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## Bradford Hansen-Smith

### Polygons to Circles

I was given a box of scrap die cut leftovers. Finally decided to explore the possibilities in these truncated hexagon star throw-a-ways. When working with polygons of various shapes there is always the limitation by what has been cut off.

*Below)* the two-sided maroon and blue fractal shapes are limited to a flat “2-D” tessellating unit.

2-D objects are inherently 3-D, which always makes better sense to my mind. So that is the next step.

*Above left)* four units joined together in tetrahedral arrangements. The possibilities are limited in joining these units.

*Above right)* The next obvious thing is to fold unit in half three times and rejoin units in the same tetrahedron arrangement. It became more interesting for the increased possibilities of joining multiples.

*Above)* making three folds across the shortest two opposite points is an equal hexagon division. Folding one crease in half to itself reforms the flat shape into two tetrahedra joined by common edge, a “bow tie” reformation. There are other ways to crease the shape; we will get to them later.

*Below)* four of the above “bow tie” units when joined the same way, edge-to-edge, form a Vector Equilibrium system of three, four, and six (same as with the circle using three diameters discussed in previous blogs.)

*Above)* two more possibilities of joining the same “bow tie” units in different ways. The one on the left has limited rotational movement between the two sets of two. The one on the right is stable.

*Below)* shows variations of reformations and joining going in other directions.

*Above)* two views of a particular development using the reformations above.

*Above)* two views of using the preceding units; twelve sets of two form an edge defined cubic arrangement. The movement in the units shows a distortion from the regularity of the cubic pattern. This is another example of the non-structural nature of the cube without diagonals to structurally define it. This is not a stable model.

*Below left)* four units are arranged in a cubic relationship showing two intersecting tetrahedra forming the diagonals of the cube. This is structurally stable.

*Below right)* a couple of models using one crease and three creases joining only two units for each.

*Below)* using the VE in multiples reveals the closed packing of spheres, origin to the tessellations of polygon and polyhedral arrangements.

In my August blog I talked about joining four circles with only one crease to make a vector equilibrium: http://wholemovement.com/blog/itemlist/date/2012/8?catid=140) . Here we see a very different treatment to do the same thing in a very different arrangement.

*Above left)* VE from three creases as seen before.

*Above right)* the same arrangement formed with only one crease in each unit. This time the creased shapes are positioned differently forming two opposite square planes out of six, showing none of the eight triangle planes. It is a collapsible system that becomes stable when standing on the square opening.

*Below)* sequence of vertical closed position opening to VE arrangement collapsing down to horizontal closed position. There is a 90 degree twisting in the movement. The system becomes fixed when the two square planes are stabilized as tension is applied to each end of the four folds.

This will be explored more shortly.

There are two more ways to symmetrically divide by folding this truncated star.

*Below)* shows three ways to crease symmetrical divisions consistent to the shape.

The last two ways of divisions are much more limited in what can be developed than what can be generated from using the first divisions. They become more specialized because of the greater complexity of division.

*Below)* are two VE systems folded from the first and second divisions above to see what differences occur.

*Above)* the second division does not join together reflecting spherical packing. The will form a planer hexagon ring of six units, of which three joined are shown. The conformation of edges makes a difference in how they join. There was not enough interest to continue this direction

*Above)* an interesting systems using four truncated star units made from the third divisional symmetry of six creases. It has limited potential in multiple joining, but there is interest in the complexity of form.

*Below)* two collapsible systems are each made from the first and the second ways of dividing the units. The dynamics for both are the same. The configuration of perimeter does not matter; the movement is around the center pivot.

Wanting to see the collapsible system in circle form I first folded three diameters thinking they were needed to get accurate proportions before realizing it is unnecessary; one crease in the circle is consistently self-positioning to all four units.

*Below)* six different positions of reforming the collapsible VE system above made using four of my circle business cards.

*Below)* three different proportional quadrilateral openings, one rectangle and two sizes of squares. Any proportion can easily be folded with only one crease because of the symmetrical self-alignment of circles. The one pictured in the center has square edges equal to the radial length.

*Below)* a six-inch diameter ceramic bowl sitting on top of four 3-inch diameter circles with one crease in each. This demonstrates the load-carrying possibilities of four paper circles stabilized by the compressive pressure on the opposite open planes. This would work with any configuration of units arranged in this manner because of the triangulation of compression of four twisting lines of force passing close to the center pivot.

During the above exploration I was also playing with a single circle to see how it can be folded so that when four are joined forming a tetrahedron they would open the four solid planes. With a slight variation of what I have done before it began to look interesting with possibilities.

*Below)* a single paper plate circle reformed to show the triangle plane collapsed towards the center with the ends opening out.

*Above)* a plan and edge view of four of the above units arranged in a tetrahedron pattern. There is a nice opening from the edge perspective but the collapsed triangle plane has opened to a solid plane again. This movement is necessary to allowed the system to come together forming the tetrahedron. The four open end points with the six open edges makes ten openings in the tetrahedron. Now this is interesting given the number describing the tetrahedron is ten (four points and six relationships between them.) The four open ends can also be closed leaving six openings that in the solid form are the edges.

*Above)* two tetrahedra, six units are joined sharing an open interior plane making a bi-tetrahedron. Each tetrahedron is formed using only three units. Here the solid triangle plane is collapsing to accommodate the change of added units. There are now fourteen openings.

*Below)* three views showing a left handed twisting tetrahelix consistent to the development from the bi-tetrahedron formed by adding two more tetrahedra in the same way, open-face-to-open-face. It retains the seven vertex points joining 3, 4, and 5 planes, only here the vertex points are open; they can also be closed slightly collapsing the system. This is a skeletal formation that conforms to the basic helix pattern but having twenty open spaces. The solid planes of the tetrahedron are now collapsed even more as it grows into a four-segmented tetrahelix, starting with four and growing to a ten-circle system, reflecting the number pattern of the single tetrahedra. This is what I was looking for and find delight in the possibilities.

It is not often I go back working with shapes and forms without the circle context, as with these truncated star shapes. When I do it always leads back to the circle. Only through understands circle/sphere origin is there information that allow a greater realization of the inherent potential that transcends the parts. The larger context reveals information otherwise unseen.

We now know there are at least two ways to form the VE pattern with four one-creased circles. This opens the question about other possibility in forming the VE using one crease in each circle. There is also the question of other ways to further open planes of the tetrahelix to maximize spatial flow without disrupting structural pattern; example the DNA helix.

The circle is always a circle no matter what polygon or polyhedral transformations it goes through. The circle Whole is never less than Whole, unless it starts out as a polygon, in which case potential has been diminished. Wholeness of the circle, unity does not change through reformations; what does change is the growth and self-understanding of the folder through a growing awareness of expanded potential working comprehensively within circle unity.

### Importance of Circles

A couple of weeks ago I attended the Buckminster Fuller Legacy Conference in Asheville NC sponsored by the Buckminster Fuller Foundation and the Black Mountain College Museum + Arts Center. It was good to see a couple of familiar faces and meet new people, all having a passion one way or another stimulated by Fullers work. I presented two workshops folding circles.

Before showing any of last month’s exploration I want to talk to you about the importance of the circle, it's not the first time, but given Fuller’s ideas about success for 100% of humanity, it needs restating. We live in critical times where it is imperative to have demonstration of a workable and sustainable ethical and moral ground for human survival to balance the ever-accelerating technological advancements for control. If we are going to allow future generations the possibility to achieve higher human potential, we need to talk about the sustainability of a moral position that includes as Fuller says, “all humanity.” There needs to be a discussion about survival for past generations, those presently on the planet, and all to follow. It is a discussion about the physical, intellectual, and spiritual directives necessary to move us forward in a balanced and healthy way for the entire planet.

In early nineteen-eighties Fuller caught my attention by saying, and I paraphrase; ‘with all the problems we face on this planet, we are at a critical time in human history. To survive we need to make a decision now. Once we have made that decision we have all the time in the world to figure out how to solve our problems.’ We have yet to make that decision for humanities survival. We keep arguing about our problems, trying to find solutions before making a decision. Life does not work that way. Purposeful decisions are made first, then problems are met and solved with resolve towards that end. Once the objective is defined and commitment made, finding solutions is assured, it has nothing to do with time and everything to do purpose of will. We cannot predict before hand what it will take to assure progressive survival for all mankind anymore than we could predict where we are. We lack a collective moral courage and intellectual capacity to make a comprehensive decision to do what is necessary. There is no demonstration, no model of what planetary unity looks like.

Folding circles is not just about folding, nor about what you can do with paper, not even about the math and art that is revealed. It shows all of these, but more importantly it is the only hands-on experiential demonstration of ethical interdependency and moral appropriateness of parts to Whole. Folding the circle is a concrete experience about these ideas where every part is in correct relationship to all other parts, even those not yet folded, giving the greatest benefit to all parts with the greatest possible expression to the Whole. No other shape inherently demonstrates individual part to Whole interdependency because there are no other parts that demonstrate total unity. Folding the circle is not arbitrary or random, but principled to the only process we know that is truly self-referencing, self-organizing and self-generating; demonstrable by anyone that can fold a circle.

Fuller talked about whole systems. There is only one Whole system. The undifferentiated sphere compressed into a triunity circle form giving demonstration to origin and context for progressive development and evolving spatial systems. No part or reformation is favored over any other, yet through movement each association serves uniquely in relationship with all others in an endlessly transforming of circle/sphere potential. By perpetuating the circle as nothing, a zero image used for construction, we hold future generations hostage by our ignorance about the inclusive nature of circle/sphere transformation, denying the experience of a comprehensive perspective beyond the linear and flat thinking that prevents healthy and balanced progress towards planetary, universal, and cosmic participation.

We are being held back by old ideas about the circle, preventing us from experiencing the circle/sphere shift from a fragmented egocentric culture towards one that is more inclusive where interconnections support common good for all human beings in the largest possible context. Individual growth is always in proportion to service to others through personal relationships on all levels of interconnectedness. We conceptually perpetuate linearly thinking about static concepts in a dynamic universe of changing relationships that progress through developing capacity to adapt.

Children as they draw circles would greatly benefit by cutting them out from the paper, folding and playing with them. Through observation about the dynamics of the circle, the first fold, how they do it, with discussion learn to clarify what they do and begin to discover information generated through exploration and familiarity. We have no idea of the potential of folding circles as an experiential tool for learning; we only draw and think image concepts without full understanding.

Folding circles is not about product. It has everything to do with starting from the circle/sphere Whole and observing a process of reforming, transforming, and informing what is principled, structurally patterned, and comprehensively unified in one place. Folding and joining circles inherently models a spherical process reflecting total interdependency of countless systems and parts, the Whole in multiplicity. The circle presents only what it is, a Whole shape not part of any other. It is complete within itself, countless parts uniquely interconnected through the circle. Movement creates extraordinary unexpected relationships. Synthesizing, summing, and constructing with parts will never achieve the unity that belongs only to the Whole. The circle/sphere inherently models a principled process of pattern formation that is instructive to all levels of human interaction. It requires observation, reflecting on what works for the full expression of the circle, not for the benefit of a few favored ideas or products.

It is not difficult to understand; parts do not make a Whole, they make bigger parts. Wholemovement generates countless relationships through evolving systems to give the greatest potential to associations between parts allowing full expression to unity of the Whole. Anything less and we shortchange future generations and ourselves. Without some comprehensive demonstration of origin, we have no idea of the enormity of human potential, or any idea how to approach that realization.

It is hard to miss in three folded diameters, reforming to 3, 4, 5 symmetries inherently interconnected through movement with alignment to the whole circle. With attention we see the same interconnections between all, cultures, levels of societies, families and in personal relationships. There is a human urge to find alignment to something outside of ourselves. Folding the circle is a process of alignment, coordinated patterns of the same folded grid reflected throughout. Every reconfiguration, all conceptions and information developments are increased by folding circles.

What brought me to this realization started years ago with Fuller using four paper plates and folding three diameters into to each made the spherical vector equilibrium. If Fuller could do that by folding four circles, then everything else we model can be demonstrated by folding and joining circles.

Last month I returned to the spherical vector equilibrium, (VE) the primary centered system of spherical packing. Last months blog shows how one fold in each of four circles will also form a spherical VE. Using another way to make the polyhedral form of the VE, I used two circles, each formed to four tetrahedra that are joined together. Then by folding the circumference outside rather than folded in, I began to explore the VE from a new perspective.

*Below left)* One circle folded into four tetrahedra; half of the VE system. On the right two circles have been joined showing eight tetrahedra reveling six open square relationships.

*Above)* two VE forms, both circles folded the same way, one circle with the circumference on the outside instead of folded inside (middle figure, first row in group photo below.) The two forms show identical VE arrangements of two circles.

*Above)* the two units from above are joined in the order of spherical packing, but now a different looking system.

*Below)* different combinations of variations using both the circumference on the inside and outside all formed to the order of spherical packing.

Next was to look at a number of different ways the circumference can be refored to the outside of the unit

(lower left hand corner.)

*Above)* these are a few variations with the circumference folded to the outside using the same folds to form the VE unit. Each variation has its own direction of development. I have only explored a couple of these, which follow below.

*Below)* four circles reformed to the variation in top row third over from left (group photo above.) This is the inside of half a spherical arrangement.

*Below)* a couple of views of the completed sphere using eight circles in a tetrahedron arrangement where the inside is the VE matrix and the outside is spherical in form. Bobby pins are holding them together.

*Below)* another reformation (upper left corner in the group photo above) is used in spherical arrangement showing two-thirds of an open sphere. The are joined on one half of the open rhomboid. This unit shows triangles, squares, and pentagons planes.

*Below)* two views of the completed sphere from above, a spherical icosidodecahedron arrangement using 20 circles.

Below) is close up of the sphere above. It shows something of the open spatial quality of the units used.

*Below)* another direction in joining the same units on the open pentagon faces reveals a cubic system of eight circles. Four views of the same joining shows a helix arrangement in an open cubic form with fissures through the center in opposite directions at right angles to each other. (The following pictures show changes in a sequence of opening the original units transforming a complex system to less complex.)

*Below)* Using the same eight units only joining them on the triangular faces forming an interior octahedron changes the way the form opens. Compare this to the last images above.

*Below)* continuation of opening joined parts reflects something similar we see in nature that diminish complexity to a simpler form. Here are three views of the same state of open transformation.

*Below)* Two views of the above model again modified by partial closing of what has previously been open. The transforming process continually changes the form and in this way gives the folder many option to explore, being able to then settle on any number of forms that have appeal as a fixed object.

I show you all these foldings because I want you to see what extraordinary differences there are in folding circles than from folding any other shapes. I find it exciting to see what is revealed starting with the circle and never being less than the circle and always being more than my imagination.

We all start with life and are never less than what life is. We are limited to individual capacity but can transcend those limitations by the choices we make and our ability to observe and adapt to the experience of living in alignment with our environment, partially formed by each other, that we know little about. Observing and adapting to change is growing relationships. It is important for me to realize possibilities of forms and systems that otherwise could not happen by putting prescription pieces together. I want people to see some of the possibilities by folding circles and to understand the benefits for working comprehensively in exploring and discovering for ourselves the benefits of a Whole-to-parts approach where movement reveals what is in-formation. The circle/sphere presents a different frame, yet incorporates methods we have developed over the centuries of development. Here we see diversity in the movement of change always with unexpected options for more choices. Currently Wholeness is a concept demonstrated by putting separated parts together forming systems with some idea about unity. Here the circle gives demonstration to those ideas and concepts of inclusive interconnectedness working with 100% of the circle; nothing is left over, left out, cut off or has to be added. Proportional balance and symmetry is inherent to unity, and beauty is the Wholemovement.

What is important about starting with the Whole circle and ending with the Whole circle is that reformations of creases are the history of choices we make moving through experiential relationships inherent in the circle towards realizing potential.

Distorted by fear, confused from lack of moral direction, short on ethical consideration, and overburdened with a focus on technological development for control, we have ignored the reality of the expansiveness and progressive nature of unity of the Whole by giving favor to small and self-serving using of imagination. Folding circles, as silly as it sounds, is one of the most comprehensive and expansive things we can do.

### Two Points and More

Exploring more about three diameters I again wondered about one diameter and the first fold (continuation from last month) and what else I have not seen. So I go over it one more time.

Marking two points on the circumference and touching them together creasing the circle in half forms a line of symmetry, the diameter. By connecting the 4 points showing 6 lines of distance between them reveals a kite shape. Four points and six edges is the number ten; 6 touching points and 4 spheres in a non-centered spherical order. There are eight triangle associations, six of them are right triangles. There is a lot of mathematical information revealed in this one fold of the circle.

All this changes when rotating around the diameter lifting the circle from the flat plane forming two open planes and two closed planes defined by the lines forming the kite shape. In rotating around the diameter in both directions there are two congruent tetrahedra, one inside out of the other.

*Below) *Development of kite shape by touching any two points and folding in half.

Ok, the question comes up; what if we fold the circle and mark two points after the fold?

*Above)* Fold the circle in half without marking the first two points. Mark two points arbitrary anywhere on the circumference, one on each side of the diameter. Draw the straight lines between the four points. There is no kite shape; it is a quadrilateral figure without symmetry, even though the circle is in half. We still have 8 triangles, but only two are right triangles. The quadrilaterals have varying edge lengths depending on where the points are located. It remains a tetrahedron without congruent edges or angles, except for the two right angles where the diameter functions as the hypotenuse.

* Above )* Two circles folded on the diameter showing a symmetrical and balanced tetrahedra formed to different proportions reflecting the two points arbitrarily placed on the circumference. Each forms a differently proportioned kite shape where the diameter is a perpendicular bisector of the distance between the two points. The four outside chords have been creased to better see the tetrahedron in a traditional straight edge form. This adds more elements for spatial reconfiguring not possible with other forms of modeling.

*Above)* Two examples of adding two arbitrary points on the circumference, one on each side of the diameter, connecting them with straight lines forming a quadrilateral. Each is a different proportion resulting from the arbitrary placement of the two points. Again the outside chords have been creased making it easier see the tetrahedron. Here the two circles are folded on the diameter forming the tetrahedra showing them to be irregular without symmetry.

*Above left)* The folded diameter is no longer the axis but becomes a variable edge length where as the line crossing perpendicular through the diameter is now the functional axial crease. It still shows dual tetrahedra but in different proportions.

*Above right)* The two tetrahedra remain irregular without symmetry. The reciprocal axial function is not at right angle but on some diagonal running through the diameter. Each circle shows the dual direction of movement in two different directions making the possibility four different tetrahedra.

*Below)* Four differently proportioned tetrahedral systems are arranged in a 2-frequency tetrahedron pattern, four circles each. Using the full circle changes the form when compared to traditional polygon constructions. Each of these tetrahedron arrangements has many different possible positions since the four rotational axes are in alignment and will go from open position, lower left, to collapsed flat in the opposite direction.

There are a lot of formal considerations and diverse information, reformation, and choices to be explored in that first fold of one diameter and four points. The more I look the more there is; for anyone to discover. There are images of other reforming the circle with one diameter on Wholemovement Facebook page.

The following images are a couple models left overs from last months explorations and a few more that came from this months exploring spirals and conical helices limited to using six creases in the circle, using the same reconfigurations. I will go further into spirals to give a context to theses and spirals in past entries in a future blog. In the mean time here are some recent models. They are all variation of the same folded unit exploring a few things that came to mind. It is always interesting that the circle reveals the geometric form of modeling as well as the more organic and biological forms we observe in some of the more obvious forms in nature. One model has additional circles reformed using more creases to give some specific forming towards realizing some biological expression.

*Above/Below)* Two views of the same helical model

*Below)* 25 circles with decreasing diameters from large to small.

*Above/Below)* Two views of a spiral helix that suggested a fish-like creature, so more circles were added as an example of how spirals function on a pattern/form level in complex systems.

* Above)* A conical helix stack of 25 units before it is reformed to half of the double-end spiral pictured above.

These images are part of an ongoing exploration into the spiral. There countless possibilities using any number of different reconfigurations where each formation reveals different aspects of spiral systems. The spiral is a pattern of movement through the concentric nature of circle/sphere unity. Movement is both into and out-from the circle as its own center. There will be more about spirals and helices in some future blog.

### June, July Grows Into August

June’s exploration started in May and moved into July expanding into Aug before finding resolution. Each open leaf-like growth left from last month has become a closed bud-like form. This seems relatively stable as a model so I will leave it as such. Were it growing in the world it might well bring forth flowers that themselves would have seeds closed within, and so it goes. Through three month of development I have reached what appears to be the limitation of the material and the direction of the form of this model.

*Below)* Two views of present state.

*Below)* Various stages of growth over the last three months starting with the tetrahedron and ending at the outer boundary of tetrahedra waiting for another season.

*Below)* Details showing build up of last layer of tetrahedra folded from one crease in the circle.

For instructions on how to fold a tetrahedron from one crease go to:

### https://www.facebook.com/wholemovement/photos and look for “Fold a Tetrahedron” and “One Fold Circle.”

This last layer of 115 circles, each folded to a tetrahedron using only one diameter crease, got me thinking about all I have seen revealed in one folded crease in the circle, the diameter. There must be much more that I have not seen. The vector equilibrium is fundamental to spherical packing and often the first model we make by joining four circles with three diameters each, joining them on the straight edges. Knowing that curving the circle will reflect similar configurations that come from the folded triangular grid, it then seemed reasonable to assume the spherical vector equilibrium can be made using only one fold in each of the four circles. I reasoned since the first fold is a tetrahedron pattern and four tetrahedra in the same orientation sharing the same center point is the first centered system in the closes packing of spheres, called the vector equilibrium, traditionally the cuboctahedron. So I folded four circle one time and joined them together.

*Below)* The circle folded in half. Next is folding the diameter to itself dividing the circle into two cone shapes that are attached by the diameter fold. It is held together using a bobby pin.

*Above)* Two circles with the diameter touching itself full length forms four cones, two sets of two cones each.

*Below)* To the left shows the two sets of two cones, two circles attached using bobby pins. To the right show the two sets of two cones each attached in the same way resulting in the spherical vector equilibrium. It is not accurate in circumference measurement from point-to-point (this can be achieved by moving the bobby pins to be equally spaced) but each point is equal distance to the center. This model shows the primary function of proportional triangular division inherent in the first fold of the circle. There are 6 square relationships between 8 cones pulled into triangulation.

The vector equilibrium is the primary spherically centered system of twelve equally spaced spheres around a center sphere. This model demonstrates again the comprehensive, economical, and interrelated nature of folding circles, unique beyond all forms of modeling. What happens first in the circle is principle to all subsequent development. We have seen with the spherical model over the last three months, starting with one fold in the circle as a tetrahedron pattern of movement, and now ending with tetrahedra that are formed from one fold in the circle.

The next thing I fold will no doubt start with the same tetrahedron pattern of movement.

**(To see more about one fold in the circle go to ****http://wholemovement.com/blog/item/112-alignment-in-milwaukee**

### Fractal Unity

This is a continuation from June's exploration and part of an ongoing investigation using only the nine creases of the tetrahedron in systems development. I am finding relationships in folding circles similar to fractal growth observed in nature and in algorithmic produced fractal images. Growing systems are always limited to the needs of larger systems where survival is supported by serving the larger context that is itself evolving. There are no whole systems, only systems serving larger systems. Every system functions in alignment to a larger context of purpose that is reflected in every part of all systems, showing unity throughout. There are always unique and identifiable evolving patterns directing the recognition of individuals, families, regions, cultures, kingdoms, phylum classifying similar formations, functions and individualized parts. Geometrically the context is the circle compression of spherical unity.

This ongoing model shows a systematic development using one reformations and a few variations from the numerous possibilities in the tetrahedron net as folded in the circle.

*Above)* Each individual unit is a reformed circle. Two similar parts can be combined in three of the same ways (right side.) I choose the above middle on the right since it was most like a line with two end points, points expanded to circles.

*Above left)* Arrangement of six of the above units in a tetrahedron pattern.

*Above right)* Thirty units, (sixty circles) forms the icosidodecahedron.

*Below)* I decided to double the single units making a set using four circles. There are two basic variations; one with open sides folded over and the other with sides not folded over (bobby pins are holding them together.)

*Below)* are three views of the icosahedron pattern formed using both variations of double units showing the spherical icosidodecahedron formed in a great rhombicosidodecahedron arrangement of squares hexagons and decagons. It is lopsided because long units were used on one side and short units on the other side distinguishing polar division.

Each cell is a multiple of unity; each circle reformed is consistently whole. Growing changing systems are always incomplete expressions of unity. Each circle is unity carrying potential for countless possibilities of reformations directed by specific design limitations of a given system. Multiple circles give a unique and individual expression to any association of parts. The unlimited potential lies within each component part as a unique expression. *The unity inherent in each part insures unity through out.* This is a primary fractal process of self-similarity towards finite expression of infinite potential.

Putting parts together gives form to synthetic unity, an idea about unity that is decidedly different than unity inherent in the circle that is not constructed. Both are important, it taking one to understand the other. By self-alignment in folding the circle (http://www.wholemovement.com/blog/itemlist/date/2010/8?catid=140) information is generated that is principled and directive to all subsequent fractal development. This is not figured out mathematically, it grows geometrically through a principled, patterned process.

The regularity of the icosidodecahedron pattern gives stability to the arranging of parts. The form is changed through layering recursive stages introducing human choice as a factor allowing variations to alter the forms in unexpected ways, always in line with specific pattern, not unlike what happens in genetic recombining. This self-referencing system allows for spontaneous and consistent choices purposefully made where unexpected growth can occurs.

*Above)* Three views where one half of the hexagon openings have been developed on one side showing the pentagons of the dodecahedron. The other side requires a different configuration to accommodate the difference in the two units initially used.

*Below)* shows further growth by modifying the single unit to conform to what is there. Again variations of reformation are necessary to accommodate each half. I have left the band around the middle open developing the polar ends.

*Below)* more layers of units are consistently added to areas that will easily accept them in a way that looks like natural growth coming from within. There is now greater variations in reforming the circle units giving greater complexity to the system.

*Below)* more layers are added expanding what is already formed moving more towards a spherical uniformity as the middle becomes increasingly filled out.

*Above)* the sphere as it now stands where all units on the outer layer are the same reconfigurations. I have left one decagon open to the inside as you might find with any growing systems where there is a primary connection between the inside and outside through all stages of growth. The outside spherical configuration is getting more spherical and complex. It remains open much like a sieve, without filling in the open space. I don’t know if I will continue developing this to see where it goes or get on with exploring some new formations that have come up during this process. Most likely I will continue to explore both.

The circle is Whole and part, something no other shape or form can demonstrate. There is unity throughout because every part starts with and ends with unity, giving consistency to the variety and differences of interrelated transformational parts. I understand the circle as pattern for all fractal self-similar processes.

### Alignment in Milwaukee

Having attended SENG, the Supporting Emotional Needs of the Gifted conference in Milwaukee this month, I met some wonderful people and had a delightful experience working with a variety of bright young students. One workshop was a small group of girls that had insightful questions going beyond most group discussions about the folding. They posed some interesting questions and observations that relate to previous blogs. I would like to share some of this with you.

After discussing properties of the circle (five congruent circles) I ask the group to fold the circle once. They asked how, did I mean in half or… ‘I said just one fold, your choice.’ Most folded the circle in half, the others folded less than half.

half fold examples of less than half folds

The reasons they gave for folding in half were pretty much the same; that is just how you fold a circle, or it seems like the right way. When asking those that folded less than half, there were three different responses; a) thinking about it and choosing to make the circle more interesting, b) emotionally feeling comfortable with the unequal division where part of the circle was held or embraced by the rest of the circle, and c) to make a display of being different by setting oneself apart and with the fear of being tricked in some way by what was being asked. Here are three fundamentally approaches to folding the circle resulting in less than half a fold. If we feel less then we fold less.

We talked about symmetry that comes from the alignment by folding in half, showing consistency of the circle boundary to itself. Folding less than half there is no line of symmetry, it is out-of-balance, without consistency, misaligned, and partial. (ref. http://wholemovement.com/blog/itemlist/date/2010/8?catid=140)

Each of us in our own unique way is individually different from each other and being different we do not need to make a display of it or try to convince others of our difference. We only need to work towards our own potential and capacity to reveal the value of that difference. By observing and paying more attention to others we notice and give value to the fact that we are all different. To the extent that we acknowledge that in others will we find strength in the uniqueness of ourselves; this is what we have to give each other.

The question came up about how to get back in balance if you feel out of balance, not in alignment with everyone else. In feeling your ideas are right and everyone thinks you’re wrong how do you figure out if your right or not? What laws or rules are there to know how to do this? Good questions from 11-12 year old people. There are no laws or rules for this, it is individual and maybe is more about *principles*, (ref. http://wholemovement.com/blog/itemlist/date/2010/4?catid=140) We talked about what principles might be, going back to what happens first in the circle, the qualities of that first fold. Folding in half is principled to all subsiquent folding, less than half is not. Going back to our own beginning we find the development of structurally patterned division in a fertilized ovum, demonstrated using four tennis balls in closest packing order. Further back towards origin before we got here reveals what applies to everyone without exception. The purpose and functions of unity that comes first is what holds differences together. We see it in the circle, not so easy in our lives.

Where does the circle come from beyond the conceptual image? How much more is the circle beyond what we are told? How much more are we than what we are told? In thinking about some of these ideas I realized we missed an important aspect to this discussion about alignment.

The two points on the circumference of the less than half chord represent the limitation of the folder to move across the entire circle. Folding circles is about touching two points together. By fully acknowledging the limited fold by bringing together the two points and creasing, the precessional results realign the circle to itself establishing balance and symmetry. This demonstrates the mechanics of realignment from any possible amount of out-of-alignment. Each diameter is a direct result of a uniquely individual choice both for folding in half and for less than half folding.

Alinging any chord less than half

Folding from out of balance to a balanced proportion reveals information about the first fold; touching any two points on the circumference will fold the circle in half. Bringing together and touching the furthest points of a limited fold demonstrates realignment to the boundaries of the full circle. Everyone folds a different line of symmetry whether on the first fold or there after. No two people will chose the same two points. We all have similar limitations with differently proportioned combinations. Even with a symmetrical fold the diameter is unique from the infinite possibilities.

The refolded circle on the 2-D plane shows five points of intersection on the circle plane, a reflections of the properties of the unfolded circle. Connecting the four points with a line show a uniquely proportioned kite shape from which there is a tremendous amount of information not visible before the fold. (ref. http://wholemovement.com/blog/itemlist/date/2010/6?catid=140) We miss much because we do not see everything we do. We are taught to look at what we have done, and what others have done, thus miss information in the moment of doing for ourselves.

Any two points on the circumference when joined form four points, two tetrahedra

Here we see the same thing happening when touching any two marked points on the circumference, usually we do not mark points when folding. When this kite is lifted from the flat plane on the diameter/axis dual tetrahedra are formed, a pattern of movement in two opposite directions, each has two open planes and two solid planes. This eventual leads to folding the tetrahedron “solid” (ref. http://wholemovement.com/how-to-fold-circles) and how it is used to generates the other regular polyhedra. When adding the six relationships to the four points we get the number pattern of ten for the tetrahedron, the same we observed in the closest packing of four spheres. The five points on the flat plane is in balance between spherical movement to both sides forming two opposite tetrahedra, one inside out of the other. There are many other connections we did not have time to go into.

When folding the tetrahedron, one student asked if we could make a tetrahedron from the less than half fold. I left that question with them to discover for themselves because of time limition. (http://wholemovement.com/blog/itemlist/date/2010/8?catid=140)

We use the circle in limited ways without understanding the potential and unlimited capacity of unity in circle form. In the same way we have a limited view to our own capacity and potential of what we are becoming. Just as with the first fold in the circle, we are directed through purpose; thinking we can improve on what is already in place, by looking for emotional comfort in the parts, and reacting with fear and distrust, thus missing the value and what is inherently principled. Less than half is not wrong, but it does cause misalignment creating an off balance. In order to progress towards greater stability, we need to know what is principle to achieve alignment enlarging our context and giving purpose to what we do. Even from a relavtively balanced position we miss information in the folding experience to fully advantage ourselves from what we have done, thus the danger of falling further out of alignment. The circle shows the possibility for realignment at any time using any chord less than half.

All people are gifted in different ways and all of us have emotional needs and fears. As our unique gifts are acknowledged and supported by others, our needs are then met in positive ways, our fears are soothed and we can trust more. There are often deep underlying differences when you move towards the outer boundary on both end of the spectrum seeing both inabilities and super-abilities that do not fit what society is willing to accept, especially in public education. Every culture draws narrow boundaries on what is acceptable, making it individually difficult for everybody.

What we demonstrated in a short time in this workshop is the circle is much more than what we have been told it is, and we discovered we are much more than what we are told. Our job is to look beyond what we are told and to observe what is.

### June

Playing with circles I periodically get into folding and curving as a natural expression of what the circle wants to do. Every fold in the circle is a straight edge; with polygons there are no curved edges. Folding circles has both curved and straight edges without having to construct either. In light of our conditioning towards the primacy of separate polygons and polyhedra, they all come from the circle.

Above In curving the straight edges in this tetrahedron arrangement (four circles, six creases each, three diameters and an infolded equilateral triangle) something caught my attention that connected to an old model that come out of earlier explorations of the nine creases of the tetrahedron.

Below are pictures of past models, two views of a tetrahedron arrangement (six points coming together in a hexagon,) a pentagon, and a square joining. They are all made using the same folded unit reformation of the tetrahedron net. Pictured are examples of 4, 5, and 6 point vertexes. The six immediately below is a double unit forming two parallel a single edge where there is usually only one edge changing the pattern. This became my focus last month.

Below Again using my circle business cards I reformed the tetrahedron as above because these units will slid into each other and allow a good degree of angle movement in joining. The variations and joining into sets are shown to the right.

Below Twelve units in six groups of two each showing the edges of the tetrahedron in a truncated form. This would be the 3 of the 4,5,6 joining from above models. Here six edges form a triangle arrangement at joining rather than a single vertex point.

Above is a tangent direction showing another variation of the tetrahedron using a variations in reformation of the units shown above.

Below is an edge model of the icosahedron that shows a truncated form using thirty sets of two units each. Here each of the twelve vertex points are an aggregate of fives circle end point forming the pentagon star with greater complexity.

Below are two models of the same arrangement each using the same number of units. On the left a short-folded unit is used, on the right the long-folded unit is used.

Below the model on the right above (long-folded units) has other units added. The added circles are folded using only one crease, as new growth might appear from the open ends, much as cell growth might generate tendril forms.

More growth changes the look of the icosahedron seed.

Below The two icosahedra have been joined as if the extension of one becomes growth from the other, with adding a variety of tetrahedron reconfigurations generating forms suggesting differentiated functions of interaction. The tetrahedron functions in a similar manner that we find in stem cells, reforming to specific interactions within their environment as relating to the needs of the whole system.

Another view of the same model.

Going back to the models that caught my attention in the first place, I did a single sphere in an icosahedron arrangement only using the double sets (four circles.) This changed the way the end points come together, now making a decagon arrangement rather than pentagon changing the pattern to a truncated dodecahedron.

Below are the units used in the following model. Reforming the tetrahedron to two variations; one short open cell and the other a longer open cell that can be combined in three different combinations, two shorts, two longs, and one short one long. One slides unit into the other and then joined in double sets in parallel allowing for circular joining on the ends. (Bobby pins are holding the double units together.) These are the unit sets used in the following model.

Below the icosahedron expanded to show two expose layers of information revealed by variations in the folded units. Here both are seen, short units on one half, long on the other half.

This is an odd pattern of the rhombicosidodecahedron that incorporates the truncated dodecahedron. It shows decagons, squares and triangles as a function of the double-edged units, making squares and open decagons without the triangular definition of the pentagon.

Below a partial third layer is added to show the dodecahedron edging. From the inner icosahedron grows the dodecahedron through the truncation process while growing outward, combining layers. There are three views.

This relationship of the icosahedron and dodecahedron through the truncations process brings up a question about the generalization in construction where five ends join together forming one vertex. We know that five locations cannot occupy the same place at the same time and therefore it becomes convenient to accommodate scale by saying the points become one rather than an aggregate of points. Truncation is a term that works to describe cutting corners from solid-formed polyhedra, but does not cover what happens with the circle where there is no cutting and there are open units. Truncation then becomes a movement into and stellation is a movement out from. When things get small to the human eye we tend to make a generalization when in fact the sophistication of our tools and concepts about quantum space suggest something different. Reading *Gulliver's Travels *gets me thinking* *about scale as it relates to Euclidean geometry and how limited our real world observations are about the interrelatedness of order and pattern as it relates to scaling of forms through a time and space experience where everything is inherently interrelated, as demonstrated with the circle.

Do not expect the model above is finished, only another beginning in exploring the growth from the icosahedron as a seed pattern found in many natural forms. July is already suggesting design changes in growth.

### May

This is a model from last month's exploration. Four circles in a tetrahedron patten designed to a truncated form.

This month I added another of the same system with some variation in the forming.

In going back to exploring the possibilities in reforming the tetrahedron net the helix form continues to hold my interest as does my amusement in using my business cards.

Three more units were attached to the three solid faces of one side making a partial bi-pyramid. This is the seed unit for systematically adding a helix unit from single circles. They were attached only where there was a congruent face and they did not interfere with others that were added.

**Below** I started exploring further various helix formations using variations of the reformed tetrahedron units used in the above system.

I had to make this from paper plates to look at the difference. When the open space are larger you see it differently making connections otherwise not seen.

...like this one using four circles and each unit of three tetrahedra opened in a different way.

Below is a hinged tetrahedrahelix with 2/3 or the circumference folded out on each unit of two circles joined into a tetrahedron. On the *left* the helix is to an open twisting and on the *right* is reversed into a tighter twisting.

Here is another reformation of the same helix system.

**Below** is another hinged joined helix that has very little opening movement but lots of individual isolated movement with each unit making a jiggling movement rather than twisting.

Another static helix form with the reformed units we started with.

### March Folding Into April

Picking up from last month and exploring further using the folds from last October I went deeper into reconfiguring the individual circles and using combinations I had not use in this direction of exploring open systems. Since the Pythagorean Theorem came up on one of the online math discussion groups I got sidetracked, then with traveling and conferences I had a short month.

Below is an account of my exploration, always looking for more clarity.

Two views of an open tetrahedron pattern but more like a truncated tetrahedron arrangement of four reformed circles. I have many reconfigured circles lying around that are of interest, but not having time or insight at the moment, they wait to be developed, or eventually to be discarded. This one is particularly lovely in its simplicity and economy.

Many of these formal arrangements do not fit the traditional classifications of geometric forms. The folded grid allow so many variations that fall between or outside of the few we have systematically classified

**Below** two views showing a variation using the folding from last October blog exploring the curving aspect of the folds. It has the symmetry of two open intersecting tetrahedra.

**Below** two views of fours circles in a tetrahedron arrangement. The one on the left has four open triangle planes, the one on the right the planes are closed. This system opens and closes as any system might with four, three-sided openings.

When open the form is stellated on all four sides using three elongated tetrahedra on each side.

Face view of the same system.

**Below** left shows a different reconfiguration where four circles are reformed and joined to form four open hexagon planes in the pattern of a truncated tetrahedron. To the right shows the open hexagons filled with another variation of the same folds making it a closed solid figure.

Here the above two systems have been combined. There are four different places and different combinations where one plugs into the other. This shows one of them that looked promising for continued development.

**Below** The above combination is expanded upon by adding elongated tetrahedra and other folded variations holding to a combined symmetry of three intersecting tetrahedra.

In another direction using the same folds in a different treatment of the tetrahedron pattern.

Here is a different variation coming from an octahedron center in a similar arrangement as the tetrahedron-centered model above. The individual units have been changed so it looks different. This shows a combination of two centered-related tetrahedra.**Below** is a variation where units have been added giving possibility for continued growth.

Here is another open tetrahedron arrangement of four circle in a very different reconfiguration from the same folds use above. Below layers are added using the same folded units except they have been reverse folded with inside to outside.

Seed

First layer

There were a number of other directions started but without time and clarity to explore further got put away. Sometimes months, or even years later I will see where something needs to go and then explore it further.

This is a an example of something interesting enough to sit around for a while that maybe I will figure out something about it. At the moment it simply is, and that is enough for now

### Orcas & Bellingham

Last month started with a continuation of the exploration over the last couple of months folding my business card circles. Here are a few things I explored before my attention got diverted towards preparing for workshops coming up on Orcas Island, WA and at the University of WA in Bellingham.

**Below** 3 views of four sets of three circles each arranged with an open center and then the sets joined in a tetrahedron arrangement showing an open centered system. Again these are all folded from reconfiguring the circle to the nine creases that come from folding the tetrahedron. There continues a deeper looking at the extraordinary transformational possibilities of the of the tetrahedron net pattern in the circle, not unlike what is observed with the stem cell.

**Below** Two views of another arrangement of four sets of three circles in tetrahedron arrangement

This open arrangement shows the forming that occurs on the inside

**Above** another view of three joined circles before the fourth was added, as completed above. We can see that the there is a beautiful almost coming together of the center points leaving a small tetrahedral gap when all four sets are joined.

**Below** More variations of four sets of three units where the centers are varied from open to closed.

Four sets of three with closed center arranged in a tetrahedron system where the inner points join at the center leaving the center open.

The same arrangement where the centers of each set have been folded over to form an open triangle center from the end point view.

**Below** I decided to use 9” paper plates to explore some of this in larger scale to see what the differences. Here two views of the same tetrahedron system of four sets of three circles each with an open center. The one on the right appears to have a floating cube which is in fact a plane perspective looking through the open center of one of the units of three circles.

**Below** two views of the same system with four more circles added expanding the form as it might in some biological growth. The appearance of the floating cube remains unchanged.

On Orcas Island I worked with middle school students in the morning and the high school students in the afternoon. There was good discussion with the high school students about the information and concepts that emerge from spherical compression and the first fold. The observable *principles* in that first movement clarify the structural forming of relationships observable in all kinds of disciplines, universal in application and directive to all subsequent folding. It is always gratifying to see students engaged in the math information and philosophical implications as well as seeing their excitement about the process of making things from circles. As often the case, and to my delight, a couple of students folded circles in ways I have not seen before.

Two days later I did a day-long workshop open to the public with people from Orcas and surrounding islands. Again there was also a good response from both kids and adults that also included a few teachers. It was the young people in the group that took the lead and explored beyond what others were still holding close. Again I observed reconfigurations that were new to me. People do not always reform the grid in the same way, they will explore different reconfigurations and systems that interest them. The connections that individuals make to what they are forming from the grid is often quite diverse, as was the case in the workshop in Bellingham a few days later.

There I worked with first year art students at the university. They had already done some folding of tetrahedra before we met. We explored folding the 8-frequency diameter grid. With that they moved into more complex folding and forming systems using multiple circles. As with most workshops, it challenged them to suspend their ideas about learned construction methods and traditional ways of art, to do from unity where nothing is added or taken away. Not so much about designing, rather observation, to see what is generated that stimulate the imagination. I notice art students are more product oriented, they do not work in groups the same way I see in other workshops. They do tend to explore more widely in reconfiguring the grid and experimenting with ways of joining.

While all three workshops were different, they all displayed the same excitement about what they were doing and new concepts they were discovering.