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The short answer is where everything else comes from.
For a longer explanation we have to start with the concept of origin. The most comprehensive, expansive/compressive concept possible must be in the form of a sphere; it is undifferentiated and concentric without scale. The sphere is unity, it is whole without qualification, without inner or outer boundary, in all direction. Unity contains units, where each unit inherently contains unity.
Two arbitrary diametrically opposite point locations on a unit sphere of any size, when compressed towards each other creates tension inward and movement outward at 90 degrees to the direction of movement retaining spherical volume. A sphere to circle transformation reveals differentiation without adding or taking anything away. This can be demonstrated by rolling a ball of clay and squishing it flat using your hand.
In this spherical transformation there appear three congruent circle planes; top, bottom, and the circle ring connecting the two. When considering individual parts there are two edges connecting the three planes making five individual congruent circle parts. There are seven properties to the disc when considering the internal volume and the space surrounding the circle unit. The concept of a unified whole suggest no surrounding space, no taking apart or separation, only expanding potential into and out from any location. Nothing is added or removed through compression, only a flattening spherical displacement showing form differentiation.
The sphere to compressed disc from which an image is then drawn loses the dynamics of spherical force to the 2-D symbol, a static stand-in to which we assign meaning. Images that appear on paper or screens are always a representation of something else. A holographic image projected into space does not embody energy uniqueness, the dynamics resides with the original. A reproduction will never be the original no matter what means of sophisticated technologies are used.
Circle/sphere origin is pattern to countless multiple copies from which are formed expressions of all kinds; it is a process of divisional revelation. The circle disc is a dynamic self-referencing system of spherical unity with infinite symmetry. Numbers have not appeared, only multiplicity of structural pattern in-formed with differences.
Working backward from the image towards spherical origin gives another perspective. Draw a circle and cut it away from the paper making a 3-D circle disc. It becomes dynamic that can be rolled and spun revealing countless rotational axis. Curving the circle around and touching any two points on the circumference will form a curved open conical plane. When the two points touching are diametrically opposite each other the curved plane becomes an open cylinder; a special cases cone.
Regardless of which two points are used when touching, crease the circle flat. Alignment occurs dividing the circles equally into halves. Rotating on the crease in both directions reveals a spherical pattern of movement reflecting origin. Any fold less than half (two points on circumference not touching) will lack alignment to origin.
Alignment shows duality in unity of three (1+2=3.) These numbered parts do no happen separate from the circle. The crease is a line of symmetry, a diameter, an axis of rotation; a multifunctional straight line with two end points on the circumference; a tri-unity system of three individually identifiable parts, four points, six relationships revealing a tetrahedron pattern of movement, and much more if you include all five circles. This all happens simultaneously. Numbers are used to spatially separate partial events.
This folded straight line crease is the symbol of our first number (1). Number one is the first mark of self-referencing movement of unity. From one line of symmetry we surmise an infinite number of symmetries and countless relationships in combinations and arrangements. All possible combinations of number functions are coded into the relationship of one straight line and one circle (the first of an infinite number of folds.)
Without unity of the circle the idea of one isolated from context is non existent. Yet separation is used to develop abstract relationships by adding, then subtracting, multiplying, and dividing to get a larger and smaller segmented number scale. We teach a unitized system of separated numbers in sequential ordering from simple to complex. We have assumed this is how it is.
Addition, subtraction, multiplication and division carries the assumption of building from nothing, zero (O) to one unit towards developing complex system of calculated functions using multiple units. We accept units coming from nothing with little understanding about how and why or there being no logic.
In retracing the process from sphere to circle then first fold we see an opposite sequence of functions to what we are taught. First is circle/sphere unity, wholeness through compression shows origin and with first movement is a folded division that generates a multiplication of individualized units that can be subtracted from and added to each other. Unity has not been destroyed, there is no separation.
Division, multiplication, subtraction and addition are observed when starting with unity where a process of ongoing interrelated complexity is revealed all within the circle. Through observation and thinking about what we see about what we do when folding the circle, numbers are then symbols signifying inherent differences that are acted upon in relationship to each other.
Nothing exist in separation; only in context. One is an abstraction. Two in separation does not exist. Only in context does relationship of two ones become three. Starting with unity the one line of division is not separate from the act of dividing in two parts. In context there is no separate between one and two, they are three; structural pattern. Traditionally we abstract one and two from three using the circle as a symbol for nothing from which numbers mysteriously emerge.
There is no conflict; one illustrates the abstraction of 2-D and the other reflects 3 dimensional reality. Both systems are correct. One is part of the other. Both taught early at the same time would advance understanding of geometry and mathematics with benefit to other disciplines. By introducing unity into a unit based system would expand thinking towards a more comprehensive way of understanding the nature of where we are, who we are and maybe get closer to why we are.
Looking at a larger picture is simply a bigger unit. The syntheses of many units of all sizes is not unity. There is no summation that gains the Whole. When you start with the whole it is all there, mostly unrealized and through a principled process of generation potential is endlessly revealed within unity.
There is no ultimate boundary, no conditions of confinement to circle unity. The circle demonstrates, order, structural relationship, and principles that are foundational to all subsequent evolving of information. This does not happen with polygons or polyhedra, no amount of numbers can account for it.
We can say we invented numbers, or maybe discovered them, possibly they are inherent in how the mind keeps track, or it is an abstraction of what occurs before number recognition later constructed as language, maybe numbers were given through higher insight, or by creatures from off of this planet coming from other worlds. Nobody now was around for the first couple of numbers so it is pretty much guess work about what it is that fits the accepted version of the number story. It is demonstrated that numbers have a context. How do we substantiate they came from nothing?
Where do your numbers come from?
Inherent in the circle are structural forms and proportional relationships of many design possibilities for a wide diversity of products for the human landscape along with the obvious connections to a variety of geometric and biological forms. Often people come up to me and suggest I should make this or that idea that comes to mind from them looking at models of forms and systems generated by folded circles. My response is always “you thought of it so it is yours to do.” I do not develop specific product, that would detract from my job exploring the circle, a transformational process showing what is there that we do not know, to leave some record of the comprehensive nature of folding circles, the only shape that can inclusively demonstrates unity. The development of product I leave to others to explore.
A week ago Wyman Williams sent me a video of a wind generator he made from two tetrahedra. He calls it a TetraGen, a concept he developed from having attended a folding circle workshop about eight years ago and buying a book about the folding. I want to share his work so you can see folding circles does not have to be fancy or complicated to reveal potential for a practical, efficient, and economical solution to a serious global problem. He is developing a wind-powered generator simply and cheaply.
I will let Wyman tell you his own story:
“One night in late 2012 is was using your book (Folding Circle Tetrahedra) as a guide to exploring folding the tetrahedron with circumference outside. It occurred to me that if the resulting material outside the tetrahedron were attached to three faces and stacked on another tetrahedron with the material arranged to catch wind like the one above, it should rotate around a shaft through to center. I folded and taped the tetrahedra and attached them to a shaft made from a coat hanger. It worked quite well in the wind.
I then built a more durable model out of corrugated sign board and aligned it in the same arrangement as the paper plate model. I attached it to a threaded rod and inserted it into two lawn mower wheels attached to the top and bottom of the base. It worked, but took a much higher wind speed to turn it than I had anticipated.
I bought a toy propeller wind generator kit and attached my improved arrangement of the tetrahedra made from poster paper instead of the propeller. Instead of matching the faces of the stacked tetrahedra, I splayed them into a hexagonal arrangement. You can see in the video it really works well.
I then restructured my outdoor model in the hexagonal arrangement. It now begins spinning at very low wind speed. This structure has been spinning outside for over five months and is still in good shape”. - Wyman Williams -
This generator design could not be stronger, lighter, or more economical due to the structural nature of the tetrahedron by using the entire circle. Without the circumference the tetrahedron would remain a regular solid polyhedron as traditionally defined. The circle reveals the tetrahedron to be a functional relationship of the first fold of the circle in half; four points moving in space showing six relationships in a dynamic, structural pattern that is principle to all subsequent reforming and joining circles. The dual nature of that first tetrahedron is beautifully shown as Wyman forms two tetrahedra, relocating them around one of the seven major axis, finding advantage for the circumference folded to the outside.
Aside from the traditional geometry and mathematics generated in folding and joining circles there is a practical side to the process, as we have just seen. 3-D printing revealing new uses for reproduction and prototyping complexities, with an ever enlarging choice of materials, applications and design possibilities. It is important to retain a hands-on designing experience of the imagination that is not tied to computer design tools and programs, but that is structural and self-organizing by nature. The circle is purely transformational since nothing is added or taken away, as we do with all other designed forms of production. Folding requires the observation, imagination of the human mind, courage to explore, and the spirit of discovering unknowns that come from direct hands-on doing. With some of the extraordinary complex forms and beautifully individual expressions coming from traditional origami, there is nothing that will reveal what can be folded from within the circle. Only the circle has a circumference, which can be diminished to any number of sided polygon and folded to any polyhedral form. All reformations comes from the same folded grid matrix making it a truly unique transformational process where all folds come from a single source first movement. There are three possibilities of consistent folding in half, the ratio 1:2 that generates only three option of proportional grids; 3:6, 4:8 and 5:10 (these are discussed in previous blogs.)
There is much to be discovered by folding and joining circles that cannot be anticipated by drawing pictures or constructing preconceived ideas based on traditional adding parts or of taking away excess. I hope this makes the point that there is tremendous design potential in what is generated from folding circles that is not accessible using traditional ways of designing and modeling. While products that are derived from folding circles are an important part, they are not to take precedence over the experience of working with and observing the inner-connections between all parts in an inclusive and principled transformational process. The circle is the only form to demonstrate division generating a multiplicity of endlessly diverse units and systems starting with unity, where parts can then be added and subtracted.
Wyman has discovered the circle to be instrumental in stimulating new approaches to old ideas. I am interested to see what other people have developed by folding circles and what kinds of products might emerge. If you have pursued your own direction in folding circles I would love to hear from you and add you to a list of people exploring folding circles in directions I have not taken.