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## Bradford Hansen-Smith

### Folding January Into February

As January moves into February I continue using my business cards to explore more of the primary tetrahedron configurations (refer to last months entry.) Following is some of what emerged.

Trying to be more systematic using the tetrahedron with 1/3^{rd} of the circumference folded out and putting four circles together in a tetrahedron pattern I found more possible variations than I was willing to make or keep track of. Rather I took to exploring what caught my attention.

Below*. *Here are a first few examples before realizing this would take longer than I wanted to give to accounting for all the possibilities of joining four tetrahedra with variations of circumferences.

This is one helix arrangement of eight sets of four tetrahedron from the possibilities suggested above. Each individual sets has the potential to form some manner of helix formation.

Below Five sets of another arrangement are joined in a helix of a different configuration. Each of the sets above can be joined in multiples ways to reveal a wide range of helix formations.

Two more sets of four open tetrahedra; beautiful little things.

There are too many variations and subtle possibilities. I decided to again go back to the beginning.

We are only familiar with the tetrahedron as solid T, the second T1 has 1/3 circumference out, T2 has 2/3 circumference out, T3 has the entire circumference folded out. Each folding out forms a tetrahedral pocket. With the T90 and T60 the circumference is out but folded flat to 90 degrees and 60 degrees to the tetrahedron edge. Over twenty years of exploring these individually I felt it is time to line them up and look at the options of a tetrahedron when folded from a circle. There are many other interesting ways to reconfigure the tetrahedron, these are limited to the circumference folded in combinations in to out.

Below* In the mean time I played with expanding the look of this sphere (a carry over from last months entry using eight T3 units above.) *

There are many ways to fill in a sphere without changing the pattern while altering the form and design. Here there are tetrahedral spaces and two sizes of octahedral spaces that open up many growth possibilities. How forms grow and develop is always fascinating.

Above Two views of another variation to the tetrahedron related to T3 joining four units in a tetrahedron pattern revealing the triangle in a hexagon and at the same time we can see from another view the cubic pattern as a function of the tetrahedron.

Below Two models formed using T1 units.

Above left Three tetrahedra are placed with one corner into the open pocket of the others forming a triangle set with an open tetrahedron. Four triangle sets have been joined into a tetrahedron pattern.

Above right Each T1 unit is placed into the pocket of the preceding unit in the same orientation forming a line that begins to make a helix. With eighteen units it comes back onto itself where no more of the same size can be added. If each unit either got successively larger or smaller it would become a spiral growth.

Below The tetrahedron patterned set from *above left* is a configuration that fits within the space of another sphere from last month entry, *below right*. They have been formed separately, yet one is the seed for the other as it fits exactly the interior space.

By traditionally cutting off the circumference to make polygons we eliminated possibilities to see many of these kinds of interrelated expanding relationships.

Another view of the open tetrahedron patterned sphere from *above. *

Below 24 solid tetrahedra fill the open tetrahedron spaces in the sphere from *above*. Following is a continuation of expanding the sphere by adding congruent tetrahedral units. They could have been filled with T1 or T2 units as well that would have changed the direction and look of forming.

Twenty-four more T units were added on to the ones previously added.

Twenty-four units from above have been replaced by T2 units, as if the tetrahedrons from before have opened, much as buds open to flower.

Here we see a continuation of the flowering by adding more units. Growth can continue with each new tetrahedron that adds three more open tetrahedron pockets and the possibility of three congruent triangles as places for development to occur. By sequentially reducing the scale and continuing the growth it would take on a fractal development.

I took another directions by opening the solid tetrahedron with circumference out, *below left*. It is one of a number of configurations of the tetrahedron that I did not consider to be primary to the circumference being folded in or out as listed above.

On the *right* shows two of these joined with the open tretrahedra at right angles to each other forming a solid tetrahedron with 720 degrees of circumference on the outside.

Above. Two units of two are joined one fitting into the other in a tetrahedron arrangement. There are other ways they can be joined since all the triangles are congruent.

I wanted this seed to arbitrarily grow to see what would develop. The only rule was to add T3 units attaching on the bottoms to open triangles where it would comfortably fit without distortion.

Above are various stages of adding only T3 units in random placement adhering to directives set by the configuration of the seed. There is no apparent symmetry to the developing mass, but it became clear there were repeating relationships between groups of units that was not reflected in the random placement of units. Individual clusters began to appear repeating the same options of attachment throughout no matter where they appeared. Randomly added units were limited to a given number of options by any cluster configurations. Had the seed been a different configuration the growth would have shown different clustering limitations.

There were other options to pack units even closer had I decided to use T, T1 and T2 units, but in this case I limited myself to only T3 to get a better idea of what was going on. There are two aspects of dependency; each part to the Whole that determines the relationships between parts that determines the design of the system. Parts can be added and subtracted, understanding that each added part, or group of parts, are different divisional reconfigurations in multiples of the same unit circle assuring unity throughout.

### Tetrahelix In Boston

While in Boston the first week in January and with limited space at the Joint Mathematics Conference, I began to fold my business cards, they are round with three and a half inch diameter. I wanted to explore some multiple systems and did not have the space for using 9" paper plates. As it was I ran out of space anyway, and so have continued working small this month folding my business cards.

Below are my cards and the models I started with. Each helix is made from four circles folded to a tetrahedron. Two tetrahedra were opened and joined to formed three tetrahedra. Each unit is two circles making three tetrahedra. they can be closes or open in form. Four tetrahedra are reformed and joined to make a helix form of twelve tetrahedra. One shows closed units and the other open.

Below on the *(left*) is a closed 3-celled tetrahelix seed, it has no twisting and is folded from two open tetrahedra. On the *(right)* the same two open tetrahedra are reformed by opening edges and joining points forming a 4-celled tetrahelix having a directional twisting either right or left hand. There are four different combinations of attaching a 4-cell unit to another in the same direction.

Above right shows three different compactness of twisting in helix form. There is a 3-directional rotation possible with each attachment of the seed units causing the relationships between units to change. There is also the possibility of changing right and left hand twisting at these junctures. The bottom helix is the one we are familiar in that it holds the DNA double helix formation.

Below Putting three different helix forms *(above)* together in a single strand of three angles of twisting keeping the same directional twist gives an idea a the movement that might occur.

I soon ran out of space and decided to stop the helix exploration, instead focusing on just the nine creases of the tetrahedron. As periodically happens, things appear not having been noticed before. I am not as systematic as it might appear, going with what at the moment catches my attention.

Knowing two open tetrahedra also form an octahedron and by adding two more open tetrahedra to the octahedron net (completing the original tetrahedron pattern) we have made a net for the icosahedron, which I opted to explore further. (Folding of the tetrahedron and octahedron nets have both been covered in previous blogs.)

Below* (left* ) shows the icosahedron net. On the *(**right) *it starts to immediately form the icosahedron open pentagons through the open squares and down to four open triangles.

This is a 3" diameter icosahedron made with four circles of the same diameter.

The four open triangle faces in a tetrahedron relationship suggested that maybe two icosahedra could be joined showing some kind of interwoven icosahedron type compound. Not having tried that before, they fit nicely one wrapping through the other.

This I had to do with paper plates on a larger scale.

The alternating dark and light triangles do not match up on the joined edges to continue the design. Yet without the colored in triangles the creases match up perfectly where the 3-6 symmetry is consistent to the 5-10 symmetry. This is not a math problem as much as a human problem, bringing us back to helix formations and DNA.

Next I wanted to see the sequential development of the open triangle, square, and pentagon planes. What traditionally is called truncation and stellation is a bit different in using circles in that nothing is added or taken away, it is revealed through movement and multiple circles.

Above shows development from the circle to the tetrahedron, two tetrahedra make an octahedron (here the octahedron is with three open triangle planes.) From expanding the octahedron net we have the icosahedron. The regular polyhedra from triangles; tetrahedron, octahedron, and icosahedron now shift revealing two semi-regular polyhedra, the snub cube and the snub dodecahedron. The three, four, five symmetries are open planes of relationships of circle-folded tetrahedra joined in multiples of 1, 2, 4, 8, and 20 circles. The triangles are structural, both open and closed faces; the square and pentagon are non-structural being relationships of triangles.

Below is a line up showing the stellated developmental stages of icosahedral growth. First the icosahedron, then stellated with four tetrahedra, one on each center triangle of the four tetrahedra that make up the net. They are equally spread around the icosahedron revealing a tetrahedron pattern with an open icosahedron center. The third expansion shows four more tetrahedra equally spaced (on open faces) closing the figure forming a cube pattern with an icosahedron center. The fourth figure is fully stellated showing the dodecahedron pattern with an icosahedron center.

The question came up; how many different ways are there to form the icosahedron net by joining only the edges? As it turns out there are four different combinations by attaching edges that form an icosahedron, only one is regular. Each form is chiral, right and left handed, making eight different configurations from this one icosahedron net.

Above shows only three of the four reformations from the net. The center icosahedron is regular where the one to the right and left, plus the one missing, are irregular deformations. They all have 20 triangle faces; 4 open and 16 closed planes, and 30 edge relationships between 12 vertex points.

Below are the four icosahedron stellated with a tetrahedron on center of the four circles used in the net. They are all tetrahedra in pattern; the regular icosahedron is on the *(left.)*

Below three of the four have been stellated on four open faces forming eight vertex points. The regular icosahedron is in the center, the others are deformed cubic pattern.

Below. Here all faces are stellated making them all dodecahedron in pattern, with only one being regular. It is not difficult to pick out the distortions. Here again only three are shown. The third distortion is missing because I was late to see that one.

Below is another way to close up the icosahedron net by joining edges. There are now only two open triangle planes making them 18-sided polyhedra of various configurations. Here are twelve different configuration of how many I do not know. Because of the chiral nature of the icosahedron each of these can be right or left hand. There are many possible combinations of multiple systems that can be derived from any number of these irregular 18-side polyhedra; there is much here to be explored.

Above shows a further folding down of the icosahedron net eliminating the open triangle plane altogher. Now they are 16-sided polyhedra, still consistent to edge joining. These are three of an unknown quantity which are for another time to discover.

Below are two different forms of the cubic pattern. One is with the octahedron center and the other has an icosahedron center. The third model to the *(right)* is the same as the first to the *(left)* except the tetrahedra have been opened to show the three planer axis of the octahedron. By connecting the twelve points you will see the vector equilibrium, also know as cubeoctahedron.

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** There are other options in folding circles**

The circumference can be folded to the outside as well as inside to form polygons and polyhedra.

Below are two examples of folding the circumference outside using the helix forms we started with. To the *(left*) is a 3-cell tetrahelix seed and to the *(right)* is the open 4-cell twisting unit.

Below is a sequential development of polyhedra with the circumference on the outside rather than folding in. On the *(left)* is the tetrahedron arrangement of four circles. Next to that shows the circles rearranged forming the vector equilibrium (cuboctahedron.) Adding four more circles to the first tetrahedron reveals the rhombidodecahedron. To the *(right)* is another derivation of eight circles in a cubic pattern.

Below are a few more examples of reconfigurations of the tetrahedron folds using four circles each.

Below we see the evolution from the tetrahedron (4 circles) to a rhombidodecahedron (8 circles.) By opening the six vertexes of the rhombidodecahedron more space is generated adding 6 squares and 24 triangles to the 12 open rhombic planes.

Below is the contrast between the open square in the snub cube *(left)* and the intersecting tetrahedra of a cubic relationship *(right.)* The snub cube is not structural, it has little stability.

Below is the snub dodecahedron *(left)* and the icosadodecaheron formed with the circumference outside *(right.)* Twenty units are used for each figure. The 12 open pentagons are placed differently in each case showing the same arrangement of twelve circles in differently formed configurations from the same creases. Both snub figures have little stability which is greatly increased when the circumference is utilized.

Above are two views of the icosadodecahedron.

These models show connections between flat plane, straight edge Euclidean and non-Euclidean spherical geometry that have not previously been explored simply because we do not fold circles. There are multiple ways to reform the circle that free us from the traditional construction methods using polygons and making polyhedra. There is also so much more information revealed in the circle that just is not possible in more traditional approaches to modeling. These figures can be formed because they are inherently in circle unity and can be revealed by using unit circle forms. The circle is the commonality between all the differences of endless possibilities of pattern formation. This is a very small sampling where each circle remains whole reflecting unity in the circumference whether it is hidden or not.

**As units are so formed, is unity informed.**

### Pattern And Design

Last month while playing with a few left over circle-folded reconfigurations, then picking up a discarded variation of the icosahedron (all open planes) I could see where together they might make an interesting object. For the delight of seeing what it will look like, the satisfaction of actually making it, and the joy of discovering what it will reveal, I spent some time folding more units redesigning them to fit this particular form of the icosahedron pattern.

This is the result of how those circles came together and some thoughts about the process.

There are fifty-one circles; twenty 9” paper plate circles and thirty-one 6” circle filters. They have been all creased to the same folded matrix and reconfigured differently as they are joined to form this patterned arrangement. This object has been coated with glue size which makes holding it an experience different than what you expect; very rigid and smoother than it looks.

This is not as regular as we would expect of the icosahedron. One vertex is open giving polarity to the system. The other vertexes are open relationships between each triangle that have been closed in. Each triangle face is uniquely different. Where the triangles join are open locations of local centers for twelve pentagons. The edge channels defining the dodecahedron change in relationship to which triangle they appear and how they were reformed. The primary points of connection are the intersections of triangle and pentagon edges. There is consistency to the icosahedron pattern with subtle differences in design. Each circle unit has it own unique characteristic difference, much like in real life.

This object reveals the conditional pushes and pulls of the folding process in its forming, much like we would see in nature as individual systems grow to fulfill specific environments. The richness of the surface is in the irregularity of parts adhering to design criteria towards giving form to the pattern. After choosing the icosahedron pattern a series of design decisions followed where each unit is predicated on the developing organization and relationships already in place. The unit circle follows circle unity.

Each circle is reconfigured to a 3-6 symmetry, and collectively joined to reflect the 5-10 symmetry of the icosahedron. Each circle is a uniquely different aspect of unity. Every small decision in folding was circumscribed by previous actions. There is nothing arbitrary, and yet it has none of the regularity and sameness of formulation that is so often seen in generic geometric models.

Having finished the above model I started playing with the icosahedron as an open form (16 solid triangles with 4 open planes.) Options are not possible with the traditional icosahedron net, since this net is structurally principled it opens endless design possibilities. I wanted to keep to the same process going but in a different direction. By using the same folded units in the above model, with reforming variations, they revealed different optional fits to the configuration of the open icosahedron. The option taken shows a tetrahedron arrangement extending beyond the icosahedron in a more open form. This combines both 3-6 and 5-10 symmetry, proportionally balanced in a way not obvious in the above model.

Two views of this model using 8 paper plate circles ; 4 open tetrahedra form the inner icosahedron and 4 circles form the individual tetrahedra vertex locations.

It is extraordinary to see over years of folding circles how many varied reconfiguration can come from reforming the same 3-6 folded triangle grid. When everything is folded from the same three-diameter grid everything is interrelated and inter-transformable in ways that are unique to folding circles. This means any configuration can be flattened to the circle and reformed into any of number of other units, recombined and joined into a variety of different symmetries and systems without adding any new creases. Once the grid matrix is folded into the circle there are an infinite number of unique possibilities for reforming and joining them. All this is possible because it is in the circle to begin with; all is revealed through keeping an eye to alignment in folding and reforming. No tool in the design world comes even close to the possibilities that come from folding circles.

Two view above is another exploration using the open icosahedron form reconfigured from only 4 paper plate, folded to the same 3-6 grid. The circles are reformed so the four vertex locations of the tetrahedron extend into forming a centralized inside open icosahedron, revealing that the two symmetries are combined in the single tetrahedron/icosahedron pattern. The open icosahedron form is a variation of four open tetrahedra. This reflects back to the tetrahedron as primary structural pattern. The four remaining regular polyhedra are patterned formations in different symmetries of the tetrahedron opened and joined in multiples. This model can function as a unit in a variety of larger systems through small design variations of form changes.

We have gone from using 51 to 8 and now 4 circles. These models went through the same process, all folded to the same pattern, revealing the same symmetries arranged to different forms where each individual system requires a given number of circles to fulfill a uniquely designed expression.

These models can not be made using traditional methods without a preconception of design and a plan to instruct the assembly of each part which would be extremely labor intensive, time consuming, with a lot of frustration and to little purpose. These were revealed in process by following what developed from the specific forming of pattern down to individual designing of elements, as revealed, each in turn, giving expression to that pattern. This can only happen with folding circles. What is in the circle is there for anyone that will take the time to find out.

### Transforming Systems

Folding the circle in half is a transformation. The entire folding process is transformational. Without adding or taking anything away the form of the circle changes without changing the nature of the circle. Creases are the result of the self-referenced and self-organizing sequential folding, but the circle does not move by itself. You must be an active participant.

That first fold is a right angle movement forming a perpendicular chord half way between any two points on the circumference. This is the pattern for all subsequent movement because it happens first. In the following months we will look at various transforming systems by reconfiguring the circle in different ways and joining in multiples to this right angle pattern.

**Open Torus Ring**

You will need four paper plate circles, four bobby pins and some 3/4" masking tape. Folding the circle in half, then folding three diameters, reconfiguring and hinge joining all four circles together into a circle makes a torus ring. Eight tetrahedra are formed and joined at right angles to each other allowing the ring to move rotationally through the open center.

See the following site for instructions; http://www.wholemovement.com/index.php?option=com_content&view=article&id=51&Itemid=43

Above) After folding it in half and then thirds, open it to the circle and see three diameters.

Below) Refold it to the cone shape and fold the top curved edge on one side over between the two end points and crease. Turn it over and do the same thing folding over the top curved flap on the opposite side. The four curved edges between the two folded over ends remain unfolded.

Above) The open circle shows two opposite sectors having straight edges. Refold the curved flaps to the inside (as shown) in the opposite direction of the original fold.

Below) Fold the diameter, the one parallel to and between the straight edges, to itself and use a bobby pin to hold it. This forms two open tetrahedra joined by a common edge. The length of the joined diameter, two radii become a singe edge of joining, is at right angle to the two straight folded over edges.

Do the same folding and joining diameter using the other three circles.

Attach two of the above reconfigured circles together taping with a hinge joint.

To make a **hinge joint** bring two circle units together attaching on the straight edges with flaps folded over. Rotate one unit to one side where the adjoining surfaces are touching. Tape along the joined edges. Fold all the way to the other side keeping edges together and tape along the opposite side of the edges. This way both sides of edges are taped together making a strong connection with maximum rotational movement between the two units.

Below) Join the other two circles in the same way making two sets of two.

Join the two sets of two in the same way as before by making a hinge joint on each end. Bring straight edges together and taping on both sides of adjoining edges. You have to roll the ring to tape the last pair of joining edges.

To make a solid tetrahedron torus ring, the movement pattern is the same, you will need eight circles, each folded into a regular tetrahedron.** The instructions how to do this are on my site; http://www.wholemovement.com/index.php?option=com_content&;view=article&id=51&Itemid=43**

Put the eight tetrahedra together in a circle, edge to opposite edge using hinge joining. You now have a version of the open torus ring made with closed tetrahedra forms.**Elongated **** torus ring**

Folowing are another and simple way to make a differently proportioned torus ring.

Fold three diameters. Then fold each end point of the three diameters to the opposite end point and crease. This generates three more diameters making six equally spaced diameters dividing the circle into twelve equal sectors.

Below) Using the creases fold the circle in half and then into quarters. You will see the folded quarter circle is divided into three equal sections.

Bring the two edges together and tape along the edge forming an elongated tetrahedron with a equilateral triangle open end.

Above) Fold and tape another tetrahedra the same way. Bring the two tetrahedra together as shown and hinge tape the taped edges together (on both sides) with the short open ends of the tetrahedra in opposite directions. Make sure to fully rotate the units as you tape on each side. this will give you the greatest movement.

Fold and tape all eight circles the same way, making four sets of two tetrahedra each.

The two open ends of each set of two will be hinged on the curved edge opposite the hinge joining on each set (the length of the shorter taped edges will be at right angle to the length of the longer taped edges.) Even thought the connecting edges are slightly curved and not straight, they can be taped and it will be strong with tape on both sides. This makes a right angle pattern of movement between the two sets.

When tapping the hinge of the two adjoining tetrahedra make sure the surfaces are face to face.

Rotate hinge to the opposite open face to open face and tape on the other side for greater strength.

Make two sets of four each. Join the two sets of four together using hinge joining on both ends completing the torus ring circle.

Here you will find videos of a variety of torus rings: http://www.facebook.com/wholemovement#!/wholemovement?sk=app_2392950137 One is the one you have just made and there are a number of others that might be a challenge using more complex tetrahedral units.

Explore and enjoy the movement.

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### Developing The Icosahedron

One of the icosahedron variations from last months exploration interested me to develop it further. I was curious to see how two with many possibilities would develop. By introducing elements that are consistent to the icosahedron form (not arbitrary design elements) I wanted to see at what point would possibilities run out, and growth stops.

No individual systems is self-contained, they are all parts of larger systems in the consistency of pattern development. In development of two different directions added-on forms are different in configurations and locations where they are placed. The direction for both is determined by the information within the icosahedron form itself. My choice from all the options was that which looked more interesting to spend time pursuing.

Above are two views of the same icosahedron. Left shows the triangle face forward with folded trapezoids joining the other three units at one place on the edge. On the right shows the open space forward where three units come together. There are four of these openings equally spaced reflecting the tetrahedron relationship.

Left below shows 12 forms of I/3 bottom layer of the tetrahedron division placed on each trapezoid surface.

Right below shows filling in three triangle open corners of each opening at the 1/3 top layer mark of the tetrahedra; there are 12 tetrahedra.

Top Left shows the addition of full tetrahedra attached to each closed triangle plane.

Top Right shows a full triangle attached to each congruent open triangle plane.

Left show the above attached tetrahedra opened as if to make room for an appearance from inside.

Right shows the 1/3 layer of the tetrahedron used above on left side development. The smaller triangle side is folded in to accommodate fitting onto the perviously added tetrahedra. I decided to leave it as is with 20 equilateral triangle planes facing outward reflecting the icosahedron of origin. When holding it is easy to recognize the tetrahedron, which positions the octahedron and reveals the cube pattern all inherent in the expansion of the icosahedron in a unique expanding development.

The Left photo above shows development continues with some elongated tetrahedra attached as if coming up through the open tetrahedra. Remaining are12 open small triangles from the original four openings.

Here the openings have been covered with elongated tetrahedra folded to the 1/3 proportion with the angle naturally occurring for alignment with the icosahedron stellations. The cube and the tetrahedron relationships are apparent in this form, the octahedron is there but difficult to find. This is a uniquely irregular stellated icosahedron.

In both cases we have by adding units eventually closing off the inside making them “solid.” Further expansion could be done by adding more units to surfaces, as if generated from the inside. Eventually the surface would diminish to where it would no longer be possible to add on in the original scale and what is left becomes textural.

Starting with two open tetrahedra forming the octahedron and by adding two more tetrahedra we form the icosahedron. The seed tetrahedra allows movement between the 3, the 4, and the 5 as it develops through various stages of growth in the icosahedron form. Any one of these stages can take different directions by the form of and how we arrange, or what constraints we set up for continued growth.

### Regular/Iregular Icosahedron

In thinking about the icosahedron and the various possible truncations using equilateral triangles in a tetrahedron patterned net I began to wonder about irregular triangles in the same icosahedron arrangement. Here are a few variations in exploring the same process using the tetrahedron folds to form a right angle tetrahedron; 3 right angle triangles and one equilateral triangle.

Using 4 right angle tetrahedra opened and arranged in the tetrahedron net pattern (one in the middle and 3 off of each side) reveals many interesting variations to the icosahedron. This tetrahedron net pattern for the icosahedron is in a pinwheel shaped net and uses 16 triangles rather than the traditional 20 triangles. The remaining 4 triangles are open planes of relationship due to off-setting the edge joining.

All the models use only 4 circles each in the same patterned arrangement, some using different combinations of the same folds, others using additional creases. This will give you an idea of how the same patterned net can generate a diversity of forms; some are made using a higher frequency folded circle.

**Forming the right angle tetrahedron**

See website for tetrahedron folding instructions; http://www.wholemovement.com/index.php?option=com_content&;view=article&id=51&Itemid=43 ( 2. Make a tetrahedron.)

Below is the tetrahedron net, the folds from which the right angle tetrahedron is formed. There are nine creases; 3 diameters and 6 chords dividing the inscribed triangle into four equilateral triangles.

The black lines are the creases traced for better viability

There are two points on the circumference on each side of the diameters, three sets of three.

Below) Fold each point individually to the center and crease. This makes 6 more creases. Having folded the tetrahedron, open it to the circle showing the triangle net.

This will form two more inscribed congruent equilateral triangles.

Fold one of the 3 inscribed triangles, either to the right or the left of the center triangle. Notice the center triangle is now off center as if it has been rotated where each division of each side in no longer evenly divided.

The second line in from each corner is at right angle to the edge. When all three corner points are folded behind on this line, 3 right angle corners are formed around the center equilateral triangle.

Bring the right angle corners together in the same way as folding a regular tetrahedron. This forms a right angle tetrahedron. The folded over right angle corners will fit one into the other to hold it closed

**Variation in forming the icosahedron net **

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Fold 4 of the right angle tetrahedra units and open them to the net and tape them together as shown. Face side up using right angle tetrahedron in icosahedron net.

Turned over with bottom side up.

We can see the proportional difference between using the equilateral tetrahedra (left) and the right angle tetrahedron (right) to form the icosahedron net. The net pattern for the icosahedron is the same, each showing twenty triangle faces.

Here is a more open and irregular form of the icosahedron by changing the proportional attachments of the right angle tetrahedron in the icosahedron net.

Below are a few more variations using the patterned icosahedron net. These are made using tetrahedral units folded from the 4-frequency diameter net. Go to “Unity Origami” blog entry from Saturday, February 26, 2011 for folding the 4-frequency diameter circle. By reconfiguring those 12 crease into the inscribed triangle and exploring truncated possibilities and using then in the icosahedron net, there are many variations to be discovered.

This variation above is formed using four 8-frequency diameter circles in the icosahedron net.

Here are 2 units of the above variation joined together forming a single unit. The concave depression of each end can be pushed out to the same configuration, thus one fitting nicely into the other. Multiples of this unit can be joined to form a benzene ring, an open sphere, the tetrahedron/octahedron matrix and a variety of other complex systems.

Another variation in the form of a tetrahedron reconfigured from the same four circles in a tetrahedron patterned icosahedron net.

There is no other form of experiential modeling that will demonstrate this kind of transformational process. Even though some of the models take the form of a tetrahedron and a variety of other truncated and open systems, they are all from the same icosahedron net in a tetrahedron pattern. Were we to make each model using individual polygons it would be a difficult construction job of measuring, cutting, and gluing, where some of them you would not know to be able to make.

Above shows a few directions to be explored using the right angle tetrahedron patterned to the icosahedron.

In the photo below we see the same pattern in different forms. The model on the left is formed using equilateral triangles where the one on the right is from triangles of a higher frequency folding with the circumference folded out rather than in.

### Jury Duty And The Circle

Last month I had jury duty for the first time. It was an extraordinary experience that gives me hope for the future that all people can join together in agreement. We were twelve randomly picked strangers that came together to make a decision. During the trial we were not to speak of the proceedings that was on our minds, so there was little else to talk about. We did not trade personal stories or chitchat. Social was not our purpose for being there, rather it was to find fairness in a conflicted situation.

On the fourth day we were given instructions to come up with a judgement and could not leave the room without agreement. A first straw poll showed ten agreed and two did not. It took about seven hours before the two differing views came to see a larger context allowing them to change perspective without making personal concessions. We all walked out of the jury room feeling we had all made the right decision. Then parted probably never again to see each other.

In that jury room was the influencing present of truth, the beauty of human struggling for goodness in a relative factual situation. Millions of years of evolving civilization and each culture and individual have had to come to some understanding about these concepts that have so much to do with the choices and decisions we make.

We looked for truth in the facts we were given. Only within an enlarged perspective were we able to find meaning that allowed each of us to get beyond our personal interpretations and bias positions. All had to participate, expressing how each felt, so we could identify the social judgement we were assigned to make. That larger context towards greater value revealed what was fair, moral, and ethically right. The appropriate decision was reached by all participants for the millions we represented.

There was beauty in finding the harmony and balance, the rhythms and proportions of human interactions by acknowledging the greater context that embraced individual conflict. Beauty was not apart from the truth giving meaning to the facts. By enlarging our perspective conflicting parts become co-ordinates revealing a perspective necessary to discovery agreement.

Goodness is a term associated with volitional action suggesting personality of will; a consciousness of well-being towards others. In society it is ethics; individually it is moral direction towards appropriate action. Not only was there good in identifying the beauty in truth that underlies conflict, but in so doing we felt that we did the right thing in finding fairness in what had been given us to judge.

There was a joining of moral responsibility and social ethics through the triunity of truth, beauty and goodness, serving as unspoken yet practical guiding for what needed to be done. These Divine values are infinitely reflected in so many ways to move our lives towards something of a finer nature than what we are used to.

Compressing the sphere to a circle disc reveals a triune surface showing the triangulated nature of pattern, principle to all movement. The first fold of the circle, a 1:2 ratio, shows balance in the dual nature of forming the tetrahedron. Two more folds proportionally generate three equally spaced diameters increasing triangular relationships that branch out from there. Truth, beauty and goodness are self-evident qualities in the folded circle/sphere unity as principle root for all subsequent folding. There is greater meaning and purpose when thinking of unity as boundless potential rather than the restrictions of a defined circle unit. I could not help but to think how these qualities are abundant in our lives even when not noticed and over shadowed by conflict of one unit against another. The only resolve is to up-step the context towards unity. The circle functions as both part (simple to do) and Whole (difficult to recognize.)

### What Is So Important...?

Often I have been asked, “what is so important about the circle?” To ask what is so important about anything means we have lost some capacity to be curious by failing to observe the larger context of what is in question. Things often become static without interest and familiarity becomes a crutch. This question says more about our lack of attention and a need to separate and abstract things from context than we are asking about.

All circles are multiple reproductions of the Whole and function as individualized parts. Each circle gives demonstration to the comprehensive nature of the Whole. The physical circle/sphere is the only form that demonstrates the concept of Wholeness. The Whole is self–organizing where multiple parts are generated through interaction. All potential of possibilities are realized through the three fold triangular grids formed through information revealed from the folding. The self-organization circle indicates endless potential in the forming process through the reconfiguration of and joining multiples in various combinations where there is no end to the number of parts that can be generated. This is important. There is no conflict about the circle as Whole and part. It appears a paradox only in concept.This is obvious with all the ways the circle can be reformed into triangles, squares, pentagons, hexagons and all manner of combinations, all from within the circumferential boundary. We can no longer afford to ignore the implications of folding the circle as it relates directly to geometry and the more abstract concepts of mathematics found in the organizing principles of pattern formation. Nor can we deny the comprehensive nature of the information that is generated.

**Folding**

The alignment in folding the circle in half proportionally generates three, four and five folded diameters from which three fundamental triangular grids are folded. Through the symmetry of forming, transforming, reforming and joining multiple circles, the forms and systems change, the grids are constant and the circle remains consistently Whole.

Mathematically we know three diameter lengths are always short of equaling the circumference. As we extend and diminish the length of the diameter the circumference increases and decreases revealing concentric circles The limitation of chords is that they are all contained within the parallel scaling of the circle, infinitely out and in. This includes all tangent lines and circles. The diameter is circle alignment, a symmetry revealing division without fragmentation or separation. The circle contains all expansion, contraction, converging and diverging of individually separated and isolated parts.

**Sphere/circle unity**

Spherical compression into a circle requires revision in thinking about the dimensionality of the circle and the importance of spherical origin. Through right angle compression nothing of the sphere is lost or added to, only the form has been changed; the transformation to circle remains Whole. No matter how we use it or what ideas we have about the circle it will faithfully reflect back what ever our limitations of our own thinking about it. Reconfiguring the circle in all ways tells us about the properties and limitations of the forming of individual expression, not about the nature of the circle. This is true, for there is no logic or demonstration to prove the circle otherwise than Whole.

Experiential, hands-on folding is necessary to move beyond the concept of bringing separated things together to make a "unity." Unity is not made, it is realized. Everything is already together even as we perceive what is individually formed. The concept of unity, of God and creation is discussed in all religions and philosophies. There is no physical demonstration of what those ideas mean until they become personalized. The idea of unity as a structural, principled, and sustaining reality can be demonstrated by the self-referencing and reconfiguring of every fold in the circle. Yet these ideas have no meaning until personally realized in some experiential way. This means the circle as Whole is not conceptually knowable until you become personally engaged and spent time folding. What other physical models do we have that demonstrate the transformations of unity with such far-reaching implications?

The circle is the only form that remains Whole through endless forming and reforming of surface. It demonstrates the interconnecting matrix that generates all classifications of polygons and polyhedra and many other unclassified formations. If these things were not in the sphere in the first place we could not make them, even with a circle. These forms are there as pattern before any fold. We do not create diameters or symmetry, they are revealed by aligning and creasing the circle; we simply identify what becomes visible and give them names. This brings up a profound question about the circle, taking us back to those early childhood questions about existence, origin, what are we supposed to do, and for what purpose?

We explain using polygons to “prove” using construction methods of reasoned logic. The circle has no reason only an identifiable logic of self-organizing structural pattern revealed through a self-referencing Whole; first observable in sphere/circle compression, consistently realized in all folding thereafter. Reason proves facts where logic combines knowledge with revealed information through experience and insight. Facts of information, reconfigurations in the folds and in combining circles, have no meaning in themselves; the larger reality is in understanding the comprehensive nature of the circle. That is what I find important about the circle.

**Whole-to-parts**

Life is both from bottom up and top down. We have formalized education as primarily bottom up, parts-to-whole, teaching only parts and expecting unity to somehow be understood. An aggregation of facts together does not make a Whole. The reality of our lives reveals both bottom up experience and top down revelation that provides direction and moral insight, a decidedly Whole-to-parts perspective. The word Whole is capitalize referring to inclusive; distinguished from the idea of separated parts joined to make a whole. Parts in number will always fall short of number of parts in unity.

Is the circle a force of nature, a form in nature, or an invention of human nature; did it evolve through practical usage to become a symbolic representation for the mathematical concept of zero? Or possibly the circle is a contextual creation of relationships that is the only possibility for infinite time/space expression of a sustaining pattern? In line with many religious traditions the sphere/circle compression demonstrates an individualized tri-unity disengagement from a trinitized monotheistic unity, which we have no other word for but God. Possibly all are true and relative within the Whole, as observed modeled in the form of sphere/circle compression and circle/sphere decompression by folding.

This detail is of the model below. It shows individual unit circles each reformed and joined in one of many possible designs. Each circle is uniquely formed from the same creased grid; an aggregation of paper plate circles.

Imagine this model, a limited number of multiple circles, combined with all the other possibilities of reforming and arranging endless multiple circles as the expression of one circle in all time/space instantaneously revealing all possible potentials at once. There is purpose in being given the advantage to experience through time, to spread out over space, to explore and discover with frustration, delight and with wonderment all the unimaginable possibilities and unseen potentials towards some far distant fulfillment.

### Movement

Having over the last month revisited some of the transforming systems I have explored by folding and joining circles has me rethinking movement. You can see videos of a few of the torus systems at; http://www.facebook.com/wholemovement?sk=app_2392950137 There will be more videos coming about variations on the torus and other kinds of moving and transforming systems inherent in the folds of the circle. Until I get a better understanding of moving videos around, here are a few thoughts.

We are all familiar with any number of geometrical transforming toys, many that appeared in the market after the rush of the Rubik's cube in the 70's, followed by a glut of “transformer toys,” and of course now the simulation of transformation through computer imaging. There is a long history of geometric and folded paper transforming and movement systems; the hexaflexagon and origami cranes commonly known to many. We all have curiosity about and are fascinated with movement. What is it to move, to be moved, to change and be changed? What happens as forms change is intimately connected to our ideas about time and space and form as the ordering of relationships that reveal interconnections between local and other.

We can trace movement through layers of infinitesimally smaller organized subsystems of individualized moving parts; beyond the atom to even smaller imagined parts and systems, about as successfully as we understand movement at speeds beyond our capacity to comprehend. Movement appears relative, so we assume a binary position of movement and non-movement, giving ourselves a standard to measure the space between things, which is only scaffolding to move from the physical mechanical to the transcendental mechanical.

As all children no doubt, I remember looking at trees moving and wondered where the wind comes from, who started it; what was that first push? What is it that is moving? I still wonder about movement and where things come from. Most explanations are imagined stories, elaborations in science about how things happen, not where it comes from or why. This moves into a larger context, into religion which has been around as long as we have been aware of forces existing outside of ourselves. Are we not suppose to ask these questions? There is no dogma in questions, or in faith; only in the formalize answers and the fixed stories we tell to others, expecting them to believe our story when they have their own stories to believe. No two pictures can have the same frame; and every picture tells a different story. You can bet we have been framed, for movement is not caught in static images.

I am intrigued that we have used the circle as a static symbol for both everything and nothing. Today we agree on mostly nothing, zero, yet desiring everything. We really don't understand either. Together is suggested an evolutionary integration towards understanding from limited movement towards increased capacity to move. A circle is a complete, inclusive, and self-referenced concentric movement system indicating there is no inner or outer boundary. There is no such thing as a one way movement except in the static image of conceptual framing. The circle is the compressive movement of the sphere from a spherical form to a planer form without a break in unity. The sphere looks the same weather moving or not, unless it is compressed into a form change. Only in origin will we find necessity and understanding the need for movement and change.

Compressing the sphere changes its form in two primary directions of symmetry; perpendicular to the expanding centrifugal plane, similar to how galaxies are formed, and the decreasing depth of dimensionality. Three circles are revealed in transforming spherical unity to a planer triunity. There is no separation of surface only a redistribution of volume. Nothing is added or taken away, the circle/sphere is Whole. Between the two circle planes is a circle ring, the dynamic agent of differentiation. Triunity of the circle is structural pattern revealed through precessional movement and is principle for all subsequent realization of potential formation; thus is prologue for another creation story:

Wholeness through movement causes division becoming duality in triangulation with each part consistent to movement and totally inner-dependent to the Whole. The individual nature of part to Whole regulates the interactions between all formed and unformed parts on all scale, in all time, with purposeful movement. This is how one story begins.

This still does not answer my question about the first push of wind, it only shows that movement is the first action that happens out of what we imagine to have no movement. From a supposed spherical non-movement compression into a time space continuum followed by the line of division into the triunity of been, doing and still doing. That first

movement can not be understood by describing a diameter or drawing line between there and here, the information is in the conscious experience of moving.

*Folding is not to just demonstrate what you can do with the circle, it is about what you can discover by observing movement and what information is revealed in movement of the circle to itself. *Folding is not about the skill, inventiveness or creativity expression of the folder, rather it is in understanding the nature of what the circle can do and what is revealed that is instructive for increasing folder capacity. Only through observation of moving do we know what we are doing. Circle unity does not change, but movement changes the form, so it is not about the form, rather to understand the movement between the forms, directional and with alignment. The circle is both context and the content; it is the folders perception and understanding that changes and moves.

As we observe the initial sphere/circle transformation where information is generated, that upon reflection and consistent development of systematic symmetrical folding leads to an equilateral triangular grid of creases. This triangle grid is the primal matrix for moving the circle; for forming, reforming, transforming, and in-forming the circle/sphere. After all the circle is a generalized flat plane, albeit the only one of its kind. Reconfiguring this grid is the means to informing the possibilities inherent in the circle. Endless interrelated movement and joining in multiples gives expression to what otherwise remains potential.

This is only part of the story. It does not answer my question about the first push of wind, but it does move the story forward off of the flat plane and out into space where the movement from one location to another requires time and consideration.

### Properties Of The Circle

It is important to understand the origin and the properties of what we are working with. In this case the 3-D circle needs to be differentiated from the circle image we draw. The circle, the subject of the image, has spatial properties that are unique from all other 3-D forms. We need to observed the differences in properties between 2-D and 3-D if there is to be any clarity and understanding about each.

If we do not know the properties of what we are working with we do not know what it is or what to do with it beyond arbitrarily imposing our will that frequently ends up violating the nature of what it is, often being counter to expected results. Lack of understanding properties has proven over time to cause unforeseen problems. Properties of the circle set the foundation for all subsequent folding (see previous blogs).

This picture shows a circle and the image of a circle.

The circle originates through spherical compression. Both circle and sphere demonstrate a dual function as individual unit and unity simultaneously. The non-differentiation of spherical surface is transformed by a right angle movement to the direction of an centrifugally expanding circle that reveals a triunity of three circles

The image shows one circle where as the 3-D circle shows three circles, one on each side and a circle ring. (Think of an extremely flattened cylinder.) There are two edges where the three planes meet. There is an inside volume and an outside space. Three planes, two edges, two spaces; (3+2+2=7.)

The circle is a triunity of three interdependent circles that can not be separated one without the other. In order to conceptually take them apart unity is destroyed, being left with three abstracted, isolated, and imaginary units. The association of three anythings is a structural pattern and reflects unity. All number of units will never equal unity, for unity is always singular. Units is always plural and infinite in number. Three is the first active number and seven is the most possible associations of three.

One set of three (ABC)

Three sets of each individually (A) (B) (C)

Three combination of sets of two each (AB) (AC) (BC)

Drawing a diameter divides the image in two halves. When folding the circle in half the diameter changes the properties where instead of two semicircles on one plane there are six semi-circle planes; six half circles. While this make no rational sense using a 2-D model, it is observationally logical to the folded circle. The circle remains whole, retaining unity even as folded into six half circles. ( If we decided to count the two edge circles it would change the possible combinations of associations.)

There is no conflict between folding and drawing circles; they are two very different systems; one is an image/idea of the other. Knowing the difference in properties helps clarify some confusion and greatly expands our understanding of the circle. It introduces a new area of dynamic exploration that in no way denies the theoretical or 2-D mathematics that has been developed. There are well over a hundred relationships, functions, and math concepts in this one fold of the circle into six halves. This is not to suggest one is better than the other, but rather to understand the difference and benefits of both folding and drawing circles and the connections between them. We know the value of drawing circles but there is no precedent for folding the circle and that means we have no experience or understanding about it. Only through the direct experience of folding will we understand the difference.

Besides the information and the beautiful objects that are revealed by folding, it is fun, interesting and engaging. We have a prejudice of not wanting to have too much fun learning something we have already decided should be difficult; if it is serious we must work at it. We are at our most open to learning when we are having fun and engaged in what holds our interest and simulates curiosity. Long ago we decided that mathematically the circle as image is a symbol for nothing, a place holder to later be replaced by something of value. It is now time to look at the information value of the circle beyond the image and the mechanical advantage we find in using it.

The information and reformation possibilities by folding circles demonstrates we can no longer afford to disregard circle/sphere unity. Because we have not done it before is no reason to continue to ignore it. I am writing these blogs in an attempt to give some understanding about the importance of the circle and that it might possibly simulate you to want to fold the circle and to find out for yourself and discover things there to be observed, and connections to directions not yet seen.