Thursday, 01 November 2012 02:36

Polygons to Circles

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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.

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

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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.


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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.)

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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.

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


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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.

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

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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.


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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.

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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.

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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.

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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


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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.

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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.

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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.

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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.

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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.

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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.


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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.


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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.







Friday, 12 October 2012 12:41

Importance of Circles

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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.

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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.


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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.

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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.)

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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.

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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.

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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.

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

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

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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.)

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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.

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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.

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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.

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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.