Monday, 11 January 2010 12:32

Think About It

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If there are over one hundred and twenty mathematical functions and relationships to be observed in one fold of the circle in half, why don’t we fold circles to teach mathematics?

  • If math can be discovered by anyone observing what is generated by folding the circle why aren't our children folding circles along with drawing pictures of circles.

  • If the symbol of the circle is nothing (zero) and the circle is everything (Whole) wouldn't it make sense to start with everything rather than nothing?

  • If the circle is Whole, a compressed sphere, why do we continue to teach mathematics in a fragmented fashion using bits and pieces where we cut the sphere apart to show a circle?

  • If the Pythagorean Theorem can be discovered in one fold of the circle why do we teach it as an abstraction to be proved by construction?

  • If the first fold of the circles reveals the principles of pattern development, fundamental to mathematics and systems formation, why don’t we teach the principles of mathematics? They are the same principles for everything else.

  • If the circle allows us to model using straight and curved edges why do we use only the straight edges of polygons?

  • If folding the circle one time reveals the Fibonacci progression of numbers, why don’t we let young children discover this progression?

  • If the circle is origin to all polygons, why do we teach children mostly about polygons?

  • If folding the circumference of a circle reveals proportions, ratios, and relationships not possible in folding the perimeters of polygons, why don’t we fold circles?

  • If the circle is comprehensive why do we only give students parts and tell them what to look for rather than allowing them to tell us what they see making their own connections within the folded circle?

  • If the circle in space is dynamic and generates information why do we only drawing static pictures of it?

  • If the square is limited to four sides, why don’t we fold circles that are without the limitation of sides?

  • Why do we think simplicity is one thing in isolation rather than unity as a comprehensive Whole of all things revealed sequentially?

  • If the meaning of anything is defined by the context why do we take things out of context and teach isolation and separation, limited to only a few constructed connections?

  • If all information about triangles is revealed in three folds of the circle why do we want to continue to teach about triangles piece by piece without context?

  • If folding a circle in half generates six semi-circles and constructing aa image with a diameter yields only two semi-circles, why do we keep drawing circles to show only two?

  • If everything in the circle is in the context of every thing else why do we want to take it apart as if there is no context?

  • Why do we put emphasis on measurement when understanding is in the ratios and proportions of the self-referencing, self-generating, and self-distributive Whole?

  • If three is structural and seven is the most possible combinations of associations of three, why is this not taught as a basic property of numbers?

  • If all parts folded in the circle are multifunctional and interconnected, why don’t we teach students about the unity and interrelatedness of all parts.

  • If all polygons are demonstrable by folding circles, why do we limit ourselves with static constructions of polygons piece by piece?

  • If one fold in the circle forms a dual tetrahedral pattern, and nine creases make the “solid” form, why aren't students folding tetrahedra rather than constructing with templates made by someone else?

  • If all regular polyhedra can be formed by folding tetrahedra, opening and joining in multiples, why don’t we do that?

  • If the circle is both Whole and parts, not demonstrable by any other shape or form, wouldn't it be to our advantage to know that?

  • If every fold in the circle is a spherical pattern of movement revealing a straight line perpendicular to and half way between the points, why do we only draw straight lines to show the distance between points?

  • All fundamentals of geometry and mathematics are generated by folding the circle; why do we require students to construct this information when it is inherent in the circle and is there to be observed by anyone?

  • If the circle is Whole, inherently containing everything fundamental about geometry and mathematics, pattern and pattern formation, why don’t we fold circles?


We do not fold circles, we do not believe it is unity, only a unit, even though we call it whole. We only draw pictures of circles and fold squares. Neither of those activities will tell us anything about the nature of the circle and what can be generated by folding them. If they did we would already be folding circles.

Saturday, 13 February 2010 16:44

Folding a Circle in Half - Part 1

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The questions in the first post were around the advantages of folding circles. Since we do not fold circles, we fold squares and draw pictures of circles, maybe we should do a little folding for some first hand experience. Then maybe the questions will make more sense and we can have a discussion around a shared experience.

Hold a circle in you hand, you will recognize it, but you have not seen it yet. What questions do you ask? What are its properties? How do you describe it; not what you have learned about the image, but the circle that is in your hands. This is about what you do not know, not what you know. Most students start out without know anything, so they say what it reminds them of rather than what they see. A few things that can be said are that it is Whole, it shows unity, it is dynamic, it moves in space, has an edge, a circle band that connects two circle planes, three circles total. It has volume. Given this information, how do we know what to do with it? Without getting caught up in parts, we can say the Whole circle moves. With that information we move the circle to itself; touching every where. In other words, we fold it in half. Then we crease it leaving an expression of that fold in the creased line.

Open the circle. The question is; what do we have that was not there before we folded it, what has been generated? We are looking for what we have not been trained to see. Most people will say a line; even math teachers will say a line in the middle rather than calling it diameter. When asking how we know the line is in the middle, rarely do I hear the edges and areas are congruent; mostly I hear because it is in half, or they are equal, or are the same. This is a good time to start introducing new words to add clarity to describing our observations, or reconnecting to what we only know in a math context. We can talk about curved and straight lines, points, areas, volume. What else? We have folded a ratio of 1:2.

How many half circles are there, two, four? The circle showing three circles folded dividing three circles into six semicircles. Only the concept and a 2-D construction shows two halves, the physical properties of 3-D are different. There are now many things to observe and talk about. All parts are multi-functional………….

Did you notice what you did when you were folding it? Describe what you did. You will discover what I have observed most people do when folding the circle in half. We all put two imaginary points together, look to line up the edges, and then we creased it.

We now have more information; two specific unseen point locations we touched together. There are two points, one line, and two semicircle areas and two imaginary points used to fold it; we just don’t know where they are yet. Seven parts folded from three circles.

Take a new circle and mark two points anywhere on the circumference: it dose not matter where. Now we see the two points, touch them together and crease. You don’t have to even it up; if the points are accurately touching the circle will be exactly in half. From this we can explore the idea that any two points on the circumference when touched together will fold a crease half way between, at right angle to the distance between the two points, and we see this folding process is about touching points. The first things that happens is principle to all else that follows. What are the principal qualities that you see in this one fold?

We have all folded the circle in half showing different diameters. How do we know that? No two people ever pick the same exact two points. The proof is in the relationship of points to diameter. Look at the properties; the similarities and differences of the parts that are now visible. How many of what we have are there? What are specific relationships between parts? There is a lot to observe. When you think you have seen everything, then draw lines connecting all four dots. This shows the distances that already exist, this gives the unseen relationships shape and visibility. Now look again.

We have six relationships between four points in space. Four points in space is a minimum description of a tetrahedron with six edges. Movement is always in two directions; folding in both directions is a 360°spherical pattern of movement. The diameter functions as an axis. That means there are two tetrahedra, one inverse to the other; an inside and outside, or a positive and a negative. Everyone has a differently proportioned kite shape, thus different proportioned tetrahedron. If by chance someone folds two points furthest apart, the kite shape will be a square. What are the differences between a kite and a square?

This is plenty to get you started and demonstrates how a little curiosity and attention, when catching our interest, reveals a lot of information. A math trained person should be able to recognize many of the over one hundred and twenty mathematical functions and relationships in this one fold. Guidance with clarifying questions will allow your students to discover many of these functions through their own observations about what they have done. It makes no sense to tell students what somebody else has discovered when they have the capacity to discover for themselves through their own experience. If they do that, it will always be theirs. Once a process for discovering things has begun, quite possibly they will discover things other people have so far missed.

Fold the circle in half and let us know what you have discovered.

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