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This continues my exploration into three diameters in the circle; three lines the same length equally spaced as they bisect each other. This goes back to a blog two months ago, Circles From Scrap, starting with the Off-Centered Circle series of five blogs from August through Decembers in 2010.
Let's start with a quick look at an example of transforming a geometry model to a more realized expression of specific biological design. These transformations bring to my mind questions about how we separate art and math by isolating one from another.
Above) is the geometry of folding and curving seventeen circles modeling a spiral form held together with bobby pins. It is pretty straight forward in organizing reconfigured circles. I like using bobby pins, tape and other means of holding circles together as part of the forming process, easily removable, and they become informational about the nature of what is being joined. There is no need to hid what we think gets in the way of a “clean” geometric presentation.
Below) the same model developed from liking the bobby pins and adding more as they began to take on a hair like quality. It seemed like an appropriate development of the material already being used.
Below) other views of what is now indication of a possible life form that may have inhabited the shell like spiral model, were that what it was, or might be. Without underlying geometry there is no art, yet art envisions geometry.
Interdependency of parts gives meaning and value associated with the organization of forms in a process within a context of change. The moment of maximum value is maybe when no more bobby pins can be added, no more glue applied.
Above) another paper plate model that has gone through similar metamorphosis using bobby pins and clear tubing explores a specific growth formed to an imagined environment.
Above) again another model of three circles using bobby pins and twisty ties to complete an idea for biologically dressing of geometry. The mind clarifies the geometry and the heart gives expression to the art of it.
Below) four circle 6” diameter can lids folded three times and joined forms a Vector Equilibrium using clips and half covered with a black rubber coating. Sometimes I use other materials to finding the harmony that connects them giving form to pattern in specific and unique designs.
The three diameters do not need the circle boundary but will always reference the circle pattern of point relationships. Reference to blogs;
Three lines bisecting each other, equal in length and angles is the structural nature of the circle. It is principle to alignment of all systematic organization of subsequent information revealed directly through folding, indirectly in 2-D methods of construction. Concentricity in reality is the self-centering without measure or construction, eliminating all scale reference, even to the point we call “center.
Euclid stated, there are not parts to the point. From that I conclude the point-circle/sphere is total unity, completely, infinitely self- referencing. Symmetry is first expressed by unity to itself, leaving a line of movement to a self-reflective three parts relationship. Unity is reflected in multiply closest packing of spheres with consistency measured in relationship between any two spheres. All geometry resides in unity. Euclid further defines a line by points with no parts, then the circle by the lines defined by points with no parts. Unity of no parts is reflected in the largest and smallest of parts. No number of parts can show unity; is then endlessly reflected in part of the whole contained. It is through this inherent organization of that which has no parts that there is any organization at all. Value is in relationship, meaning is what we bring it.
Above) four 8” diameter circles folded in half, then into thirds forming three axial divisions of rotation; the same creased pattern as the tin lids. Each circle is reconfigured to two tetrahedra with the four circles joined together on the creased lines. http://wholemovement.com/blog/item/92-center-off-center
Above) four 8 x 8”squares folded and joined in the pattern of the circles above. Alignment is only at the point of intersection. The form has no coherent perimeter yet the pattern of relationship is the same.
Above) comparing the same expanded folded grid in the circle, square, and rectangle. The grid position to the perimeter in each is very different. Only with the circle is there complete alignment and comprehensive self-organizing.
Above) folding a hexagon star using the six creases of the 8 x11 ½” rectangle has visual interest, but is limited.
By exploring other possibilities in combining shapes of different kinds folded to the same pattern there is much greater possibilities for variations that are visual interesting.
Below) a range of shapes will be used; rectangles, squares, torn circles to approximation, hexagons, torn star-like configurations, all from the same folding with three equally spaced and bisected creases.
Above) combining two squares and two torn circles with bobby pins holding them together.
Below) examples of combinations of the above configurations forming a few possibilities.
Below) the center point of intersection have been torn out, some holes punch through for surface interest. The variations show a continual decreasing the planes by tearing parts away, increasing the length of the perimeter of each circle. The form continually changes to an unchanging pattern.
The fractal quality is arbitrary to both the inside and outside; the edges of attachment remain unbroken. The structural pattern is the triangulated relationships of the six axis holding this object together. If the bobby pins joining them were removed the model would collapse. As long as part of the planes connect the axis, it will stand. The three diameters seem to function well with any shape of paper as long as there is some connection between the six axis, thus suggesting endless design possibilities in the ways of forming modifications without disrupting the structural pattern.
The pattern is the structural nature of three and the form is the circle/sphere coordination showing a 3-4-6 symmetry. The design is what we do with modifying parts (above) without changing the underlying symmetry.
Above Top) two views of the Vector Equilibrium sphere with the creased grid matrix of triangles and squares from six folds each in progression of the 1:2 ratio of the first fold. There are two sets of three diameters, each set functions differently. (The creases have been traced dark to see them clearly.)
Above Bottom) 4 circles are joined on the star points rather than the bisecting diameters, as above, showing hexagons and octagons. The spherical form has not changed but the grid pattern shows different designs with two different ways to attach the circles that are unobserved using only 3 creases.
Above) each half of the sphere is the same grid combining the two different forms of grid expression. This breaks the symmetry on a level not seen in the form without the folded grid.
This folded grid indicates the pattern of structural relationships and stability that is observed when removing parts of the plane surface above.
Above Left) once the circle is folded into the triangular sector, then fold each point to the others (folding in half in three directions) will locate a midpoint in each of the 6 sectors resulting in the grid matrix seen in the spheres above. There is more about this folding at: http://wholemovement.com/blog/item/121-circles-from-scrap.
Above Right) when the circle is reconfigured to a quarter sector both grid functions appear sharing a 30 degree sector.
Above Left) sectors cut from circle. Cutting destroys unity of the circle leaving it to function as parts; each part showing a center point of unity.
Above Right) four parts rearranged forming a second net for a tetrahedron.
Cutting the sectors from the circle makes it is easier to fold them individually rather than with many layers of folding but we are now working with unit parts and not with unity of the circle. But then other things become apparent. There are three ways to reconfigure using the three creases to forming dual tetrahedra, two are right and left handed relationships to the curved edge.
Below) here are four sections folded to one of the three possibilities showing two open tetrahedra each.
Above) two sets of two tetrahedra are formed by connecting them in pairs on similar edges. There are four different ways these sets can be joined.
Above Left) two sets are joined in the same manner as the spherical Vector Equilibrium, open triangles and squares. Given the number of choices in building sets, this is only one of many possible arrangements.
Above Right) all open tetrahedral and octahedral spaces are regular. Four regular white tetrahedra fit into the tetrahedron openings. Two are to the longest radial measure and two are to the shortest measure forming the inner and outer regular VE boundary of the irregular yellow.
Okay, that’s enough yellow circles.
My brother Ernie passed on this month; not anticipated but expected. Seeing him live through fifteen years with Amyotrophic Lateral Sclerosis (ALS) and his support in my exploration in folding circles brought us into a friendship we did not share as kids or in growing up.
He was a gregarious sort that took energy from many people only to increase and enhance in giving back. He used those years of a slow progressing disease experience to help many people with ALS whose lives would be much shorter. He enjoyed giving to all kinds of people in diverse ways. It took most of our adult lives to begin to appreciate and love each other for our individual differences and life choices.
Over those fifteen years I would stay with him while going to math conferences and doing workshops in the area. We did conferences together. He would move through the aisles of the exhibitor’s hall in his wheelchair asking attendees, “Have you folded your tetrahedron today?” He would playfully engage them to fold as he instructed, showing them with not fully controllable fingers. Having been with Ernie people would excitedly come up to the booth holding their folded tetrahedron to see what else they can do and buy a book about folding circles. In watching Ernie, thinking his good old boy approach was cornball, yet it worked to get people interested and coming to the booth. Chalk one up for Ernie. We sold a fair number of books over the years while he sold people on folding circle and enjoying life. This worked because he was a genuine and generous man that approached people with an open heart; enjoying people every bit as much as I enjoy exploring folding circles. People saw how he delighted in showing them, how easy it was even with large hands that year-by-year became less responsive to his control. He impressed them; a big man in an equally large motorized wheel chair with attached table loaded with paper plates, folding sticks, and tape, engaging them as they passed by. He openly, honestly, directly approached people for what he could give to them, receiving much in return. He loved doing this even as it strained him, pushing him against the goodness of his decision to live a diseased life fully; and we enjoyed our time together.
Sitting around the house playing with new ideas about the circle he would sometimes ask if I had thought of doing what I was from the other direction, or maybe to count differently. I would explain to my students to fold every other point to the center and crease. This can be confusing for some people. He says fold point 1,3,5 to the center and crease. Clear to everyone. From the start he put the process of choice into the hands of the folders. I now often use this in instructing students; everybody understands.
Something he said revealed another way to look at folding the circle in less-than-half that opened deeper levels of exploration. We folded the thick paper circle underneath a take out pizza dinner; we began to eat lots of pizza dinners. Some of his ideas sounded obvious at first but I learned to listen and value them. In his own way he appreciated the depth of my interest in folding circles and at the same time I developed a great respect for his courage, practicality, openness, joy in living, his humor, and the choice he made to live his life fully even as the ALS was slowly closing down his body. We were both proud to be each other’s brother. I carry him with me and I with him where he goes.
Ernie was a man with many activities and interest. The disease did not lessen the value of his life or limit giving to others as he continued cheerfully to serve in various capacities through times when it was difficult for him to care for himself. His self-sufficiency was grounded in his relationships with others. After his passing many people came together sharing personal stories enlarging everyone’s understanding of this life well spent. This helped fill in those years we were unable to find a friendship that later found us.
Ernie was a living demonstration of the value of relationships, not in an abstracted, generalized, or conceptual way as I observe in folding circles, but in the everyday encounters he had with people. Whether between points, lines, areas as they shift and change in the circle or the living relationships expressed with many people that come into and out of our lives; the same unity, principles, order and consistency of individual potential is revealed and realized in the human spirit. Ernie showed me a living geometry in his choice to live life as best he could with his capacity for love and service, despite his disabling body.
Below are a few pictures of fold circles done with Ernie.
Four 14” pizza circles
Ernie liked transformational torus rings and movement systems using paper plates.
A few objects around his house.
Four 25” diameter circles upped the scale.
We made soccer balls from party plates
and a number of off-center foldings.
An 8-frequency collapsible tetrahedron
and a spherical cubeoctahedron column served as conference displays.