We hear concerns by people about thinking outside of the box. Yet nobody wants to leave the comfort of the square plane. A box is constructed with 6 squares. Give up our obsession with the square and there is no box.

There is no excuse not to fold circles. Pick up any scrap of paper, wrinkled, stained, any shape or size. Fold and tear it and you have a circle good enough to model the 2 and 3-D geometry found in any math book plus much that is not in books. It is all there in any scrap of paper.

1. Not to be put off by step-by-step instructions (words are often difficult to interpret) below are videos showing how to fold and tear a circle from any shape of paper and a few first steps. There is no need to be overly concerned about accuracy in tearing circles, just pay attention to what you do as you are doing it. I use a throw away catalogue, it has lots of thin pages, easy to tear and easy to fold.

**The following videos have no sound, this is a visual experience.**

It is important to accurately fold the half-folded paper into thirds; this is done by proportionally adjusting edges until they are even, this insures the folds are angled equally. Measure and mark a point the same distance up from the corner point on each edge insuring the edges will be the same length. Accuracy in tearing the arc between points does not matter as long as you start at right angle to each side and tear towards the middle. When tearing use your fingernail to accurately position where the tear will start.

*Above left)* here is the vector equilibrium sphere made with four torn circles. The thirteen points are all the same distance apart. The paper images add to the seeming disorder of an organized association of points.

*Above right)* is the same model with the edges folded over between points to show the regularity in geometric form we are familiar with. The circles are held together with bobby pins.

A tip making for when using multiple circles. Make an accurate traingular folding, mark this as a template. After folding another paper in half, line up the folded template with the edge on the first fold from the approximate center out. Bring the first fold to the edge of the template and crease knowing it is accurate. Remove template, fold over, line up edges and crease. Using an accurate template eliminates proportional adjustment and saving time in folding.

2. This video shows folding a tetrahedron folded from a ragged-edged circle without losing accuracy to the polyhedron form. Weather the circumference is ragged or not has little to do with the regularity of relationship between points.

3. This video starts with the torn circle folded to the triangle and expands the folding to a 4-frequency diameter grid of twelve creases by making three more folds. All twelve creases are full chords as a function of the relationship between points as the layers are folded.

Make a comparison of this folding to folding each chord individually as shown in a past blog Unity Origami

The two approaches reveal different information about the same grid. With heavier paper it is best to fold and crease each chord individually because of inaccuracies created by the thickness of folded layers. There is a consistency in direction when folded one at a time. With the folding in the video each cord is a combination of forward and backward directions of folds. Since every fold is an axis moving in both directions, it makes little difference except for thickness and amount of folding.

*Above)* two different sectional divisions have been colored to show the importantce of diameters as coordinate aspect of the 4-frquency grid. The left triangle relies on one point on the circumference to divide the radial bisector in two equal sections, where on the right the diameter is the edge showing the same division. There is an unequal division of the center bisector on the right showing reciprocal division between edge and diagonal. This reflects the primary hexagon division. These two triangles can be individually separated, but together are the means of infinitely scaled division of circle unity to any unit measure. The difference between them is important to the interrelated dynamics of the grid and foundational to geometry.

*Below) *this exploration started with a yellow pad of paper, wanting a circle and having no scissors. All I needed were three equally spaced diameters and I could tear the circle from the paper. With six folds in a rectangular piece of paper this is what I ended up with, a fundamental in/out folding of the four-frequency grid.

Notice the hyperbolic dynamics talked about in recent blogs; In/Out Hyberbolic Surface

and Folding Hyperbolic Surfaces

Circle unity has no points, lines or separated areas. The circle unit can have these properties. A point is a scaled down circle. Euclid defined “A point is that which has no parts.” He is talking about unity then uses it as a part of construction. We start with a center by way of the compass. There is no center to unity, it just is. The circumference gives meaning to the point in a formed circle unit, not to unity. We do not make unity; we rearrange parts and work with proportional relationships between units within unity.

Tearing the edge of the circle from point to point is relative as is using regular curves or cutting straight lines. Edge boundaries are always a bit ragged. Each edge changes the form and relationship between parts but does not affect the consistence nature of structural pattern of arrangement. We have straight edges, regular circumferences, and now arbitrarily torn edges that gives us opportunity to see something different about the circle and giving some clearity about the idea of pattern formation.

*Above left)* both are a tetrahedron. One has the circumference folded to the outside and visible while the other has the circumference folded to the inside and invisible.

*Above right)* is a folded tetrhedron with the edge on the outside. There was little attenion to the configuration of how it ws torn. This is no less a terahedron but much more in-formation.

*Above) *you can see the beautiful symmetry in the folded tetrahedron net regardless of the forms it takes. These are the two tetrahedra fron above opened flat. This reflects a process we see in nature.

*Above)* six tetrahedra, four in a tetrahedron arrangement and one to the side in line with the others. The tetrahedron with the circumference folded out forms three open tetrahedra cavities. When tetrahedra are fitted into the cavities it will hold the tetrahedron in opposite position. This inside and outside is an interesting and efficient way to build a tetrahedron/octahedron matrix.

*Above left) *a two-frequency tetrahedron, four circles torn form scrap paper.

*Above right) *the same two-frequency tetrahedron with the circumference folded to the outside. The two are exactly the same arrangement of four tetrahedra with an open octahedron space.

*Above) * using the two-frequency tetrahedron from above and adding four more tetrahedra to the four open planes reveals a cubic pattern of eight tetrahedra in a spherical rhombidodecahedron form. This a function of the inside circumference being on the outside.

*Above)* the six vertexes of the rhombidodecahedron have been opened to the outer creases in each tetrahedron net. This introduces six squares and twenty-four open triangle faces that are tetrahedral and another twenty-four open triangles combined to form twelve smaller rhomboids that are octahedron cavities. The twelve rhomboids have opened to reveal a complex spherical division of tetrahedra and octahedron cavities.

As far as I know this is not a classified polyhedron because you cannot get it through truncation. This a transformational process by opening the six vertex points that open 24 edges to a very regular spherical division. This can not be derived by cutting away corners because there are no corners on a sphere. Circle unity is inherent to all possibilities of unit configurations.

I consistently find levels of information in folding circles that surprise and delight me that I do not observe elsewhere. To share some of this is why I write this blog each month.

Now you know paper circles are as cheap as bending down and picking up scraps of paper. By simple folding and tearing the circle away from the paper you can explore the beauty and mathematical relationships we never see in the paper trash we throw away, or in the books we are given to read. Everything I have done with paper plates can be done with torn circle scrapes. This is yet a beginning towards understanding the perfection of unity and the ragged nature of boundaries in division.