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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.

On a number of occasions towards the end of a workshop a child will ask how to make a cube. Then I would have to figure out which one; how much time is left and can we do it without diverting the class from what they are doing.

I tell them yes, but not with the folding we are doing. I show them an easier way using the 4-8 folding rather than 3-6 we have been doing. A cube is made, others see it; very quickly half the class is making cubes without distracting the other half.

Here are five cubes made from both the 4-8 grid, right angle triangles and the 3-6 equilateral triangle grid. They have been colored to the creases used for each folding.

We will explore folding cubes with both the 3-6 and 4-8 symmetries of the circle.

*Below left)* fold the circle in half, and in half again; four points on the circumference, five with the point of intersection from the two perpendicular bisectors. Alternate areas have been colored in to show areas of division.

*Below right)* the inscribed square has been creased and two lines of division parallel to the sides have been folded and alternate areas colored. There are five squares, one inscribed and four divisions of smaller squares.

*Above left)* Two circles of the net above are reformed and joined forming the cube. It is the same as joining two open tetrahedra to form an octahedron; (see http://wholemovement.com/how-to-fold-circles. Scroll down to #3.

*Above right)* is the same forming and joining only with more creases showing more areas colored in. This is pretty straightforward to what we understand about divisions of and construction of the square and cube.

*Below)* is another way to make the cube; it takes a little more folding but is richer in transformational and 2-D designing possibilities.

From folding three diameters we fold the 4-frequency diameter grid. http://wholemovement.com/blog/itemlist/date/2011/2?catid=140.

This grid divides the circle into 12 equal parts that show three differently positioned squares.

*Above)* the grid is shown with one of the squares colored to show that part of the grid that lies inside the square net. Two extra creases were made from the grid to show the axial division perpendicular to the sides. Two folded circles have reformed to a cube. The cubic folds are the same for the 3-6 and 4-8 symmetries, different in context making the divisions from the 3-6 gird uniquely different.

*Above)* the flat circle shows two radii that become the diagonal and edge of the square in different formations. Folding under one-quarter of the circle a right angle tetrahedron is formed. Starting with the folded square the same occurs. They are interchangeable to different scales and forming of the cube.

*Above left*) the circle is formed to a right-angle tetrahedron with perpendicular bisecting diagonals.

*Above right)* the tetrahedron edges are pushed in becoming the diagonals of the squares formed by the three edges of the tetrahedron. There is a reciprocal function between edges and diagonals. The right angle tetrahedron is one-quarter of forming a cube and the three squares are one half of a formed cube.

* *

*Below)* is another design of division from the 3-6 folded symmetry.

*Below left)* the 4-frequency grid with one folded square. The areas have been alternately filled in to make a more interesting proportional surface design. With these folds there are many possibilities for designing the surface.

*Above right) *two circles joined forming a complete cube.

*Below)* the circumference is folded to the back to show only the square. With out the folded circle it would be difficult to come up with this proportional design.

*Above)* all three creased squares colored to show the three square-compound. By adding creases for the three right angle axis to each square. It generates 24 division in the circle.

*Above)* the circle with circumference folded behind showing only one square. Two squares of the same design are reformed and joined. There is the option of putting six squares together into a cube.

In both the 3-6 and 4-8 folded grids the primary points of intersection represents centers and tangent points of a circle matrix. Curved lines are used to redesign straight creases. These show only two of many possibilities.

*Above left)* the 3-6 triangle grid has been colored using the curves that resemble the closest packing of spheres in a traditional arabesque design.

*Above right)* the 4-8 square grid has been designed the same way.

*Above)* The same design can be “centered” differently in how it lies within the circle. This goes for any grid

*Below)* right and left shows the cube formed from each of the above designs.

*Below)* primary folding possibilities of one circle folded to a 4-8 symmetry where the edge of the inscribed square is divided into 8 equal segments. this is different than starting by folding the diagonal into eight equal divisions.

*Above middle right) *is one half a cube.

*Above bottom right)* shows ¾ of a cube formed from this square division.

A “full” cube can be formed from one circle with the 8-frequency division folded on the diameter/diagonal. Out of infinite possible diameters in the circle the square uses two. We can then see root function is the diameter of the circle which goes for the square as well.

*Above)* Two diagonals/diameters divide the circle into four equal parts. Each diameter is folded to 8 equal divisions. The circle has been reformed into a complete cube where the four part colored areas shows a volumetric division in surface design different than what we usually see.

*Above)* two cubes; one from four circles and one from a single circle are combined making a beautifully proportioned cubic system from five circles. Pattern blocks are proportional systems of relationships that have origin in the circle.

The square and cube while they can be constructed separately are relationships of triangles inherent to the circle. There are a lot of possibilities for making your own set of pattern blocks from circles.

*Below) *some three-quarters cubes and some partially formed right-angle tetrahedra make up another kind of proportional cubic system.

Let’s look at other possibilities by folding the circumference to the outside of the cube.

*Below left)* two perpendicular diameters and the square relationship are folded to the 4-8 folded symmetry where the alternate areas in the square are colored.

*Below right)* four circles folded to the right-angle tetrahedron with the circumference folded in, then joined forming a cube.

*Above)* one circle is reformed to the three squares (half cube) with the circumference folded to the outside.

*Above)* the cube from above has been reformed with circumferences on the outside. It reveals the spherical tetrahedron, one of the two tetrahedra forming the cube. The four circles are held together with bobby pins.

*Above)* two views of a different division of proportional folds with circumference on the outside. The cube is smaller with more circumference revealing a spherical truncated tetrahedron.

*Above)* a circle folded to the 3-6 symmetry with alternately areas of one square colored in.

*Above)* Two views of four circles, shown above, have been folded and joined with circumference on the outside. This time part of the second spherical tetrahedron is revealed as they intersect crossing on two opposite faces.

There is another way to form a cube solid is to fold the tetrahedron net; http://wholemovement.com/how-to-fold-circles.

*Below)* the tetrahedron net. Fold the two end points of the three creases that form the inner triangle to the center point and crease (one at a time.) This gives you two more congruent triangles, one on each side of the first folded triangle.

*Above right)* using the creases from either the right or left side of center inscribed triangle, fold the circumference behind forming the triangle. The center triangle will be off center to the just folded triangle. Fold over on second line in from each end point forming a right angle hexagon.

*(below left.)* bring triangle points together forming a right-angle tetrahedron as you would form a regular tetrahedron Tuck triangle flaps inside to hold the right angle tetrahedron together.

*Above right)* the tetrahedron is opened with flaps folded out forming right angles that shows three sides of three-quarters of an open cubic arrangement.

*Above)* two open units joined on surfaces and taped together.

*Above)* two sets of two, above, are joined. The backside cavity is half an octahedron.

*Above)* two sets of four units are joined with the eight open cubic units on the outside. The 24 external points reveal a truncated cube of eight octagon and eight triangle planes. The center is an octahedron axially divided by three intersecting square planes.

There are many variations of cubic division and arrangements, some with circumference folded in, some folded out, and some combined. There is much to be discovered. A cube can be made from one circle or with many, from the 3-6, 4-8, or 5-10 folded circles.

Sometimes students will make a transforming system by hinging four solid cubes together in a square arrangement.

A full cubic transformational system takes eight cubes, all hinged in the same right angle system reflecting the torus ring where the surface rotates through the center opening. You can see an example at:

https://www.facebook.com/photo.php?v=1870781780855&;set=vb.125203137500210&type=3&theater

The surfaces in this torus system were designed to show the closest packing of spheres as well as a few straight-line possibilities. There are other torus systems in the same album made with other forms.

Within the circle are the proportions for blocks having a variety of measures with lot of interesting surface design possibilities.

enjoy...