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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; 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.
It is important to understand the origin and the properties of what we are working with. In this case the 3-D circle needs to be differentiated from the circle image we draw. The circle, the subject of the image, has spatial properties that are unique from all other 3-D forms. We need to observed the differences in properties between 2-D and 3-D if there is to be any clarity and understanding about each.
If we do not know the properties of what we are working with we do not know what it is or what to do with it beyond arbitrarily imposing our will that frequently ends up violating the nature of what it is, often being counter to expected results. Lack of understanding properties has proven over time to cause unforeseen problems. Properties of the circle set the foundation for all subsequent folding (see previous blogs).
This picture shows a circle and the image of a circle.
The circle originates through spherical compression. Both circle and sphere demonstrate a dual function as individual unit and unity simultaneously. The non-differentiation of spherical surface is transformed by a right angle movement to the direction of an centrifugally expanding circle that reveals a triunity of three circles
The image shows one circle where as the 3-D circle shows three circles, one on each side and a circle ring. (Think of an extremely flattened cylinder.) There are two edges where the three planes meet. There is an inside volume and an outside space. Three planes, two edges, two spaces; (3+2+2=7.)
The circle is a triunity of three interdependent circles that can not be separated one without the other. In order to conceptually take them apart unity is destroyed, being left with three abstracted, isolated, and imaginary units. The association of three anythings is a structural pattern and reflects unity. All number of units will never equal unity, for unity is always singular. Units is always plural and infinite in number. Three is the first active number and seven is the most possible associations of three.
One set of three (ABC)
Three sets of each individually (A) (B) (C)
Three combination of sets of two each (AB) (AC) (BC)
Drawing a diameter divides the image in two halves. When folding the circle in half the diameter changes the properties where instead of two semicircles on one plane there are six semi-circle planes; six half circles. While this make no rational sense using a 2-D model, it is observationally logical to the folded circle. The circle remains whole, retaining unity even as folded into six half circles. ( If we decided to count the two edge circles it would change the possible combinations of associations.)
There is no conflict between folding and drawing circles; they are two very different systems; one is an image/idea of the other. Knowing the difference in properties helps clarify some confusion and greatly expands our understanding of the circle. It introduces a new area of dynamic exploration that in no way denies the theoretical or 2-D mathematics that has been developed. There are well over a hundred relationships, functions, and math concepts in this one fold of the circle into six halves. This is not to suggest one is better than the other, but rather to understand the difference and benefits of both folding and drawing circles and the connections between them. We know the value of drawing circles but there is no precedent for folding the circle and that means we have no experience or understanding about it. Only through the direct experience of folding will we understand the difference.
Besides the information and the beautiful objects that are revealed by folding, it is fun, interesting and engaging. We have a prejudice of not wanting to have too much fun learning something we have already decided should be difficult; if it is serious we must work at it. We are at our most open to learning when we are having fun and engaged in what holds our interest and simulates curiosity. Long ago we decided that mathematically the circle as image is a symbol for nothing, a place holder to later be replaced by something of value. It is now time to look at the information value of the circle beyond the image and the mechanical advantage we find in using it.
The information and reformation possibilities by folding circles demonstrates we can no longer afford to disregard circle/sphere unity. Because we have not done it before is no reason to continue to ignore it. I am writing these blogs in an attempt to give some understanding about the importance of the circle and that it might possibly simulate you to want to fold the circle and to find out for yourself and discover things there to be observed, and connections to directions not yet seen.