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Home schooling comes home to folding circles

The paper plate circle holds food we eat, and it holds the information that we would like to feed the minds of our children.

Folding circles is about touching points and creasing. If the points are accurately touching the creases will be exactly where they need to be to reveal the extraordinary information inherent in the circle. Points are small circles and circles are Whole. Only the circle has a circumference and is undifferentiated unity; it is simultaneously both Whole and part. Anything that can be constructed using polygons can be revealed by simply folding and joining circles. A child, if they can fold a circle in half, can experientially discover and learn mathematics through guided observation by a teacher or parent. It is time to start folding circles along with traditionally drawing pictures of them.

Folding the circle gives young students a hands-on experience with the patterns, organization, and arrangements of basic relationships and transformational functions that are required when learning algebra, geometry, trigonometry, and calculus. They are the same patterns found in the sciences, the arts, and in nature. Systematically folding the circle to the direction of information that is inherent in the circle reveals concrete demonstration of functions that only need to be observed and then talked about in order to begin to understand the abstract concepts and generalizations we call mathematics.

Observation and reflection are the keys to understanding what the circle reveals when folded. There are over a hundred and twenty mathematical functions that can be observed in the first fold of the circle in half. It is impossible to imagine that amount of information and the beauty of the models that can be formed by folding and joining circles when all we know is drawing pictures of circles and using that image as a symbol for zero. All images are static; they are only representations of things experiential and dynamic. Without the experience from which to make connections, pictures have no value and little meaning beyond what we give to them; and so it is with drawing circles.

Observation and curiosity is the first step in the process of learning. Education by extension is sharing that experience of interest with others. Over twenty years working with both teachers and students starting from four years old, I have observed these qualities become more active when folding circles. The circle is a natural curiosity because there is nothing else like it, and we so take it as common place until we start folding it and realize it is far more than we had ever imagined. This is an open ended process that is organizational beyond individual learning; the circle is Whole and synergetic in nature. It crosses all cultural and educational boundaries.

The circle being the most comprehensive and inclusive of all forms, gives the greatest context, thus the greatest meaning, also provides the greatest opportunity for the greatest number of students to make individual and meaningful connections from their own experiences. All polygons are circles that have been truncated; the circumference has been cut off and discarded in favor of fragmented and isolated parts. To fully understand the function of any part or polygon, it must be seen in the full context of the circle. That means all parts are seen in the context of all other parts; nothing is in isolation from anything thing else and everything becomes multifunctional, just as we observe in real life. We are not overwhelmed by the endless possibilities of parts because the circle is Whole; it is only one self-organizing unity to consider. All formal education is a parts-to-whole fragmented process. The circle is the only tool that demonstrates a Whole-to-parts process of understanding.

The history of mathematics is in recognizing patterns and relationships where generalizations are made about these observations. These structural patterns and relationships found in nature are also the same ones observed in folding the circle. Because someone made a discovery long ago and gave it a formulated mathematical expression is no reason to deny a child from making that same discovery for themselves. Making ones own discoveries is far more meaningful and will not be forgotten. One does not need to know mathematics to fold the circle or to make these mathematical discoveries since math is simply an exacting language that describes what is revealed in the folding process. Discovery is what folding circles is about; the more we discover in this deceivingly simple form the more there is to explore.

Folding circles requires individual, interactive observation, which means doing something with the circle, observing what happens, using that information to continue the process, reflecting on, talking about, and sharing our experiences and our observations with others. This does not happen in formal education because there is always a limited curriculum of specific information leaving little opportunity for self motivated learning that comes from individual interest to know.

The sphere

The sphere is the only form we know that is Whole, complete within itself; totally self-referencing, self-generating, and self-replicating in a way that nothing else can demonstrate. Understanding the Wholeness of the sphere means literally everything is in the sphere, nothing can be added or taken away, otherwise it would not be Whole.

We live in a dynamic universe where most everything is spherical on both micro and macro scale. Everything spherical seems to be moving in circular orbit around some other sphere. Our physical origin is a spherical egg cell and we are born onto a spherical planet circling about a spherical sun, all part of a larger circular movement quite unexplainable on a cosmic scale. The sphere is the largest context that holds all information fundamental to everything that we want to know about anything.

If we take a clump of clay and roll it into a ball we have a sphere. By flattening this ball it transforms into a circle disc. Nothing of the sphere has been lost; everything within the sphere has been transformed by compression into a circle. Only the form has changed through compression, showing the movement of the sphere perpendicular to the circle plane that is formed. Nothing has been added or taken away from the unity and self-referencing nature of the Whole.

We now have the form of a circle disc in space through the transformation of the sphere at a right angle movement to itself. By folding the circle, itself a right angle function, we are decompressing all that spherical information. We are giving expression to the implicate order of the Whole through a sequential fold by fold process.

The advantage of using the circle is that it reveals spherical movement demonstrating both 3-D and compression, 2-D, at the same time in the same place showing the dynamics of functional relationships without destroying unity or changing the self-referencing nature of the circle/sphere.

By observing the changes that occur in compression we see what happens first, which is principle to all folding and reconfigurations and joining of circles. The more we observe the more connections we are able to make. The invisible is only one observation away from what is visible and it is through talking about those observations that reveals what otherwise seems not to exist at all.

If you are still reading this you are getting the idea that folding circles is about observing what we do not see and talking about what we do not know, otherwise we learn nothing and are wasting time. The circle is the only form that demonstrates a totally principled process that is comprehensively and inclusively Whole, and we do not know this because we do not fold circles.

What is important is to observe what we see in front of us as we fold, not to overlay what it looks like, or what it reminds us of, or what we imagine it to be, or have learned about circles. Observation exercises our ability to see the information that is actually there in front of us; to reflect on it and to identify what is otherwise obscured by what we have been taught. The question is always what is there that was not there before, what changes has the movement generated?

If we go deep enough into what has been formed in the first fold of the circle we can discover what is fundamental to everything else the circle will do. Folding in half is always the first movement of the circle. The more we know about what is principle, what comes first, the more we will know what can be done with the circle later on. We will not see all of the information that is generated at one time. There is too much information, with too many levels, and it will take time to uncover. What is important is to observe, make some generalization about that first movement of the circle and to periodically come back to observe more of what is there. We can not eat all the food at one time that is necessary to sustain the body over a life time; we must periodically go back to the plate for continued nourishment. This is just as true for the mind; and the paper plate serves both very well.


New in-formation

What is being presented is new information backed by twenty years of folding circles and many observations that have come from this exploration. There appears to be no historical precedent for any serious, in-depth folding of circles other than this process I call Wholemovement; the movement of the Whole to itself.

After thousands of years of drawing circles not one picture has generated any information; nothing.  I figure this is why we call the drawing of a circle zero. All we have gotten from the circle is what we have put there through means of developing geometry construction. One fold of the circle in half and over a hundred mathematical functions are generated. We can know this only by observing what we actually do in folding it. The information begins in the doing, not in what we have done.

Circles have been used as metaphors, analogies, illustrations, symbols for abstract ideas and concepts. We have used the circle for mechanical advantage in an extraordinary number of useful ways. So continuously has the circle been used that no one has asked what the circle is or where it comes from. We all assume the circle is the picture we have been drawing since we were small children. Okay, draw another picture of the circle. Then cut it out, look at it, feel it, move it, play with it. How is what you hold in your hand different than the image; compare the similarities and differences. Folding circles is experiential; it means nothing to talk about it if you have not spent a little time folding the circle. The information and understanding is in the folding, it is not with the idea.

Every reconfiguration of the circle/sphere is patterned to the first fold of the circle and is inter-transformable with all other reconfigurations. Every reformation comes from one of three differently proportioned triangle grids, the only possible options to continue folding consistent to the circle folded in half.  The equilateral triangular matrix is primary before the other two proportional grids. They are all proportioned to the same triangulated ratio of one Whole to two parts. We are only working with three-six, four-eight, and five-ten fold symmetries, everything else is an extrapolation.

Any paper circle will work. I recommend paper plates, they are cheap. The only other tools needed are a straight edge for creasing, a roll of  ¾” masking tape and some bobby pins for joining. Folding circles is the most direct, economical and most comprehensive tool for understanding the principles and the patterns of formation that we identify and recognize as foundational to everything we know about anything. We have no idea about the magnitude of the circle as Whole, or even what that really means beyond the extraordinary demonstrations that come from folding it.

Go to my web site and look at the gallery of pictures of models made by folding and joining circles. They represent only a small number of what is possible by folding circles. The number of models gives indication about the amount of information that is also generated. There are folding directions on the site so you can fold a few basic models giving you some experience about the simplicity of this principled and profound process. I have written seven books and have a DVD that documents, fold-by-fold, something of my exploration and discoveries about the circle. This material is available through  to those that want to find out more about folding circles.

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