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Folding any piece of paper by touching two points together reveals a tetrahedron pattern. All circles inherently carry the properties of this pattern in that first fold, which can then be reformed in countless ways. The tetrahedron is principle because it is first in alignment.
The following pictures show a variety of many possibly variations of tetrahedra formed to an equilateral triangle grid using only four 9” paper plate circles each. They are all folded the same way, reconfigured differently, and joined to the same pattern in forming a wide range of designs.
The forming of the regular “solid” tetrahedron formed from one circle using the same folds can be seen at; http://wholemovement.com/how-to-fold-circles. Higher frequency folding of the circle grid can be seen at; http://wholemovement.com/blog/item/97-unity-origami.
Below top) shows the “empty” circle representation we all carry in our minds.
Above bottom left) shows the 8-frequency grid (much like an octave in music) inherent to the circle, though not seen until folded. Moving from right to left we see each folded stage within the context of the inherent non-changing grid. This grid can be taken to much higher frequencies through the same ordered process of touching points and folding.
Above) a traditional tetrahedron solid from four circles with the circumference folded to the inside. The four vertex points are the only connections to the folded-in circumference of each circle.
The properties of the tetrahedron are 4 points in space defining 6 edge relationships revealing 4 triangle planes. The 4 points and 6 realtionships between points is sufficient to fully define a tetrahedron; a number 10 (one circle one diameter, the first fold.)
Above) the same tetrahedron with the circumference folded to the outside.
Above) the same tetrahedron where one sector of each circle is folded out leaving the remaining circumference folded in.
The following pictures are a selection of tetrahedra done over a few years using 4 circles folded to the same 8-frequency triangular grid and reforming the creases, joining in different ways to a single pattern. My intention in showing so many pictures of the same pattern is to get across the idea that forming is ongoing and there is more to the circle than we imagine.
Below) exploring reconfiguration of 4 circles in a variety of inside and outside combinations.
Bobbie pins and tape hold them together. Others are held together using glue.
Above 3) the 8-frequency folded grid in three closely related variations.
Below) 4 circles joined in a tetrahedron pattern forming an icosahedron. Some surprisingly aren’t in recognizable tetrahedron form or symmetry, yet are the same pattern from same grid.
Above right) holes punched in each corner allowing it to be twisty-tied together.
Below) random selection of variations in forming tetrahedron pattern.
Above 2) two variations in star designs from tetrahedron pattern. The three projections on each of the four sides are reconfigured from the same grid. There are slight differences in reformation showing variations.
Above) another type of star with slight variations in configuration.
Above right) the model to the inside is folded using four 22" diameter circles and the one on the outside of the same design is made from four 9" paper plates.
The tetrahedron is not the only possibilities in joining triangles nor do these even come close to the number of reformations possible from this one of three fundamental triangle grids inherent in the circle. This gives you some idea of the tremendous breadth of design possibilities by folding and joining circles and indicates extraordinary possibilities in higher frequency grid folding, reconfiguring, and joining multiples. Any of these objects can well serve as a single unit for developing greater complex of fractal systems in different symmetries.
Imagine all of the above models folded from the same 4 circles where each of the four circles is a multiple of one circle folded to one grid. This is all a transformational process of a single circle without the limitation of sequencing locations through time.