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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.
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.