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Bradford HansenSmith
Folding A Circle iIn Half: Part 2. Principles
Let’s get back to the first fold in the circle (Feb. entry) and talk about the principles behind the parts that determine the interrelationships between those parts and all subsequent folding with a circle. In deciding that the circle is a mathematical symbol representing nothing, and using fragments to draw 2D constructions to prove abstract formula, we have failed to understand that the circle is both Whole and part. Understanding the circle is Whole allows us to observe what is principle. Principles happen first; they affect all parts and all folds, reconfigurations, and joining of multiple circles. If we do not know what is principle, what comes first, we do not really know comprehensively what we are doing.
The sphere is Whole before all other forms. Compression transforms the sphere to circle. Spherical unity is principle to the nature of the circle; it comes first. It is triunity; a disk in space showing three circle planes. By adding the two edges there are five generalized parts. We can add the inside volume and the external space, seven individualized associations of unity. There are seven observable qualities that happen first in this act of compression and again in decompressing spherical information by folding the circle.
Stating with the WHOLE, there is MOVEMENT that creates DIVISION forming a DUALITY in TRIANGULATION, where there is a CONSISTANCY of all parts to the movement of the Whole, and each part is INNERDEPENDENT to the Whole.
Every fold in the circle reflects these seven principle qualities. They apply to all aspects of our lives.
Notice the first five observations are about the mechanics. They reveal the manifest functionality of structural order. The last two observations are relational. They give us the most trouble. We are not consciously consistent in developing progressive habits, and we do not like the idea of being dependent on anyone or anything. We have trouble connecting with the inner, the unseen intention of first cause, the absolute pattern that regulates all purposeful evolutionary formation. The degree of clarity about these two qualities has everything to do with how we relate and interact with each other, and the connections we make that give meaning and value to our lives. These ideas about principles are not just a philosophy illustrated by folding a circle. It comes from direct observation about what happens when a sphere is compressed and again when the circle is folded.
Cutting the circle into parts violates the principles and destroys unity causing disruption, and confusion; then breakdown occurs. We are left with separated pieces and must rely on the inconsistency of the human mind and construction methods. Unity can not be constructed; it is a function found only with the Whole. Principles are not a function of parts, but of the Whole. We only have to look at what we have done to this planet and the condition of people’s lives to know there is destruction, lack of clarity, and little understanding of principles or purpose. Humanity lacks knowledge of unity, is confused about the Whole, and has become addicted to fragmented construction using bits and pieces for our own shortsighted pleasure. We are under the illusion that we create unity with parts. Yet we do make extraordinary bigger parts from smaller parts, and incredible smaller and useful parts.
Understanding these principles help us to recognize unity, support the beauty of endless differences and to progressively benefit from such diverse expressions of inner goodness. We lack responsibility to the interconnections between all people for our lack of understanding our dependency on the common source of all being. To better understand what is inclusively principled can help clarify and reconcile the confusion between widely differing forms of social and religions cultures and our individual experience.
The Wholemovement approach to geometry and pattern development differs from traditional understanding based in a mixture of false assumptions, inconsistencies, and an over abundance of self interest. Ask mathematicians what the principles of mathematics are and keep tract of all the different answers you get. We do not have an understanding of what is principle beyond what we think is more important than what somebody else thinks. Folding circles offers a demonstration of principles that are inclusively dynamic to all pattern development of formation, in all fields of study as far as I can tell. Knowing what is principle helps to increase our capacity to a clearer understanding about consistency of appropriate behavior about our place in the universe. The sphere/circle reformation and folding is the only principled, experiential, handson activity that demonstrates anything about the idea of a comprehensive and inclusive Whole.
Origin and Meaning of Wholemovement
Over twenty years ago, from curiosity, I went to my first math educators’ conference to see what people do there. I ended up looking at lot of geometry books to see how the education industry defines geometry. Surprisingly over half the books gave no definition. The others gave ‘earth measure’ and ‘measuring things of the earth.’ Geo means earth and metry meaning measure. So this is why measurement is important to teach; because the word says so. That made no sense to me. Measurement is simply a way of keeping track from one location to another. Geometry had to be more than that.
I thought about how the word is broken down. Geo means earth; the earth is spherical, and the sphere is fundamental to everything we know about this universe. The sphere is Whole beyond all other know forms. Metry is simply to measure and measuring is about movement from one time in space location to another. That is when it made sense; geometry comprehensively and inclusively means Wholemovement. The movement of the Whole; complete, selfreferencing, revealing through movement everything that is, is not, or yet to be; everything we know and do not know, and all unimaginable potential we shall never know. The movement of the Whole generates endless parts. No amount of parts will ever realize the Wholemovement. Now I had a context from which to better understand geometry, mathematics, and all other patterns of order and organization of the things in our universe.
Geometry is about relationships, dynamic interactions between everything in time and space. It is not as presented in static, isolated, and abstract images. The concept of Wholemovement was so much more interesting, made more sense, without denying anything of what is in all the books. Wholemovement gave a new meaning to geometry and mathematics that I was unable to find in all the overwhelming amount of fragmented and abstracted information in a discipline defined by its own definition.
We talk about unit in math and geometry; isolated facts and functions we are suppose to learn and the isolated unit sections in which we are to learn them. Units are a function of a linear concept from what appears simple to complex. Nowhere do we talk about unity except maybe as a collection of single units brought together to make a bigger unit, usually call sets. Whole numbers are no more whole than whole math, whole language, the whole family, whole systems, the whole planet, or the whole universe, all in an unimaginably large cosmic environment. All are systems of relative size and complexity of parts, bigger and smaller, endlessly forming systems, moving, changing, transforming, all interconnected and innerdependent to the movement of the Whole. No part or combinations of parts, no system can ever equal or reveal the Whole. Without the absolute of the Whole there is little meaning in the relative relationships between parts. Unity belongs to the Whole; unit is separation.
When we consider the Whole we are looking at origin to all possible locations, relationships and interactions. To know the origin of anything is to better know what you are working with. To know origin is to know what happens first, and what happens first is principle to all that follows. There is no meaning without context and the larger context gives the larger meaning. If we do not start from the Whole we will not get there by constructing, there are an infinite number of parts. Without context we are lost in a confusion of constructing what will eventually fall back onto itself. Starting with movement of the Whole, everything is all in the same place; simply spread out over time and throughout space. Thus, through observation and reflection we learn about this extraordinary existence of being.
Learning to act appropriately within the largest context is the most practical thing we can do to survive locally. We all have in common spherical origin. Individually we come from a spherical egg cell on a spherical planet, in a universe of spherical objects, moving in ways too large or small to see. None of this we understand, it just happens and we create informational stories about it. Our experience is not constructing life one unit at a time; it is more comprehensive and dynamic than that. Wholemovement is much closer to our individual experience of the innerrelatedness and outer interconnections that we experience and observe about all life. All of this gives meaning and value beyond just that of local concern.
When using the word Whole as inclusive I capitalize the “W”, otherwise when using whole referring to a big part or what seems coherent, a lower case “w”is used. Whole used to mean health, unity of parts, and unity without parts. It has recently come to be used as a modifier to give greater meaning to what ever we think is important and want to emphasize. The word Whole has been corrupted and currently holds little meaning.
The word geometry reflects a partstowhole process of endless measuring of parts where the Whole is never reached. This has become the direction of formal education in this country. On the other hand Wholemovement is a Wholetoparts process of modeling that has allowed me to find a way to make sense of the all the bits and pieces of fragmented information that keep us on the unit level of separation. Understanding that the word geometry comprehensively means Wholemovement can help us move towards health, greater clarity, and unity.
Think About It
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 semicircles and constructing aa image with a diameter yields only two semicircles, 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 selfreferencing, selfgenerating, and selfdistributive 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.
Folding a Circle in Half  Part 1
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 2D construction shows two halves, the physical properties of 3D are different. There are now many things to observe and talk about. All parts are multifunctional………….
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.