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.