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Wednesday, 20 April 2011 18:40

Transforming Systems

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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;




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;;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.  


ts-8  ts-9

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:!/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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