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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.