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By folding the circle in half there are, to my count, over one hundred and twenty individual mathematical functions and relationships to be observed (see list below.) There is nothing in the image of the circle to indicate this or in holding a paper plate circle to indicate this. Only through movement is there change. What we observe about the change is what we call information. By isolating what we think is important from the experience we create isolated facts. Folding the circle give an experiential context that contains the facts and informs, giving greater meaning and increased value to the facts. Fold the circle in half and crease it, observe what you are doing. All these functions happen in one place and at the same time, are interconnected within the unity of the circle.

Without having to construct anything, something is generated  in the circle that was not there before we folded it. We did not add anything to the circle, folding simply reveals what is already in the circle, but hidden until acted upon. Observe not only the results but what we are doing, what is actually occurring when we are doing it. What we do is what informs us. Everything observed after we are done folding is separating out individual facts about what we have done, or what someone else has done. The diameter is a fact. What else is it? Where did it come from? How does it function? What value does it have?

A creased line is generated by the self-referencing movement of folding the circle. Where the diameter meets the circumference are two end points, two open planes in the folding movement continuously changing angles. The movement divides the circle in two semicircles forming definable and reciprocal inside/outside space.  If we were to fold the circumference to itself without creasing the circle, the facts would change, we would see something entirely different and unexpected about the circle. If we placed the creased diameter on to the circumference, then something else would happen.

Understanding the properties of the circle is to count similarities of parts describing the physical form. This allows us to go deeper. The circle has two circle planes (top and bottom) with a circle plane or band connecting the two. The circle is a triunity, three circles without separation. This means the circle has the same volume as the sphere, of which it is a reformation through compression at right angle to the circle. We can separate parts into similar and different functions making two individual sets. One set is the open circle band and the other is the two circle plane surfaces. This makes a separation of 1 circle and 2 circles where all three shapes are congruent. Through making this separation we have lost the volume, and more importantly, disregarded unity for the convenience of making an image to illustrate a conceptual generalization  The concept of two halves does not hold true when folding the circle as it does with constructing a diameter through the circle image. It is not logical to divide a circle with one diameter and get six half circles, but this is a fact substantiated by folding the circle and counting planes. There is a consistency of movement from an undifferentiated spherical Whole, to triunity of three, to six halves.

In folding, the circle remains Whole, nothing is added or taken away. The line, points and areas come from the self-referencing, divisional right angle movement of the circle. These parts could not have appeared if they were not already there. It took interacting with the circle to reveal the diameter and six semicircles and all the rest of the information to be discovered. By making a full revolution of the folding movement, the diameter now functions as an axis. The circle shows a full 360°spherical pattern of movement; revealing the origin of the circle.

Let’s go back to how we folded the circle. Everybody does pretty much the same thing. We touch two imaginary points on the circumference together, look to line up the edges and then crease when they are even. (With points together it is unnecessary to even the circumference.) Now there are four points to consider; the two diameter end points and the two imaginary points on the circumference that we touched together. There are four points and six specific relationships between the points. This one fold reveals a tetrahedron pattern of movement. It also tells us that all subsequent folding of the circle is about touching points. If the points are accurately touching then the crease will be where it needs to be. Accuracy is in the touch, with the eye, not the measure.

This time mark two points, anywhere on the circumference; touch them together and crease. Now all four points are visible with six unseen relationships between them. The diameter shows one of the six relationships, the other five are determined by the circumference and the four points. The areas are a function of relationships between the four points. To view the relationships, connect the dots with a pencil. All of this comes from the right angle movement by touching two points on the circumference. There are countless mathematical functions and reconfigurations that come from these first four points; but then genetics coding is also based on a difference of four. The circle is much like a steam cell; the pattern and order of sequential development of endless possibilities of formation and information generation is inherent in every circle.

As you look through the gallery of photos on this site, understand these models serve only to indicate the enormous diversity and range of objects that can be generated from folding and joining multiple circles. How much more there is as we begin to follow the options revealed in this first fold. This is only the beginning in understanding the difference between the circle and the conceptual image we are familiar with.

Some of what is in the circle can be seen and described to the level of any child as soon as they are physically able to fold a circle in half. This is about observation, from which comes understanding.

You can see more about this project on my blog

Following is a list from over one hundred and-twenty mathematical functions and concepts that are covered in the “One Fold Circle” book: