Sunday, 28 December 2014 20:40

## Folding Concentric Circles

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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.