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Over the years I have disregarded the impressed inner circle that gives form to paper plates. This deformation of the circle is not a property of the 3-D circle and is always flattened during creasing. Yet paper plates do come with this circle pressed into them, and concentricity is inherent in circles.
Jose Albers accordion pleating concentric circles has always intrigued me. There are been a few interesting developments coming out of Albers folding exercise. Erik Demaine and Martin Demaine at MIT have pushed it more towards art by cutting the circle, removing the center to give greater movement to the ribbons of folds as they are joined in complex curving systems of equilibrium. It is not uncommon to see paper engineers scoring curved lines and cutting paper moving away from the straight edges of traditional paper folding. Cutting the circle reveals many intriguing directions but the beauty of the spherical hyperbolic reconfiguration curving to itself is lost. It seems Albers point was the unexpected nature of the uncut circle plane by reforming through folding concentric circles that reveals a balanced movement around the center circle. http://wholemovement.com/blog/item/119-in-out-hyberbolic-surface.
In exploring this I went back to the great circle divisions of the spherical vector equilibrium, octahedron, and the icosidodecahedron, wondering how circumference sectors would change using concentric circles.
Below left) Four circles with three folded diameters systematically joined on edges form the spherical vector equilibrium where each circle shares a center point. This forms eight open tetrahedra. The impresses concentric circle shows a smaller scale spherical VE nested within the larger.
Below right) Four circles with three folded diameters plus three more creases to form an inscribed equilateral triangle that when joined has no center. The folded center on left moved to four places on the circumferences forming a single enclosed tetrahedron. Here we see both spherical and polyhedral forms of tetrahedra, reflecting the first fold of the circle in half forming both spherical and tetrahedral patterns of arrangement in the movement.
Below) Using the figure above on right, one folded tetrahedron http://wholemovement.com/how-to-fold-circles is joined to the center of each face of the large tetrahedron forming another tetrahedron of equal size. The circle sectors are flattened to the edges of the added tetrahedra forming vesicas showing the six faces of the cube resulting from two intersecting tetrahedra.
Below) Being consistent with all circumferences folded to the outside now allows for opening the cubic arrangement to spherical form revealing unusual divisions. On the right is a variation where one stellated tetrahedron is folded to a higher frequency grid enable to generate smaller triangles.
This is interesting but not where I want to go.
Starting again with one circle I fold the concentric inner circle that comes with the paper plate.
Above left) Creasing the inner circle shows the hyperbolic nature of concentric circles as the circle is folded and curves into itself. Right angle tension is created in the inner ring; a bobbed pin holds two opposite sides together.
Above right) Two circles have been folded (on left) and opened enough for one to fit into the other forming a tetrahedron arrangement where the outer circle of one joins the inner circle of the other at four points. Were the two circles joined on the outer circumference the two circles would lie flat one on the other.
This again substantiates the primacy of the tetrahedron but yields little formally.
Not to complicate things I decided to add one fold of the circle in half with the one concentric circle.
Above) This is one example of reforming the circle to one folded straight line and one folded circle line. Each half outside of the folded circle is folded in opposite directions from the concentric crease.
Below) Two views of one possibility in joining three of the units above.
Below) Adding more units further compounds the complexity of curving surfaces.Two views of six units joined in an octahedron pattern.
Below) Adding four more of the same units the three axis of the octahedron become more apparent.
Below) Four units attached in a tetrahedron pattern showing different arrangements of the right angle relationship between opposite edges.
Below) Using the model in the last picture above right, two are joined forming a dual tetrahedron pattern. One tetrahedron intersects the other showing six rhombic relationships in a distorted cube arrangement.
Below) Using another reformation of one straight and one inner circle, and joining multiples units reveals a variety of open systems in various polyhedral patterns. Two views of six circles arranged in a tetrahedron relationship bring out the octahedron relationship inherent in the tetrahedron pattern.
Below) Multiple concentric circles have been added increasing the level of complexity; again two views of each.
Below) The difference between the concentric circle folded by Albers and folding a diameter in the circle is the congruent concave and convex as each semicircle fits the other. When diameter fold is opened tension develops along the center line.
Below) On the left are two circles folded in half with four accordion pleats. In these units the circumference has been partially separated by folding them in opposite directions along the outside ring allowing space between them. On the right is a seven ring circle.
Above) Two examples of concentric circles reformed into spirals.
Below) A system of four circles joined showing different views.
Below) Two circles are folded to the same off-centered division to the creases inner circle. Joining the straight edges forms two intersecting circles where the vesica piscis becomes spatial.
The folds inherent to circle pattern are also found in all irregular parts of the circle. http://wholemovement.com/blog/item/130-order-without-boundary
This means concentric circles can be folded at any location on any shape of foldable material and there will always be a unique reconfiguration in relationship between placement and individual boundary configuration of the paper as it conforms to the circle effect on the plane surface.
Below) An example of concentric folding using an irregular rectangular piece of paper showing both sides standing in different positions.
I continue to delight in the exercise Jose Albers gave his students by exploring the beautiful forms that are produced in concentric circle unity, and the possibilities to combine with other ways in reforming circles.
To change a paradigm is to change one's frame of beliefs about the world to another. Is it possible in a world of infinite parts to think about an inclusive Whole? Is this even desirable in a world seemingly bounded by infinite boundaries?
The circle is fundamental in math as a 2-D image concept. Can we believe the circle is more than this image we draw, more than a defined circumference? Can we equally accept the circle as a 3-D form that comprehensively demonstrates through folding the idea of spherical unity that is inclusively whole?
Traditionally the idea of the circle is demonstrated by cutting a sphere in half. By compressing a sphere a 3-D circle is generated without destroying the wholeness of the only form that represents absolute unity. When the sphere is compressed the volume of the circle remains equal to the sphere. The surface boundary of the sphere changes properties. The sphere/circle is whole; nothing is taken away or added, it is transformed through movement from a spherical form to a circular form through compression. The circle/sphere functions as both Whole and discrete part simultaneously.
Euclidean construction of a circle is based on the definition of a point. Using a compass to draw a line that is considered as a set of points on an imaginary plane at a given distance from a single point, we draw and thus define a circle, calling the center point 'origin'. The small circle point contains even smaller equally concentric circles in the same way circles get progressively larger through opening the compass. There is no fixed outside or inside boundary, only unseen circles in alignment. The radius measures the openness of a compass used to draw a circle. It does not fully measure a circle, although we generalize and use it that way in the abstract.
The property of the image shows one continuous curved line defining a given area; we imagin individualized points. The properties of the 3-D circle show five distinct circles with volume. Think of a disc, such as a coin. Circles are dynamic, they move, can be moved; and through a consistent and systematic sequence of folds will self generate proportional relationships that are not possible with other shapes, yet contain all shapes.
If there is uncertainty about the difference between a 2-D and a 3-D circle then do the following:
Draw a circle, use a compass or trace around a circular object.
Cut the circle from the paper you drew it on.
Observe the difference between the image you drew, the hole that is left in the paper, and the circle in your hand.
Each of these must be understood for what they are in able to understand the interdependent nature of one to the other. There is nothing to suggest throwing out traditional understanding about the circle; on the contrary we are moved to enlarge what we believe by embracing the full nature of the object the picture represents.
This shift of perspective can be understood by looking at the word ‘geometry.’ Geometry means earth measure, measuring things of the earth. Geo refers to the earth. The earth is spherical and the sphere is whole. Metry means measure, keeping track through movement in space. We can now understand geometry comprehensively as wholemovement; a self-referencing system of the whole. This better describes, giving demonstration of our presently evolving worldview in a universe among many where much of what we see and know now was a short time ago unimagined. We can no longer afford to hold a geocentric view about ourselves, anymore than we can sustain the belief that the sun and celestial bodies revolve around the earth. We are a small part of something much larger and far more complex that yet imagined.
This suggest maybe it is time to consider shifting from a commonly held parts-to-whole thinking to a Whole–to-parts perspective. To start with the Whole in the form of the circle/sphere and observe information as revealed through movement, the order and proportional arrangements, the appropriateness of interactive systems, is perhaps worth considering. Through observation of what is generated from the circle, through folding and joining reveals what is not possible using other shapes or forms. The circle is its own center; an alignment of inner and outer boundaries. The whole is origin to all seen and unseen parts, realized and unrealized, revealing ongoing potential. The origin, both center and outer boundary, is already whole and cannot be constructed or deconstructed.
Broadening our perspective does not deny any mathematical value. What temporary benefits progress will eventually drop away being replaced by greater knowledge that elevates value towards a greater realization of human potential as we begin to see the finer reality of further abstraction. We need a more inclusive and comprehensive way of thinking about our universe, our place in it, and how we negotiate the inequalities and fragmentation, the separation that are no longer sustainable given our present understanding of the interrelated and interdependent nature of parts and systems. There is a fear of the unknown that isolates and keeps us from reaching out on a cosmic scale. Fear keeps us from reaching outside of the circle we have draw around ourselves.
Comparing properties of the image and the circle/sphere compression shows differences between 2-D & 3-D. The 3-D circle shows five discernable circles; three circle planes and two circle edge lines that contain a volumetric location. It all starts with a spherical point. Tradition shows two points making a relationship formed by connecting with a line. Three non linear points form three lines forming the edges of a triangle area. Similar components in both 2 & 3-D are points, lines, and planes. Folding the circle by touching any two points will generate two more points at the end of the crease, an axial line of division perpendicular to the distance between points that shows six relationships. These four points can be form into six edges that define four solid triangle planes. There are two solid planes and two open planes defined by five edges you can see, one edge you cannot see. Minimum description of a tetrahedron is for points in space. Edge relationships and planes are inherent in this spatial movement of two points of the circle. The first fold of the circle around the creased line/diameter/axis, in both directions forms two tetrahedra, one the inverse of the other at the same time showing a spherical pattern of movement as origin.
For those that like pictures with their words, I have included some photos of an old folded circle piece that bares relationship to this discussion. These pictures, “The Transient of Venus” show different views of an idea about Venus as it moves in a slow spiral around the sun observed from earth where for a short time appearing as a dark spot moving across the sun. This object is made using seven paper plate circles, four are folded to an open tetrahedron and joined with string holding them together, and the fifth is left unfolded, the other two form Venus and its path. It is painted conforming to the folded equilateral triangular grid matrix, indicating the activity between the surface and interior of the sun as Venus passes giving a temporary linear alignment of three spheres in space.
This form becomes an expression of multiple parts and by adding string, the black square box, and paint gives design to the discrete parts of the folded matrix. Here we have made in “parts” a multiple expression of the whole. Parts and whole are folded one into the other.
The circle/sphere is the only form that demonstrates, with some degree, the idea of a comprehensive Whole. In that regard no other form can be observed that is principle to all symmetries, and all subsequent generations of parts. What is principle is what comes first, not what is believed at the time to be most important. The whole, even defined as “nothing,” for lack of boundary, binds all potential to the principles of first movement. Does our compass open wide enough and will it close small enough to construct all circles contained in the one that can not be drawn?