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Last month I posted a few pictures of folded circles on Wholemovement Facebook page. Some of you may have wondered how the complexity was achieved from paper plates circles. I thought it might be of interest to see the process of folding the triangle grid matrix that generates the forms used in assembling those and many other complex systems I have recently been exploring.

This process is similar to unit origami in that you are joining similar paper units to form complex systems. The similarity stops there. Because we are using circles and the circle demonstrates unity beyond all other shapes and forms, we then might call folding paper circles “unity” origami. This is not the unity of adding things together, it is a singular unity wherein all things are already together as potential. It is then only a matter of giving individual expression through reformation of the circle and various arrangements in combination. Squares, triangles, and all other reformations of the circle are never less than the Whole circle. They are all formed from the same folded grid, simply reformed differently in multiples and joined. Unity origami does not carry the limitations and restriction that come along with folding square units. The circle demonstrates spherical unity revealed through patterned movements of reformation that are inherent only with the circle.

Steps in folding an equilateral triangle grid matrix.

First fold the circle in half by touching any two points on the circumference together, a simply process of self-aligning of the circle.

Fold one end point of the half folded circle to a position half way between the two edges as you change the length by moving along the circumference arc. Do not try to measure; use your eyes to see the proportions. It may be easier to look for equal angles keeping the circumference even.

Do not crease yet.

** **

Fold unfolded part under and line up the end points; one fold over and one fold under like a “Z”. When the points are touching the edges will be even. After turning over and checking both sides to see that everything is even, t**hen crease**. (This is the hardest part of the entire process, it requires using your eyes coordinating with your fingers.)

Open to the circle; there are three diameters dividing the circle in six equal sections, seven points. There are three choices to continue folding point to point. We will take one; the other two will also lead to the same complete 8-frequency grid.

Fold end point of one diameter to the opposite end point and crease. Do that the same with all three diameters; end to opposite end. This generates three more diameters that now divides the circle into twelve equal sections.

Fold one end point to the center point and crease.

Open the circle. The radius of the folded diameter is now divided in half and with the fold between radii of two diameters forms an isosceles triangle. Fold one end point from the first fold (the isosceles triangle) to the center similar to before and crease.

Continue around the circle touching every end point of each isosceles triangle to the center and creasing (this is every other end point to the center.)

When you are back to where you started there will be a folded hexagon star of two intersecting inscribed triangles and three bisecting diameters. The three diameters form six star points with three bisecting diameters half way between each star point.

Each diameter now has two new points of intersection not there before. Each star point diameter is divided into four equal sections. I call this a 4-frequency diameter grid. The hexagon diameters are divided equally, the bisecting diameters are unequal in division. Does this look like the circle you just folded?

Start with the one diameter; two end points, the center point and two new points that divide the diameter into four equal segments.

Fold the end point to the furthest new point on the same diameter and crease.

Fold the same end point to the closest new point on the same diameter and crease.

There are now three parallel folds that divide one radius into four equal sections. Do this to each star point diameter (six times.) If you go from one to the other in sequence you will know when you are finished coming to the first crease.

You have folded a grid of three diameters where each is divided into eight equal sections.

This I call an 8-frequency diameter grid. The three bisecting diameters are now part of the grid of three sets of seven parallel lines each. It is like an octave in music. All the notes you need are there to form endless arrangements between any combination using endless possibilities of intervals. The other two choices with the first three diameters that we did not take are now folded within this 8-frequency grid.

This octave can be further divided by following the same process of folding the same six end points to the new points of intersection on each individual diameter dividing each section again in half. It goes from the first fold 1, 2, 4, 8, 16, 32 and so on, until you reach the size limitation of the circle. If you want to take it to a higher frequency start with a larger circle.

The following pictures show the development from the first, a 1-frequency grid to a 32-frequency grid. The higher the frequency the greater complexity can be generated from folding one circle. It is the 8-frequency octave that is fundamental to scaling in and out, all to the pattern of three.

The pattern does not change while the possibilities in reconfiguring a single circle build in complexity with the increase in frequency. Kids in workshops sometimes fold a 32-frequency grid after showing them only the 8-frequency. They did not know what to do with it when done because of the enormous amount of information, but they felt acommplished doing it. You will get the most out of a higher frequency grid if you have worked with increasing levels of frequencies first. Some students find it a challenge and engaging to just fold and explore the possibilities of reconfiguring a high frequency grid of creases.

You can see this process is straight forward always generating information to continue the folding. Each of these individual frequency levels have different directions to explore and will reveal very different reconfigurations of the circle and when you start combining them the possibilities become endless. This is not different that any other kind of frequency modulation except you are doing it with a circle.

The more you understand the tetrahedron as basic to all pattern formation and that the other four regular polyhedra are patterned arrangements for reconfiguring and joining the more you will get from various frequency gird levels. Go to the previous five "Center Off-Center" blogs for more information about some basic instruction for folding the tetrahedron, octahedron and the icosahedron.

There is another useful folding for all frequency levels. I call it an in-folded hexagon rather than an inscribed hexagon since we are folding it in; unless you figure we are drawing it out from the circle by creasing. The hexagon comes directly from the in-folded equilateral triangle.

Here is a 16-frequency folded grid circle. Fold the circumference in forming the equilateral triangle. Each side is folded under the previous side locking one into the other.

The equilateral triangle is folded in half forming a right triangle. Fold each diameter, the three perpendicular bisectors, one at a time.

Open the triangle and put the circumference folds on the outside making three small vesicas. This is a another way to see the proportional difference between the diameter length and distance around the circumference. This proportional folding demonstrates what we call pi.

Open to the circle and fold in the circumference informing a hexagon shape. This is a more familiar representation of pi. There are now many more possibilities for using the circumference in exploring all levels of reconfiguring the circle.

For your first time in folding the grid do not be over concerned if all lines are not exactly parallel, some will be slightly off, but with attention to touching points they will all be close enough. Remember when points are touching the lines will be where they need to be, and subsequent points will be in alignment.

This is not a difficult process and it will become clear as you begin fold by fold. I would like to hear about and see and what you come up with. Feel free to contact me if you have questions or need help with folding.

Enjoy the exploration.

Folding the circle in half seems intuitive or at least a well conditioned first response. So let's fold it in less than half.

Fold an off-center crease (below top left.) Line up the long part of the circumference with itself so the angle that is made on the off-center fold and new edge looks divided in half (below top right.) Turn over and fold the unfolded part to line up to the edge just formed dividing the folded circle into thirds; even up the edges and crease (bottom left.) The dark lines in the opened circle are the resulting creases (below bottom right.)

My brother asked what would happen if the fold was aligned to the smaller fold of the circumference when folding into thirds instead of the larger outside edge as pictured above. So the next step was to align the same right hand point to the smaller circumference edge dividing the new angle in half. Then turn over and line up all the straight edges and crease, again dividing the off-center folded circle into thirds. The lines (below) show a symmetry of folding the same end point to both the larger and smaller parts of the circumference, just as if we folded form both ends exactly the same to the large section. One point of crossing is both a right and left handed fold just by turning the circle over and doing the same thing. Orientation is an important and curious factor.

There are two diameters crossing at the center of the circle that intersect at two places with the first off-center fold. This forms two off-center points and five chords with two triangles on the first folded line where one point of one triangle is to the center of the circle.

Using information from the folds we can make parallel creases by accordion folding in all three directions forming an equilateral triangle grid (below left.) Using the triangle pointed to the center for position we can then fill in all triangles of the same orientation to see better how this triangle grid lines up with the circle (below right.)

As we see the grid does not line up with the circle.

We have previously established that folding the circle is about alignment, not about the center. Here we have a triangular grid centered to the circle without alignment. We can see the consistent symmetry of the grid and that a point of intersection is to the center of the circle but there is no true relationship of the dividing grid to the circumference. The first fold was arbitrarily off-center and still is.

Below are two more arbitrary off-center folds showing the same equilateral triangular grid to a different scale, depending on where the crease is off-center. A grid developed from an off-center fold will never be aligned to the circle.

Alignment comes from the concentric nature of the circle not from the triangulation or regularity of a grid. For this reason no polygon or polyhedron can be whole and will never reveal as much information as the circle to itself. Adherence to alignment between the furthest out and furthest in boundary of the circle reveals an order that far exceeds all other relationships since the movement in both directions is for all practical purposes infinite. When we start out misaligned it is sometimes difficult to discern when the reference is less than a circle. The accuracy of alignment with that first move within the established boundary has everything to do with determining subsequent development.

Let's see how it works folding the 4-8 symmetry. The process of folding is the same, the proportions are different.

The 4-8 symmetry shows one diameter and a change in proportions of triangles (above left.) There is no long or short circumference in the division, the folding is the same from both ends of first fold. Again there is information to accordion fold the right angle triangle grid matrix. Another diameter can be located by lining points of intersection perpendicular to the center crease which will place the center of the circle, thought it is not formed to this grid level. As before the grid is out of alignment with the circle boundary.

Folding the off-center crease to a 5-10 symmetry shows five chords where two are diameters. Again folding on both sides of the off-center line (below top left) show again differently proportioned triangles. By folding the triangle grid we see a very different division of creases (below top right.) By coloring in the triangles of the same orientation shows an out of alignment to the circle boundary.

We have seen folding the circle in half reveals alignment (previous post; Sept 20, #2 and Oct 19, #3.) Folding the half folded circle into thirds consistently forms three equally spaced diameters. This happens with the same consistency but with different proportions that correspond to the 4-8 and 5-10 symmetry. Let's look at the grid from folding the circle in half and how that is different from what we have just seen with developing the grid from the off-center folds.

Fold in half and then fold into thirds showing three diameters. From this folded information we can fold the equilateral triangle grid similar to what we did previously, only in this case we are folding all possible combinations of touching points and then creasing. This reveals the triangle grid showing the enfolded hexagon star and three more diameters where there is exact alignment to the circle

The circle is equally divided into twelve sectors. There are three sets of three parallel creases and three diameters. Twelve lines in the grid reflect a pattern formation of three. This is not arbitrary, there is self-organizing and order that comes from alignment of the inner and outer boundary of the circle. This alignment is critical for the full functioning continuation of folding the circle (below bottom.)

This is not so much about the triangular matrix or symmetry as it is unity of the circle. There is an order of proportional organization, balance and symmetrical arrangement of finite parts that only occurs with alignment. Any fold out of alignment and off-center will always reveal a consistent grid of triangles from which the center of the circle can be located. But there is only one way to align the circle to gain full benefit from the inclusive nature and potential of the circle and that is to fold the circle in half.

This one fold aligns the circle, in a proportional ratio of 1:2 that is directive for all that follows. Sequential development reveals three possibilities of symmetries; 3:6, 4:8 and 5:10 (post Oct 19, #3.) The one Whole two parts ratio sets the structural pattern of three, a triunity that happens first with the compression of the sphere to a circle form, thus reflected in the first fold. Consistent developed from that first fold is a true expression of circle/sphere unity. This embraces all limited expressions from off-center folding and truncations into polygons.

The off-center folded grid, centered but not aligned, can be brought into alignment when you cut back, or move out to boundary towards the concentric nature of circles corresponding to the local center and the primary points of intersection of the grid. Even with alignment missing, there is always information to get realigned.

This in-the-hand demonstration seems to have direct implications of how we might think about our off-centeredness and misalignment and how we might bring principled organization and balanced to the symmetrical and infinitely concentric proportional nature of life.

This picture is not unlike the disorder of the planet we are living on. It is how I find my couch at the end of the year, with not even a place to sit down. This coming year I plan to find ways towards aligning my personal off-centeredness (a self-centered perspective) to a more inner and outward concentric balance in this most extraordinary existence.