Monday, 23 August 2010 17:02

Center Off-Center



Accuracy in touching is seeing with the eyes, with the mind, coordinating with the body. To not see is to be off-center. This is not about measurement, it is about movement through space with purpose. Without awareness towards the Whole can we know the center? Can we know off-center? By starting with the circle as Whole, and through action of folding we will explore center and off-center locations.


Axiom for folding circle: Any two of an infinite number of points on the circumference of the circle when touched together and creased will equally divide the circle where the linear distance between points will move spherically at right angle to the diameter diminishing the distance between those points as they join on the circumference, where the linear distance between points becomes half the original distance.

This gives symmetry to the division of the circle


The corollary to this: where any one point on the circumference with any other point not on the circumference (or two points not on the circumference nor aligned to a diameter) when touched together and creased will generate a chord less than diameter where the right angle circular movement will diminish from one point to touching the other, and the line segment distance between the two points is reduced to half the original distance. There is no symmetry to this division of the circle.


The axiom is the only way to symmetrical fold a circle, assuming it to be the first act of folding. The corollary holds true for the circle and applies to folding any shape. Two circles will be used to demonstrate the same process for finding center and off-center locations. Points are touching or they are not; there is proportional harmony or disharmony to the circle. Center is always sustainable, what is not sustainable has no center. Inaccuracy in putting points together becomes arbitrary, random movement without attention to finite boundaries.


(left) axiom; (right) corollary. There is no right or wrong in observation

coc2  cocc3

We see the difference in touching two points on the circumference and touching two points with only one point on the circumference. How does each develop as we apply the same folding process to both?


The next two folds are generated by folding the semicircle into thirds around the circumference. Fold one third in front and one third is folded to the back behind the middle third. It is not necessary to measure; use your eyes to know all points are touching by sliding the folds bringing them into alignment. Don't crease until all the edges are lined up and exactly even. Fold the off-center circle into thirds the same way; sliding circumference sections back and forth until the edges are even (points will not match up, you might look for division of central angle.) When edges look even, then crease. This folding process can be seen at the “How to Fold the Circle” page on my website: www.wholemovement.com

(Left) All chords are diameters. (Right) Only one chord is diameter The lines have been traced with a marker to clearly show the creases in each circle.


coc4   cocc5



Both circles show three evenly spaced chords revealing a point of intersection, where each is differently located to the circumference. If the points of intersection were a rotational axis, one would wobble and the other would show steady alignment to concentric movement.




To further develop folding in each circle in the same way, reform each by folding one of the chords to itself. The center is even with two opposite radii touching. With the off-center circle the division of chords is uneven, the end points will not meet. Use a bobby pin to hold the creases together. There are three folds, therefore three possibilities for folding a crease onto itself. Only with the off-center circle does it make a difference in the configuration. This joining a crease to itself is called a “bowtie” configuration.



.coc6   coc7


Fold another circles as before, making another “bowtie” for each. Join together the two individual sets of each, straight edge to straight edge, using bobbie pins or other means to hold them together.


coc8   coc9



This makes obvious the difference between center and off-center forming and joining. Each of these sets of two circle each is one half of a spherical pattern. (In the off-center the circles are creased to one-quarter distance of a diameter of the circle, to hold to some proportional consistency. Otherwise they could be folded in anyway.)




Make a duplicate set of each ( two sets of two circles) center and off-center, and join them together respectively on straight edges. (Again described on my website in the How to Fold the Circle section). The off-center is now getting more difficult to work with and the center system goes easily together.


coc10  coc11


Each is shown in same orientation so the difference is apparent. The one diameter in each of the four off-center circles has been assembled to reveal the one composite circle for comparison to any of the four composite circles seen in the center joining.


Below. Other options in joining off-center circles do not show a composite circle. 

coc12  coc13


All center and off-center spherical systems show alternate open triangles and squares. The off-center of each circle displaces the circumferences showing no regularity of a patterned spherical form. The form is deviant, while having an invariant pattern of three equally divided chords within a local center, not obvious in looking at the form.




Here one half of the centered sphere is joined to one half of the off-center sphere. This does not show balance, only that the pattern of developing three equally space chords for both is consistent and allows irregular joining of center and off-center.




There is only one of an infinite number of proportional symmetries in touching two points together in that first fold of the circle. Everything starts form this ratio of 1:2, one Whole two parts. This is the first expression of symmetrical pattern. Anything short of that will eventually cease to generate. The off-centered asymmetrical forms, while they can be interesting and many, are limited and will not sustain generation through multiple joining.



Below are  examples of off-centered variations of spherical forms.


coc15  coc16

coc17  coc18





Below; the edges of each, center and off-center, are folded in-between the six points on each circumference before joining, this is more in keeping with the straight edge flat plane truncated look we are traditionally familiar with. The vector equilibrium, traditionally called the cuboctahedron, is pattern for both; clearly identifiable in center folding, not so easily seen in off-center folding.


coc21  coc22


Where is the center off-center? There is no outer boundary or inner boundary to concentric circles meaning there is no center to the circle; the circle is the center. The circle demonstrates the center is never out of center; there is only awareness about when inaccuracy slides into random movement and becomes misaligned. The mind functions as a balance, a connection, between seeing a finite center in physical form, and possible insight to perfection of infinite center regardless of form. This folding is a demonstration to seeing what works, therefore is sustainable, and what does not work and is unsustainable.


Over the next couple of months I will continue this center off-center exploration in folding and joining circles.


Published in Blog
Tuesday, 03 August 2010 16:56

Math Word Problems

Years ago when first reading word problems in my math book I felt I was reading about things on another planet. They did not make any sense to me. Why on earth would anyone try to figure out answers to the questions that book was asking. I entertained my self by drawing pictures in the margins, pictures that had a relationship to where I was sitting. The wording of those questions was as strange as the thinking I was supposed to do to get the answers. They always seemed to have a number of different answers even though we were told there was only one right answer. That just didn’t fit my young life.

Now I know there are questions that are not questions at all; where there are no right answers other than the answers that make sense. Traditionally they are called parables. Why didn’t we have parables in my math book? Why aren’t there parables in math books today? Too many right answers I guess.

Here are two contributions for a math book, if anyone cares to use them. They are certainly something one can draw pictures to, even if you can’t come up with the right answers.

Problem #1

The square when resting had a dream of triangles, pentagons, hexagons, trapezoids, octagons, and circles. Upon awaking he was troubled finding this dream disturbing.

The triangle when resting had a dream of squares, pentagons, hexagons, rhomboids, octagons, and circles. Upon awaking the triangle was also troubled finding this dream disturbing, but not as much as the square.

The circle when resting had a dream of triangles and squares, pentagons, hexagons, and all manner of polygons. Upon awakening the circle had a slight bitterness at the delightful recollection of this dream.

The sphere dreaming of all those things, upon awaking did not know it, and so continued dreaming on.


Problem #2

A man comes from a sphere; he does not remember. He grows up on a sphere, surrounded by endless spheres of all different sizes. He looked around and it appeared to him flat. He holds a ball in his hands; bounces it on the “flat” to entertain himself, then for profit, and possibly for the enjoyment of others.

Another man comes from a sphere; he does not remember. He holds an imaginary sphere in his hands; no one can see it. He compressed the sphere into a circle and then sticking his head into the circle his mind was consumed knowing spherical reality.

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