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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.
What came up this month is numbers negotiate between 2-D and 3-D.
Euclid wrote, a point has no part. The point is whole. All parts inherently carry the whole. The point and circle are forms of unity scaled up and down beyond our limited perception of concentricity. Unity is singular; one unit is always in plural. Both inherently carry unlimited endless potential in generations of parts.
To start, DRAW a point. Draw a single regular curved line ending at the starting point. Make a second point approximately centered in the circle. Starting from the second point draw a single curved line, the same size of the first, around the first point ending at the second point. You will have two intersecting circles from two points showing four points.
Above) the two circles where the circumference is part inside to the other, both sharing the same measure. They do not have to be exact, you can draw free-hand for the same results.
Above) to FOLD a circle (mark any 2 points on the circumference, touch them and crease) two more points are generated on the circumference (end points of diameter) making four points, the same number of points as the drawing above. Two points on one circle show four points. To the right all four points have been connected with straight lines showing the 6 relationships between them (10 parts.)
Below) continue drawing on the circles by connecting all four points with straight lines, each starting and ending on the circumferences. All connections will be chords showing 6 diameters (realtionships.) There are 10 equilateral triangles, 5 pointing up and 5 in opposite direction. There are other equilateral triangles of different sizes within the two intersecting hexagons along with many other unmarked shapes and relationships between the points.
Above right) by shading one triangle in opposite orientation on the two ends reveals a rotation-slide symmetry.
Above left) a drawing of two circles with 4 triangles shaded to show a reflective symmetry on the vertical axis of both ends.
Above right) are two images of the icosahedron. One shows the solid and the other the placement of 4 triangles equally around. The properties of the drawing and the folded icosahedron are the same but with different symmetries; one is 3-6 and the other a 3-5 symmetry.
2-D as compression of 3-D opens unseen connections. 10 points are counted on the viewed side and 2 more points on the underside of the drawing, 12 points in all. There are 10 individual equilateral triangles showing and 10 underneath making 20 triangles planes. There are 19 lines (edges) on one side and 11 on the underside making 30 edges.
In the drawing are encoded the properties for the icosahedron; 12 points, 20 planes and 30 edges. When using four circles folded into a tetrahedron, opened and joinied in a tetrahedron net, then reformed to an icosahedron will show four open triangle faces (see http://wholemovement.com/how-to-fold-circles) The four shaded triangles in the drawing corresponds to the open triangles of the folded icosahedron triangles.
Below left) fold two circles to a 4-frequency grid (blog http://wholemovement.com/blog/itemlist/date/2011/2?catid=140) and join them to the arrangement of two intersecting circles. This shows two large triangles reformed and joined to show a symmetrically opening on each side through the inside space. This is one of many possible combinations by joining these two folded circles.
Above right) is a 2-D grid of two intersecting circles with shaded openings seen in the folded circles to the left.
This reveals full information in the first construction by Euclid in Proposition No.1 (proving an equilateral triangle.)
Above) folded four-frequency diameter circle. The infolded hexagon is optional in the folding.
Below) two views of the configuration from above with 16 triangle surfaces (front and back) stellated with 16 tetrahedra made from 8 circles. The 4 open triangle planes are left open.
Below) two more reconfigurations from the same circle grid are added to one end. All the triangle planes are congruent giving many options for attachments developing a variety of systems, this being only one possibility.
Below) different views as another unit is added to the opposite end extending the system.
Below) two views of an open spherical unit from four circles that is added.
Above) another two circles make an open icosahedron (previously described) that is joined to the original body of the stellated system. Because of the congruency of triangle planes from a single folded grid, reconfigurations can be attached in many different places creating a wide variety of complex systems.
Below) are different views and repositioning between two segments.
Above) there is a folded hinged unit in the system allows for some articulation in changing positions between two sections.
Above) are five basic reformations of the 4-frequency folded grid that were used to reform the units used for developing this system coming from drawing two intersecting circles. This just happens to be where my interest at the time took me. There are many hundreds of directions to explore with this kind of developing process. Possibilities are different for each folder since no two people will respond in the same way to make the same choices or reformations, yet all coming from the same folded grid.
By understanding how drawings are abstract representations that hold compressed information from 3-D configurations we can use counting to help make connections between 2-D and 3-D about what otherwise goes unnoticed, where each is held in separation from the other. Reality is not flat, nothing is separated; it is spatial and we flatten it to make a simple 2-D conceptually organized system, using numbers to give location and meaning. Numbers represent groupings where every fold is circular movement that changes the numbers. To get the most from a circle we must fold it, from numbers we must count. To get the most from nature we must pass through the experience necessary for understanding. Looking at the picture does not count as the experience