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Last month while playing with a few left over circle-folded reconfigurations, then picking up a discarded variation of the icosahedron (all open planes) I could see where together they might make an interesting object. For the delight of seeing what it will look like, the satisfaction of actually making it, and the joy of discovering what it will reveal, I spent some time folding more units redesigning them to fit this particular form of the icosahedron pattern.

This is the result of how those circles came together and some thoughts about the process.

There are fifty-one circles; twenty 9” paper plate circles and thirty-one 6” circle filters. They have been all creased to the same folded matrix and reconfigured differently as they are joined to form this patterned arrangement. This object has been coated with glue size which makes holding it an experience different than what you expect; very rigid and smoother than it looks.

This is not as regular as we would expect of the icosahedron. One vertex is open giving polarity to the system. The other vertexes are open relationships between each triangle that have been closed in. Each triangle face is uniquely different. Where the triangles join are open locations of local centers for twelve pentagons. The edge channels defining the dodecahedron change in relationship to which triangle they appear and how they were reformed. The primary points of connection are the intersections of triangle and pentagon edges. There is consistency to the icosahedron pattern with subtle differences in design. Each circle unit has it own unique characteristic difference, much like in real life.

This object reveals the conditional pushes and pulls of the folding process in its forming, much like we would see in nature as individual systems grow to fulfill specific environments. The richness of the surface is in the irregularity of parts adhering to design criteria towards giving form to the pattern. After choosing the icosahedron pattern a series of design decisions followed where each unit is predicated on the developing organization and relationships already in place. The unit circle follows circle unity.

Each circle is reconfigured to a 3-6 symmetry, and collectively joined to reflect the 5-10 symmetry of the icosahedron. Each circle is a uniquely different aspect of unity. Every small decision in folding was circumscribed by previous actions. There is nothing arbitrary, and yet it has none of the regularity and sameness of formulation that is so often seen in generic geometric models.

Having finished the above model I started playing with the icosahedron as an open form (16 solid triangles with 4 open planes.) Options are not possible with the traditional icosahedron net, since this net is structurally principled it opens endless design possibilities. I wanted to keep to the same process going but in a different direction. By using the same folded units in the above model, with reforming variations, they revealed different optional fits to the configuration of the open icosahedron. The option taken shows a tetrahedron arrangement extending beyond the icosahedron in a more open form. This combines both 3-6 and 5-10 symmetry, proportionally balanced in a way not obvious in the above model.

Two views of this model using 8 paper plate circles ; 4 open tetrahedra form the inner icosahedron and 4 circles form the individual tetrahedra vertex locations.

It is extraordinary to see over years of folding circles how many varied reconfiguration can come from reforming the same 3-6 folded triangle grid. When everything is folded from the same three-diameter grid everything is interrelated and inter-transformable in ways that are unique to folding circles. This means any configuration can be flattened to the circle and reformed into any of number of other units, recombined and joined into a variety of different symmetries and systems without adding any new creases. Once the grid matrix is folded into the circle there are an infinite number of unique possibilities for reforming and joining them. All this is possible because it is in the circle to begin with; all is revealed through keeping an eye to alignment in folding and reforming. No tool in the design world comes even close to the possibilities that come from folding circles.

Two view above is another exploration using the open icosahedron form reconfigured from only 4 paper plate, folded to the same 3-6 grid. The circles are reformed so the four vertex locations of the tetrahedron extend into forming a centralized inside open icosahedron, revealing that the two symmetries are combined in the single tetrahedron/icosahedron pattern. The open icosahedron form is a variation of four open tetrahedra. This reflects back to the tetrahedron as primary structural pattern. The four remaining regular polyhedra are patterned formations in different symmetries of the tetrahedron opened and joined in multiples. This model can function as a unit in a variety of larger systems through small design variations of form changes.

We have gone from using 51 to 8 and now 4 circles. These models went through the same process, all folded to the same pattern, revealing the same symmetries arranged to different forms where each individual system requires a given number of circles to fulfill a uniquely designed expression.

These models can not be made using traditional methods without a preconception of design and a plan to instruct the assembly of each part which would be extremely labor intensive, time consuming, with a lot of frustration and to little purpose. These were revealed in process by following what developed from the specific forming of pattern down to individual designing of elements, as revealed, each in turn, giving expression to that pattern. This can only happen with folding circles. What is in the circle is there for anyone that will take the time to find out.

Folding the circle in half is a transformation. The entire folding process is transformational. Without adding or taking anything away the form of the circle changes without changing the nature of the circle. Creases are the result of the self-referenced and self-organizing sequential folding, but the circle does not move by itself. You must be an active participant.

That first fold is a right angle movement forming a perpendicular chord half way between any two points on the circumference. This is the pattern for all subsequent movement because it happens first. In the following months we will look at various transforming systems by reconfiguring the circle in different ways and joining in multiples to this right angle pattern.

**Open Torus Ring**

You will need four paper plate circles, four bobby pins and some 3/4" masking tape. Folding the circle in half, then folding three diameters, reconfiguring and hinge joining all four circles together into a circle makes a torus ring. Eight tetrahedra are formed and joined at right angles to each other allowing the ring to move rotationally through the open center.

See the following site for instructions; http://www.wholemovement.com/index.php?option=com_content&view=article&id=51&Itemid=43

Above) After folding it in half and then thirds, open it to the circle and see three diameters.

Below) Refold it to the cone shape and fold the top curved edge on one side over between the two end points and crease. Turn it over and do the same thing folding over the top curved flap on the opposite side. The four curved edges between the two folded over ends remain unfolded.

Above) The open circle shows two opposite sectors having straight edges. Refold the curved flaps to the inside (as shown) in the opposite direction of the original fold.

Below) Fold the diameter, the one parallel to and between the straight edges, to itself and use a bobby pin to hold it. This forms two open tetrahedra joined by a common edge. The length of the joined diameter, two radii become a singe edge of joining, is at right angle to the two straight folded over edges.

Do the same folding and joining diameter using the other three circles.

Attach two of the above reconfigured circles together taping with a hinge joint.

To make a **hinge joint** bring two circle units together attaching on the straight edges with flaps folded over. Rotate one unit to one side where the adjoining surfaces are touching. Tape along the joined edges. Fold all the way to the other side keeping edges together and tape along the opposite side of the edges. This way both sides of edges are taped together making a strong connection with maximum rotational movement between the two units.

Below) Join the other two circles in the same way making two sets of two.

Join the two sets of two in the same way as before by making a hinge joint on each end. Bring straight edges together and taping on both sides of adjoining edges. You have to roll the ring to tape the last pair of joining edges.

To make a solid tetrahedron torus ring, the movement pattern is the same, you will need eight circles, each folded into a regular tetrahedron.** The instructions how to do this are on my site; http://www.wholemovement.com/index.php?option=com_content&;view=article&id=51&Itemid=43**

Put the eight tetrahedra together in a circle, edge to opposite edge using hinge joining. You now have a version of the open torus ring made with closed tetrahedra forms.**Elongated **** torus ring**

Folowing are another and simple way to make a differently proportioned torus ring.

Fold three diameters. Then fold each end point of the three diameters to the opposite end point and crease. This generates three more diameters making six equally spaced diameters dividing the circle into twelve equal sectors.

Below) Using the creases fold the circle in half and then into quarters. You will see the folded quarter circle is divided into three equal sections.

Bring the two edges together and tape along the edge forming an elongated tetrahedron with a equilateral triangle open end.

Above) Fold and tape another tetrahedra the same way. Bring the two tetrahedra together as shown and hinge tape the taped edges together (on both sides) with the short open ends of the tetrahedra in opposite directions. Make sure to fully rotate the units as you tape on each side. this will give you the greatest movement.

Fold and tape all eight circles the same way, making four sets of two tetrahedra each.

The two open ends of each set of two will be hinged on the curved edge opposite the hinge joining on each set (the length of the shorter taped edges will be at right angle to the length of the longer taped edges.) Even thought the connecting edges are slightly curved and not straight, they can be taped and it will be strong with tape on both sides. This makes a right angle pattern of movement between the two sets.

When tapping the hinge of the two adjoining tetrahedra make sure the surfaces are face to face.

Rotate hinge to the opposite open face to open face and tape on the other side for greater strength.

Make two sets of four each. Join the two sets of four together using hinge joining on both ends completing the torus ring circle.

Here you will find videos of a variety of torus rings: http://www.facebook.com/wholemovement#!/wholemovement?sk=app_2392950137 One is the one you have just made and there are a number of others that might be a challenge using more complex tetrahedral units.

Explore and enjoy the movement.

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