inner and outward concentric balance in this most extraordinary existence.

### Center Off-Center # 4

This is a continuation of exploring center off-center folding and polyhedral development. It appears polyhedra can only be folded from circle alignment first to a centered circle.

There are two primary ways to see the circle. Traditionally where the circle has a center and concentric circles radiate out from the center; defined by using a compass. The second is where the circle has no center, being itself center, where concentric circles go infinitely into and out from themselves without boundary in both directions. One seems practical because we are familiar with it and the other is conceptual because we are not familiar with it.

Folding the circle demonstrates both ideas. The unit circle has a describable boundary. Through alignment of folding a center is reveled, through more folding more local centers are revealed; eventually the entire circle can be filled with centers. The circle is both unit and unity, one center and infinite centers.

Below left) Concentric circles show one point center, like one point perspective in drawing, it is a perceptual illusion that happens on a flat plane.

Below right) Folding the tetrahedron net pattern reveals six center points through a principled folding process (this is not counting the nine tangent points.) The circle is the quintessential fractal pattern of self-similarity infinitely revealed throughout. Remembering the circle is a compressed sphere; by folding we are decompressing spherical information that represents interference patterns of energy radiation.

Above left) a tetrahedron folded from a one center circle shows one of four sides with a center and the other three sides with partial rings of concentric circles. Reforming the centered circle to a tetrahedron does not consistently accommodate the 2-D image.

Above right) each of the six points of the tetrahedron net generated by the circle are each a vertex and center point. There is an equally distributed, wrap around organization that reflects the order of spherical packing in a cut away polyhedral form.

The folding for both tetrahedra is the same and comes from the pattern of alignment generated from folding the circle in half. Drawing in the concentric rings shows a different spherical organization for the same reconfiguration; the difference between the circle with one center and with multiple, or local centers.

Below left) tetrahedron arrangement of four tetrahedra made from circles with a single center bias all facing in the same direction. This arrangement is consistent to the tetrahedron in a single orientation. The concentric rings are on parallel planes without a common center and do not reflect any consistency to spherical arrangement.

Above left) a tetrahedron folded from a one center circle shows one of four sides with a center and the other three sides with partial rings of concentric circles. Reforming the centered circle to a tetrahedron does not consistently accommodate the 2-D image.

Above right) each of the six points of the tetrahedron net generated by the circle are each a vertex and center point. There is an equally distributed, wrap around organization that reflects the order of spherical packing in a cut away polyhedral form.

The folding for both tetrahedra is the same and comes from the pattern of alignment generated from folding the circle in half. Drawing in the concentric rings shows a different spherical organization for the same reconfiguration; the difference between the circle with one center and with multiple, or local centers.

Below left) tetrahedron arrangement of four tetrahedra made from circles with a single center bias all facing in the same direction. This arrangement is consistent to the tetrahedron in a single orientation. The concentric rings are on parallel planes without a common center and do not reflect any consistency to spherical arrangement.

Below left) Drawing concentric rings around each of the six center points on the net and reforming into the tetrahedron in the same tetrahedron arrangement shows ten spherical centers indicating concentric spherical shells as a slice through spherical packing. Each edge length is divided into 16 equal segments. The edge division depends on the intervals of wave frequency used between center points. These ten spheres reflect the four spheres and six touching points seen in the alignment of the first fold by touching any tow points.

Above right) one more tetrahedron and two octahedra have been added to show how filling in tetrahedra and octahedra further reveals the closest packed order of spheres of the same size. Of course each sphere is a movement into and out from itself creating complex interference patterns.

All observable spherical systems seem to develop from a local center within a universe that itself seem to be centered within others of centered universes all filled with countless moving local centers within what can only be called Whole. The center is everywhere of endless scale; unity containing everything down to the smallest single unit. Alignment of any size circle will demonstrate a similar process of inner-relationship of center points.

Only a circle that is concentrically in alignment into and out from itself can demonstrate something called “true” center. All other centers are off-center, just as we find with less than half folding. Intension generates movement that brings change as dynamic forces of time and space works towards alignment (accuracy is time with experience.) Movement revolves around pattern, forming generations of multiple centers, off-centered systems.

Folding the circle is a practical demonstration of movement from off-center to center, from planetary to cosmic, from a one to the many, from unit towards unity. Between the concept of the centered circle and the circle as center are truths that lie at the center of this exploration. By calling the smallest visible concentric circle a center point we create a conflict with our observations about spherical movement. The value of any unit lies with the context of unity.

In thinking about this I made a five-fold system using concentric circles with the off-centered folding and joined them in an icosidodecahedron arrangement. As you would expect from previous post, there is boundary distortion. (Center Off Center #2)

Above) Three symmetry views of the off-center folding with the concentric circles colored in where the black ring is the furthest out boundary of the circle that is common to these particular off-centered folds.

Above) making twenty elongated tetrahedra and filling in the tetrahedral openings changes the configuration closer to a more familiar polyhedron form. Each tetrahedra is individually designed to the creased lines necessary for the folding of the circles.

Above) all twenty open tetrahedra are filled in with elongated tetrahedra.

Below) Here pentagon pyramids were folded to fill in the open pentagon spaces leaving the tetrahedron spaces open. Again each pentagon was individually designed to the creases necessary to reform the pentagon pyramids. We can see a nucleus beginning to give form to the distorted formation.

Below) Views of evolving polyhedra where both open pentagon and triangle spaces have been filled; each form designed differently. Were they to again expand to the same level we would have a “solid” form identifiable as the icosidodecahedron.

The initial off-center distortion being absorbed by filling in what was missing forms towards a traditional polyhedron. In uniquely designing each surface to the information of the folded creases there is a proportional consistence through out the complexities within each individual unit and together as individual systems combined with other systems all made from multiples of the same circle., same folding process, differently designed. There are profound implications of this centering process and the transforming from off-center to center aligning of the boundary of the circle that is itself center. There is beauty in the proportional consistency and harmonizing of individual relationships that reflect rightness, appropriate interaction between all parts within the circle that can be viewed by the relationship of each individual part to the Whole of the circle.

These posts are to share some of my exploration into the nature of the circle and what we can learn from observation about the consistency of information that is generated through folding. Feel free to make your own models and add your thoughts and ideas about this.

### Center Off-Center #3

This is the third post exploring the alignment of concentric circles as spherical shells starting with folding the circle in half. Folding less than half of the circle causing misalignment that will locate an off-center position and peripheral distortion. We will continue with the center off-center concentric circles and how they relate to polyhedra.

The touching points are not connected except in relationship to the circle as a dynamic system. Aligning two points on the circle boundary will provide consistency in continued development of folding and combining circles with a common spherical center. We see this in the symmetries of the three primary spherical systems: 3-6, 4-8, and 5-10 (below.)

Concentric circles are self-organizing and self-generating structural pattern. The circumference and center are simply the largest and smallest definable circle units. Movement into and out from each circle boundary is a right angle movement. This is reflected in movement by touching any two points forming a crease at right angle to the movement between points. Centering of the circle comes from this alignment of the circle to itself.

Below left) are equally spaced concentric circles. The intervals are arbitrarily ½ " apart; consistency of intervals is important. This will give an idea of spherical shells as the circle is reformed.

Below right) shows two arbitrarily placed points on the circumference showing concentric circles from each point. The kite shape is straight-line connecting of intervals between the circles as they intersect. This is the same as folding the circle in half. An interference pattern is inherent between two locations of concentric circles in a right angle relationship.

Above) Combining the touching points and the center point of the circle show the interference pattern of three center locations. Three is structural, two by themselves are not. Alternate concentric rings are shaded to make them easier to see. The right angle intersection of the kite shape comes from the intervals inherently patterned to circle movement. Folding the circle in half shows alignment of the circle/sphere context.

Below left) concentric circles with intersection of three folded diameters.

Below right) Using the same 1/2” intervals and shading alternate rings shows the interference pattern with three diameters. Each of the six points is a local center point with concentric circles. Only two rings of each have been shaded.

Above left) Six primary points of the tetrahedron net (a two-frequency triangle) where the center points have two concentric rings each. Expanding the circles to all fifteen points would have been too dense to clearly see the net.

Above right) This wave pattern shows the 13 primary points of intersection forming the hexagon star pattern. Each diameter is divided into four equal sections where each point is a center point. The level of complexity is determined by establishing design constraints, in this case two shaded rings around each local center point.

Above) Three diameters have been divided into eight equal sections, a consistent development in folding the circle. There are twenty-four creases with two concentric rings around each of the nineteen primary points forming a limited interference pattern. The intervals and shading of rings is consistently 1/2" to keep it simple. Each small white triangle interval is a point of three intersecting creases that coincide with circle intersections. This is a fractal design, often called a pattern.

The interference patterns of concentric circles reveal the polyhedral nets that are inherent in the dynamic ordering of circle division. This is where the polyhedral forms relate directly to circles and to spherical packing.

Below left) A tetrahedron unit folded from the tetrahedron net (above with six centers) indicates four spheres in closest packed order. In this case the edge length is an eight-frequency division of a two frequency tetrahedron. A single tetrahedron unit does not exist in spherical order, only as an organization of four spheres that have been truncated.

Below right) A two-frequency polyhedral form of four tetrahedral units showing ten spheres with the open octahedron space. The edges shows a sixteen-frequency division.

Above) The octahedron relationship is a formed unit of two open and joined tetrahedra. It shows six vertex points, six tangent spheres in spherical packing. The concentric circles defined on each face wrap around showing the spherical pattern in a polyhedral form.

Below left) Adding eight tetrahedra and an octahedron to the vector equilibrium sphere we see spherical packing by filling intervals between the thirteen points with concentric rings. There is an unseen interference pattern created between the concentric circles of the centered vector equilibrium sphere and the thirteen local-centered spheres in the closest packed order.

Above right) A polyhedral representation of spherical packing of the tetrahedron/octahedron matrix showing eight equal divisions of each unit edge. This association of polyhedra shows patterned spherical origin in the form of truncated spheres.

Above left) The octahedron is opened on three edges with two points joined forming a dual pentagon cap arrangement of four triangle faces and one open triangle plane; five triangle planes around the two opposite vertex points, ten triangle planes. Seven spheres are tangent on the plane surface, but unlike the matrix above, there is interior spherical distortion by reduction of the radial measure.

(Above right) By adding two open tetrahedra to the octahedron net in a tetrahedron pattern and bringing edges together an icosahedron is formed: sixteen triangle faces and four open triangle planes. This is a non-centered arrangement of twelve tangent circles on the surface planes, but again spherical distortion occurs on the inside. It is not an expression of spherical order.

Below) Another reconfiguration showing two circle each reformed to 1/2 a tetrahedron using the 12 creases (shown above.) Each reformed circle reveals the square dividing through the octahedron within the tetrahedron.

Above) The two units joined on open square faces form the tetrahedron. The concentric circles surface design shows a higher frequency division of each edge. More creases generate more complex reformations, still keeping wrap around consistency of surface design.

(Below left) This model of the off-center folding of the vector equilibrium from last month will be used to look at how the center off-center come together.

(Below right) Eight tetrahedra, from above, fill each of the open tetrahedron spaces. The tetrahedra come together at the center point showing a consistency of outward facing triangles that form the open square relationships. The completion of spherical packing has been drawn to indicate an interference pattern of interpenetrating spherical shells.

Above) Two views of octahedra filling in the six open square spaces This forms a large regular octahedron that begins to approach the off-center periphery. There is a lot of information about the interrelationships between spheres, circles and polyhedra in this system.

Continuing higher frequency development suggest the off-centered periphery will eventually be absorbed and become aligned to the circle center in a polyhedral form. Every location of local phenomena within circle/sphere unity is center, albeit local. It is then a matter of consistent higher frequency development towards bringing the center and off-center into alignment as one.

We will go more into that next month as this exploration continues.

### Center Off-Center #2

This month we continue the center off-center investigation. Can a local center be re-centered to spherical alignment? We have seen any point on the circle can be a center point by folding in half or less than half. Folding comes first before centering a location. This is not like drawing a picture of a circle where the compass first sets the center point. Concentrically scale and perspective determines which circle becomes center, usually the smallest in any system gets to be center.

Fold four circles with three diameters each (see last month’s entry.) Do the same with four less than half folded circles. Before joining the circles into the vector equilibrium arrangement, draw out concentric circles from each point of intersection on all circles in both sets. (Below) The circles are drawn on both sides, on one side every other ring was filled in to keep track of different sides of the circles. Be consistent with the intervals. Spacing does not matter.

(Above) concentric circles in a less than half folded circle.

(Below) join each set of four circles using bobby pins to hold them together. Look at the similarities and differences between the two systems.

The concentric circles in both are in alignment to a center location. The off-center folding shows gaps in the planes; parts of circles are missing. If we filled in the gaps, completing each circle, the spherical periphery would then be aligned to the local center in a concentric form. The local center would be realigned to the spherical center showing an increase in the size of the spherical form.

By filling in with parts the sphere would not be Whole. Nothing can be added or taken if it is Whole. This might seem a bit obtuse, but it is an important distinction to make between the Whole and parts that look whole. We don’t want to go around calling things Whole when in fact they are coherent parts in alignment. There is confusion enough figuring out about centers.

To give some perspective to all of this; …*“Actuality exist centermost and expands therefrom into peripheral infinity; potentiality comes inward from the infinite periphery and converges at the center of all things.”*

There is a center of all things and then the multitudes of local centers that through peripheral or boundary alignment go to the same place. Knowing that concentric centering works for the 3-6 symmetry, the vector equilibrium. Will this also work the same for 4-8 and 5-10 symmetries?

The 4-8 symmetry starts with the same folding a circle in half. Fold the half circle into quarters by touching points and creasing. Then fold one point back to the opposite point; touch and crease. Turn over and do the same to the other side; point over to point and crease. This folds the circle into four diameters, eight equal sections. Fold four circles this way. The dark lines are creases.

Do this with four less than half folded circles in the same way. The proportional division of ½, ¼, ⅛ will determine the angles for off-center location the same as it does for the half folding. Do not measure, use your eyes, this is about seeing proportional relationships, not measuring.

Overlap two eighths, reforming the circle into a right angle tetrahedron with an open triangle face having three curved edges and three right angle straight edges. Use tape to hold the circle together. Join two on the straight edges going in opposite directions, forming two open right angle tetrahedra between them. Both curved and straight edges make right angle crossings. Bobby pin the circles together.

Now make another set of two units the same way. Join the two sets of two, symmetrically, with two closed planes completing the two open planes of the other with edges touching. Bobby pin together as before. This makes a spherical octahedron with eight equal open right triangle tetrahedra. Do the same folding and joining with the off-center folded set of circles.

(Left) an example of the half folding showing spherical form.

(Right) the same joining using less than in half folding. Both show an octahedron centered pattern. One is spherical; the other is a distorted form. The difference is in peripheral alignment; going back to the first fold.

By filling in around the periphery, spherical potential then moves towards center that is now reflected in the outward form. This demonstrates the possibility of reforming ever expanding distortion of boundary properties to reflect spherical unity. The “converging at the center of all things” brings potential in line with a centered and balanced symmetry. The center is infinitely everywhere when there is alignment of concentricity.

In reforming the circle to a 5-10 symmetry the folding in half and folding less than half circle will be the same. The difference will be in folding five times to a different proportion and using six circles for each set.

1.) Fold circle in half. 2.) Fold one end point of the half circle to the point along the circumference showing a 1:2 division. What is left is one unit and what is folded over is the two units. When it looks correct, lightly crease. 3.) Fold first point back to point of the last fold, reversing the proportions to 2:1, leaving two units with one unit folded over. 4.) Then fold second end point of diameter to the edge of the previous fold. They should look equal with the fold edges down the middle. Lightly crease. 5.) Fold the two sections to the back and together. Everything should line up, if not, go back and make adjustments before giving a strong crease to all folds.

Open the circle to five diameters, ten equal sectors. As with the 3-6 folding, bring one diameter together joining opposite radii with a bobby pin. This forms two open pentagons with two open tetrahedra intervals separating the two pentagons (below left). Three and five are odd numbers making these reformations different that with the 4-8 symmetry.

Above right) Make another unit the same way and join them together. The sides of the pentagons of one unit will close the intervals between pentagons of the other unit.

Fold another circle as before and reform to a double pentagon. Then add that to the two already joined in the same way, completing five open tetrahedra around one of the open pentagons. This will be obvious when your see how the three reconfigured circles fit forming five pentagons around the sixth centered pentagon. As before use bobby pins to join the circles together.

Make two of these sets of three circles each. Join them the same way with edges of open pentagons closing the open sides of the tetrahedra intervals. Putting the two halves together forms an icosidodecahedron sphere.

Above left) This has twelve open pentagons and twenty open triangles.

Above right) Six less than half folded circles reconfigured and joined in the same way showing a distorted boundary. They both have the same pattern center. The potential for distortion is endless; spherical alignment is one.

Demonstrations of the less than half folded circles are all folded about two to three inches off alignment of the circle. You can imagine that even a half a millimeter off will cause misalignment with distortion. This is not about the center or measuring, it is about alignment and symmetry. Symmetry is a quality of spherical formation. Alignment is what locates the center. Within the concentricity of the circle, which one is the center? When exactly is the periphery in alignment to the center? How close is close to be called accurate? Is anything less than spherical a loss of symmetry, or simply a distortion at the periphery of an always there center of everything?

Explore the symmetries and concentric circles of varied intervals. Next month we will continue to explore further the relationships between polyhedra, concentric circles, and the center off-center.

Another view of folding the vector equilibrium sphere can be seen at http://www.wholemovement.com

¹ The Urantia Book, Urantia Foundation, Chicago IL 1955, Paper115: section3, p.1262

Posted by brad at 7:31 PM 0 comments

### Center Off-Center

Accuracy in touching is seeing with the eyes, with the mind, coordinating with the body. To not see is to be off-center. This is not about measurement, it is about movement through space with purpose. Without awareness towards the Whole can we know the center? Can we know off-center? By starting with the circle as Whole, and through action of folding we will explore center and off-center locations.

Axiom for folding circle: *Any two of an infinite number of points on the circumference of the circle when touched together and creased will equally divide the circle where the linear distance between points will move spherically at right angle to the diameter diminishing the distance between those points as they join on the circumference, where the linear distance between points becomes half the original distance. *

*This gives symmetry to the division of the circle*

* *

The corollary to this: *where any one point on the circumference with any other point not on the circumference (or two points not on the circumference nor aligned to a diameter) when touched together and creased will generate a chord less than diameter where the right angle circular movement will diminish from one point to touching the other, and the line segment distance between the two points is reduced to half the original distance. There is no symmetry to this division of the circle.*

* *

The axiom is the only way to symmetrical fold a circle, assuming it to be the first act of folding. The corollary holds true for the circle and applies to folding any shape. Two circles will be used to demonstrate the same process for finding center and off-center locations. Points are touching or they are not; there is proportional harmony or disharmony to the circle. Center is always sustainable, what is not sustainable has no center. Inaccuracy in putting points together becomes arbitrary, random movement without attention to finite boundaries.

(left) axiom; (right) corollary. There is no right or wrong in observation

We see the difference in touching two points on the circumference and touching two points with only one point on the circumference. How does each develop as we apply the same folding process to both?

The next two folds are generated by folding the semicircle into thirds around the circumference. Fold one third in front and one third is folded to the back behind the middle third. It is not necessary to measure; use your eyes to know all points are touching by sliding the folds bringing them into alignment. Don't crease until all the edges are lined up and exactly even. Fold the off-center circle into thirds the same way; sliding circumference sections back and forth until the edges are even (points will not match up, you might look for division of central angle.) When edges look even, then crease. This folding process can be seen at the “How to Fold the Circle” page on my website: www.wholemovement.com

(Left) All chords are diameters. (Right) Only one chord is diameter The lines have been traced with a marker to clearly show the creases in each circle.

Both circles show three evenly spaced chords revealing a point of intersection, where each is differently located to the circumference. If the points of intersection were a rotational axis, one would wobble and the other would show steady alignment to concentric movement.

To further develop folding in each circle in the same way, reform each by folding one of the chords to itself. The center is even with two opposite radii touching. With the off-center circle the division of chords is uneven, the end points will not meet. Use a bobby pin to hold the creases together. There are three folds, therefore three possibilities for folding a crease onto itself. Only with the off-center circle does it make a difference in the configuration. This joining a crease to itself is called a “bowtie” configuration.

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Fold another circles as before, making another “bowtie” for each. Join together the two individual sets of each, straight edge to straight edge, using bobbie pins or other means to hold them together.

This makes obvious the difference between center and off-center forming and joining. Each of these sets of two circle each is one half of a spherical pattern. (In the off-center the circles are creased to one-quarter distance of a diameter of the circle, to hold to some proportional consistency. Otherwise they could be folded in anyway.)

Make a duplicate set of each ( two sets of two circles) center and off-center, and join them together respectively on straight edges. (Again described on my website in the How to Fold the Circle section). The off-center is now getting more difficult to work with and the center system goes easily together.

Each is shown in same orientation so the difference is apparent. The one diameter in each of the four off-center circles has been assembled to reveal the one composite circle for comparison to any of the four composite circles seen in the center joining.

Below. Other options in joining off-center circles do not show a composite circle.

All center and off-center spherical systems show alternate open triangles and squares. The off-center of each circle displaces the circumferences showing no regularity of a patterned spherical form. The form is deviant, while having an invariant pattern of three equally divided chords within a local center, not obvious in looking at the form.

Here one half of the centered sphere is joined to one half of the off-center sphere. This does not show balance, only that the pattern of developing three equally space chords for both is consistent and allows irregular joining of center and off-center.

There is only one of an infinite number of proportional symmetries in touching two points together in that first fold of the circle. Everything starts form this ratio of 1:2, one Whole two parts. This is the first expression of symmetrical pattern. Anything short of that will eventually cease to generate. The off-centered asymmetrical forms, while they can be interesting and many, are limited and will not sustain generation through multiple joining.

Below are examples of off-centered variations of spherical forms.

Below; the edges of each, center and off-center, are folded in-between the six points on each circumference before joining, this is more in keeping with the straight edge flat plane truncated look we are traditionally familiar with. The vector equilibrium, traditionally called the cuboctahedron, is pattern for both; clearly identifiable in center folding, not so easily seen in off-center folding.

Where is the center off-center? There is no outer boundary or inner boundary to concentric circles meaning there is no center to the circle; the circle is the center. The circle demonstrates the center is never out of center; there is only awareness about when inaccuracy slides into random movement and becomes misaligned. The mind functions as a balance, a connection, between seeing a finite center in physical form, and possible insight to perfection of infinite center regardless of form. This folding is a demonstration to seeing what works, therefore is sustainable, and what does not work and is unsustainable.

Over the next couple of months I will continue this center off-center exploration in folding and joining circles.

### Math Word Problems

Years ago when first reading word problems in my math book I felt I was reading about things on another planet. They did not make any sense to me. Why on earth would anyone try to figure out answers to the questions that book was asking. I entertained my self by drawing pictures in the margins, pictures that had a relationship to where I was sitting. The wording of those questions was as strange as the thinking I was supposed to do to get the answers. They always seemed to have a number of different answers even though we were told there was only one right answer. That just didn’t fit my young life.

Now I know there are questions that are not questions at all; where there are no right answers other than the answers that make sense. Traditionally they are called parables. Why didn’t we have parables in my math book? Why aren’t there parables in math books today? Too many right answers I guess.

Here are two contributions for a math book, if anyone cares to use them. They are certainly something one can draw pictures to, even if you can’t come up with the right answers.

Problem #1

The square when resting had a dream of triangles, pentagons, hexagons, trapezoids, octagons, and circles. Upon awaking he was troubled finding this dream disturbing.

The triangle when resting had a dream of squares, pentagons, hexagons, rhomboids, octagons, and circles. Upon awaking the triangle was also troubled finding this dream disturbing, but not as much as the square.

The circle when resting had a dream of triangles and squares, pentagons, hexagons, and all manner of polygons. Upon awakening the circle had a slight bitterness at the delightful recollection of this dream.

The sphere dreaming of all those things, upon awaking did not know it, and so continued dreaming on.

Problem #2

A man comes from a sphere; he does not remember. He grows up on a sphere, surrounded by endless spheres of all different sizes. He looked around and it appeared to him flat. He holds a ball in his hands; bounces it on the “flat” to entertain himself, then for profit, and possibly for the enjoyment of others.

Another man comes from a sphere; he does not remember. He holds an imaginary sphere in his hands; no one can see it. He compressed the sphere into a circle and then sticking his head into the circle his mind was consumed knowing spherical reality.

### Beyond Circle Is Circle

Folding circles is not just about geometry, mathematics, education, patterns, information, transformation, beautiful forms, and interesting movement systems. These are simply what is generated by folding and joining circles. Beyond those benefits folding the circle provides a different way of thinking; to reevaluate what we think about the circle, and by extension everything else we think we know. There is no comfort here, growing pains are real. The choices we make are all we can claim to be our own.

With many thousands of years of accumulated knowledge all we can think to do is to destroy the planet to which we have been born, that sustains and nourishes us, and to destroy other life forms, and each other, by conscious design and willful acts of choosing selfishly to consume. From a cosmic perspective this is irrational, irresponsible and totally unacceptable human behavior. We consider ourselves the most intelligent creatures on the planet while collectively we appear to demonstrate the opposite. We have developed relatively high levels of technology without developing any moral responsibility in using it.

Every culture builds on experience of human needs in a local setting. We all access the same mind spirit filtered through local cultural habits. These habits of thinking distort and undermine individual experience. There is a great need to rethink those things we take for granted, that we accept without giving thought. Each culture defines life by its own boundaries for its own survival. Observable facts combined with multi-generational stories cause information fragmentation and global confusion with little understanding of where we are or for what purpose. Reading the genetic code is only a story about local events that stops just before now.

Changing how we think is similar to rearranging elements in an equation and keeping the two sides equal. Nothing changes except the arrangement of symbols. Many of us have e-mail but none of us have e-quality. E-mail is the easy way out of the hard issue of e-quality. If only one person is considered to be more, or less, than others, there is no equality. But equality is an imaginary issue; it only functions in virtual reality.

When holding a circle, it is the same for everyone, nothing more or less than a circle. No matter how many different reformations can be folded, the circle remains Whole. When ask to describe the circle, most people will talk about the image; it has a center, it is round, etc, etc. We are then confronted with reconciling what we have been taught and what we individually experience. So often we deny our experience to uphold the stories we have been told. The circle is our experience which far exceeds the simply stories we have been told about it.

We fight for cultural histories, stories that are in conflict with human values and moral development. We think our story is better just because we happened to be born locally, here or there. Well, we are all born locally, in the same way on the same planet in the same space that is so much larger now than it was when Buckminster Fuller called this planet “spaceship earth.” Years have passed and disorganized mutiny continues. We have not yet figured out who is in control or where we are going. Science does not know how to repair the ship. Too many religious maps get in the way of being able to agree on any one destination.

We seem to be lost in our own internal space, warring against each other for ownership without much thought of a cosmic navigator or moral mind that knows the territory. Maybe the fight is not so much about who gets control, but more about flight by those who do not want to participate as crew members to keep this “spaceship earth” in working order and in keeping to the course towards human potential with an eye towards spiritual birthright.

Within the circle every movement is different; there is no conflict. Everything is interrelated and works together in unity beyond the circle/sphere shape.

Animal fear response is no excuse for separation. Fear prevents us from finding value in differences, from learning math, from approaching the unknown; the “X” factor. Fear disturbs comfort, restricts interaction, reduces meaning and fails to accommodate progressive change. It prevents from and gives us opportunity to realize our better selves. Fear inhibits curiosity, diminishes the wonderment about the diversity of life. Overcoming fear does not happen in the abstract, it is the reality of our lives, the choices we make. To overcome limitations stimulates the mind, ennobles the heart, nourishes the soul, and delights and gives courage to the spirit.

We all come up against personal/collective origin. The biggest unknown is the idea of God. We understand the fact of our beginning, yet cannot explain it. We are a purposeful part of creation and there is no agreement as to why. The creator is always more; always embedded within creation. We all are created in the same way, nobody is special, but we are all a different and unique personality. What ever story we accept about origin gives meaning to our existence and has direct impact on the choices we make. The accumulation of facts weaves different cultural stories. It is insight to what happens between stories that tie the facts together making up the true reality of our lives.

Holding a paper circle is to experience unity. The triunity of the circle demonstrates an elegant truth in the simple beauty of absolute symmetry, where goodness of movement is reflected in human striving towards these eternal values. These are qualities reflected in the movement of the spherical creation and of “spaceship earth.” Busy with local self-interest and ownership we have paid little attention to where we are and where we are going.

Only unity can withstand endless differentiation of division and the appearance of separation. To take away is to pretend there is no unity. There is nothing to add to what is inclusively Whole. The circle is never less, it cannot be more. It should not be a surprise that within endless diversity of all things is unity, the foundational, structural connective. What we call God is the only a concept of absolute unity; larger than human mind, beyond human imagination, and personal context for both. Anything less than unity is not; anything more than unity never was. So here again we come up against origin and purpose.

There are no facts that give direction, no diagrams or numbered explanations. Where in our math education do we find what makes us a better person, giving us the experience to change the way we relate to the world; to think differently? What in the abstractions and constructed generalizations of the stories is there to change our thinking towards higher ideals, to ennoble individual growth, to support personal relationships and advance human civilization? The pursuit of mathematics has much value, but very little that nourishes the human soul or elevates understanding through recognition of living truth, responding to beauty, and doing good; bringing triunity to a higher level of experiential reality. The only choice we really have is giving to faith in finding that one story that individually only we can hear. And for some that story is mathematics, but that is not the Whole story.

Everybody folds the same circle, but creases a different diameter. In that first folding there is inherent value that provides opportunity to be mindful and to experience a different way to think about these things.

Posted by brad at 5:02 AM 0 comments

### One Fold Circle Outreach Project.

Years ago when starting to fold circles and working with students, the question came up; “Why don’t we fold circles?” There is no simple answer, but it was clear we would benefit in ways yet outside our thinking by actually folding circles rather than just draw pictures of them. Children fill many pages drawing circles as they imitate writing, and draw representations of many things. We get older and are trained to draw circles in writing and with numbers; as an analogy for going nowhere (around in circles) and in some cases it is everything, most often the circle means nothing, zero. We continue to use the circle to store data, but favor bar code and pixel transfer of information. We accept the utility of circles for multiple mechanical advantages, and marvel at the beauty and perfection of circle forms and movement in nature. We are mystified by the appearance and meaning of crop circles. All mathematics is rooted deeply in the circle.

We cut the static image into parts for mathematical constructions, distort circles for topographical demonstrate of surface equivalents; always using only the idea and image of the circle. We do not have experience or directly dialogue with the circle to understand the nature and unity of what it is. We use the circle for everything except information.

Every child should be folding circles as much as drawing pictures of them. Because we do not fold circles does not mean it has no value. This is simple an ignorant intellectual position of condition that favors folding squares and making circle images. In realizing there is no in-depth information about circles, or folding them, I took it on as my job to explore and understand the nature of folding circles and give demonstration to its value.

I found much that was unexpected about the circle. First, the circle is the only form that can demonstrate the concept of the Whole (which we chose to ignore), while simultaneously functioning as a part across all disciplines. Second, it stimulates and requires stretching our mind beyond the conditional limitations we have accepted. Third, the circle is inclusive to the physical demonstration and development of fundamental patterned information that is not possible with any other method of modeling. In short, folding circles requires physical engagement in coordination that stimulates mind function and engages the spirit of human potential. I do not see this in any other materials or process in such an accessible, direct, principled, and comprehensive way.

Folding circles is a process with far more educational value than any tool we have constructed. This sounds ridiculous to say in an age of extraordinary technological development with such a focus on increasing simulation education. I make this claim from twenty years of folding circles. We have lost sight of a larger perspective about purpose in favor of proving ourselves above all else, even above what we do not understand. We don’t spend much time deeply contemplating things we discover, we are more interested in trying to figure out how to use these things for our own rewards.

From the beginning of this exploration I have envisioned folding circles as an integrated educational activity for all students. We have no idea of the difference this would have in the practical, purposeful, progressive, and meaningful effects of educating future generation

A year ago I decided to count the number of mathematical functions and concepts in the first fold of the circle in half. This was to tie curriculum based information required in formal education directly to folding circles, which would help teachers in making connections. This is not to illustrate math concepts and learn vocabulary, but to observe and discover what is inherent in the circle. More mathematical concepts continued to appear and now there are close to 150 individual math functions, all in one place at the same time, where nothing is added or taken away. This also seems like a ridiculous statement given that we accept learning bits and pieces spread out over a lot of years using many books.

Enable to see this information we must first fold the circle and observe what we are doing. Information is revealed in how we do things, not in the facts about what we have done. We train teachers to teach other peoples after-the-fact experience, thus robbing ourselves of our natural ability to observe and learn from our own experience. Folding circles provides a tool and supports a natural way of learning. This does not preclude guidance from those more knowledgeable in specific areas. The circle is not hierarchical nor favors those with more education. If you can fold it in half, the information is there for anyone that will spend the time.

People that teach teachers tend not to look beyond their own understanding of abstract prescription about math curriculum and how it “should” be taught. Even to achieve what is hoped for with greater benefit is not in their sights. The only way they will know there is an alternate approach to parts-to-whole education, is for someone to give them a Whole-to-parts process. Part of my job is to do that. Folding circles can not be rationally discarded just because we do not do it, or that we have never seriously thought about doing it.

I don’t know of any other way to get this information into the hands of people that teach the teachers that teach our children, other than just giving it to them so they may prove to themselves the worth by their own experience. They don’t have to pay anything, only to read a bit, make one fold in the circle and consider the possible benefits. I have twenty years of doing something that nobody else has felt important enough to do that tells me this is one educational approach we have not yet tried. This project is an effort to pass on some of this information to people that could find ways to meaningfully integrate folding circles into primary level curriculum. It is also a way that others can become involved in this effort of seeding information that will, during some future generation, take root and grow to be as common as grass on which to stretch out and observe the universe in ways yet unseen.

For more about this project go to: http://www.kickstarter.com/projects/1619831705/one-fold-circle-outreach

Thank you for your interest in Wholemovement.

### Unity And Units

The circle image is mathematically defined by a specific arrangement of infinite points all equidistant from the center point. The circle is not that arrangement, nor the imaginary points, or the line that forms the image. Nor is there a center to the circle if you think about the movement of concentric circles going out and going in. The circle is the center. Unity is not the uniformity of units. Unity is the context that generates the ordering of endless units. Units come in all descriptions, material, sizes, and arrangements; even ideas become unitized by similarities as we minimize differences. We add and subtract, divide and multiply them, cut them into parts towards constructing some kind of imaginary unification. All of our knowledge is gained in unit segments. Everything is subject to unit measure and becomes uniformity. We are born, nourished, educated, measured, live by, and die as cultural units identified by name and number. The uniqueness, the interactions, relationships, the context, the values that gives meaning to life has been eliminated in order to generalize formulations of unit construction.

The circle is a one-line symbol. We have broken the line and straightened it. Circles have become a barcode of straight-lines. The longest unit-measure of a circle is the diameter. We assume this straight line times three plus infinitely smaller units find equality with the circumference. We give proof using a straight-line formula thinking we can reconcile the measure of a finite part with the infinitude of unity. It is confusion and irrational to think any number of parts could ever equal the circumference boundary Whole.

It matters little, how many, how much, or in what arrangement we put units, we cannot construct unity. There can never be enough units to demonstrate Unity of the Whole. When something is considered a unit, it is taken from context, isolated, striped of individuality. Meaning has been generalized away leaving an abstract idea, a mathematical term, a comfortable generalization so we do not have to deal with the messiness of individual differences. There is no obligation to a unit. This takes a toll individually and in cultural progression towards any meaningful purpose of value or moral responsibility.

To reduce people, animals, environment, ideas, to units provides a way to rationalize greed and destructive self-centered behavior. For those that have seen the bumper stickers that goes something like; ‘he/she who dies with the most of what ever it is wins,’ can recognize the humor and the seriousness with which we embrace this kind of win/lose separation. This is all too evident by the condition of human culture on this planet. Every human being bought and sold, in every way possible, is a unit mark on somebody’s ledger. Disassociation and separation are cause for confusion, disharmony, and the fears that plague us all.

A unit is an idea about a part in separation from the Whole. Without equality in unique and individual differences there is no unity and we are left with sorted parts and constructed unification. The Whole is ordered to patterns of harmonies and rhythms of integrated and coordinated differences to infinite expression; not one part being better than another. To understand unity as our progressive destination we must first start with unity as our highest ideal, then with discerning mind and determined will, move towards that reality.

The Whole is singular and through movement creation becomes a plural expression of Unity. The Whole is both unity and unit without separation, without number. This brings us around to the circle, a symbol for everything and nothing. The circle is the only form to represent unity, but unity is not the circle. The circle reveals infinite parts in creased lines, areas, and points of intersection. The circle can be reformed to endless configurations. By defining isolated properties rather than interrelationships and interdependent functions between all parts within the Whole we have become confused about the nature of forms of expression of the circle and the circle, of the Whole and parts.

Our animal inheritance binds us to the particulate, abstracted by mind to individual units. Our moral and spiritual endowment motivates us towards unity. We have the capacity to make choices and influence the direction of change, individual and social, towards higher ideals of transcendent perfection. To make progressive choices we must be clear about differences that give value, without being caught in separation and the isolation of units, or in the static form of uniformity. With endless reconfigurations of the circle it is never less than Whole, regardless of the levels of reformation and multiple joining. Each circle remains whole, visibly functioning to reveal endless expression in part. In what ever Religion, and in all the ways we chose to describe and acknowledge this energetic, personally realized, creation of seamless existence; unity belongs to God as the greatest concept of a personalized reality. Without the Whole there are no parts, without the circle, no lines; without unity there are no units.