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Just as we needed at least four vertices to make a system, "four is also the minimum number of great circles that may be folded into local bow ties and fastened comer-to-comer to make the whole sphere..." (455.04). An interesting characteristic of this minimum model is that the sum of the areas of its four separate circles, which fold up to create the model, is equal to the surface area of the sphere they define, or 4pR ². A system with less than four great circles cannot be directly constructed out of that number of bowties, but in some cases the pattern can be simulated using more than the prescribed number of paper disks and doubling the polyhedral edges:
You cannot make a spherical octahedron or a spherical tetrahedron by itself... 109°28' of angle cannot be broken up into 360-degree-totalling spherical increments... . (842.02-3)
A spherical tetrahedron can only be created out of "foldable" paper circles by constructing the six-great-circle spherical cube, which produces two intersecting tetrahedra. See Fig. 14-16: six great circles will submit to a bowtie construction, similar to the four-great-circle model described above:
842.04 Nor can we project the spherical octahedron by folding three great circles. The only way... is by making six great circles with all the edges double...
In this construction, six circles are folded in half and then the resulting semicirdes are each folded in the middle at a right angle. The six bent semicircle pancakes (rather than three open bowties) are then simply pinned together to simulate the three great circles of the octahedron (Fig. 14-17).
Fuller attributes these discoveries to a "basic cosmic sixness":
There is a basic cosmic sixness of the two sets of tetrahedra in the vector equilibrium. There is a basic cosmic sixness also in an octahedron minimally-great-circle-produced of six great circles; you can see only three because they are doubled up. And there are also six great circles occurring in the icosahedron. Ml these are foldable... . This sixness corresponds to our six quanta: our six vectors that make one quantum. (842.05-6)
Both six-great-circle patterns can be constructed out of foldable circles. The star-tetrahedron version involves six bowties with two right isosceles triangles on the surface (60° -60° -90°) created by central angles of 70°32' (a) and 54°44' (b), as labeled in Figure 14-16. The six bowties are pinned together at the seams to create the 24 surface triangles of the spherical cube and star tetrahedron. And it works: the illusion of six continuous circles is maintained.
The icosahedral six great circles fold into pentagonal bowties. Fold lines divide each circle into ten equal slices of pie, carving out the 36-degree central angles. Each circle is then pinched together at one point to form a double-pentagon figure-eight. Just like the triangular predecessors, each circle folds into a "local circuit" and is connected to other local circuits to create the illusion of continuous great circles (Fig. 14-18).
Fuller experiments with the "foldability" of considerably more intricate patterns. Larger numbers of great circles intersect more frequently, and arc segments have correspondingly smaller central angles. The number of folds and the precision required for each of the tiny irregular angles make these higher-frequency models extraordinarily difficult to build, and even more difficult to visualize without a model. Each of the ten great circles of the icosahedron can fold into winding chains of six narrow tetrahedra, which then interlink to reproduce the ten-great-circle pattern. (2) Fuller claims that the fifteen-great-circle pattern (outlining the 120 LCD units) can be reconstructed with fifteen four-tetrahedron chains. (3) While a paper circle will fold into four consecutive LCD tetrahedra, it is not clear how they fit together to recreate the whole sphere. We note however that this pattern can be easily generated by using thirty paper circles and doubling the edges. Each circle folds into four adjacent LCD tetrahedra, to form one self-contained diamond-a face of the rhombic triacontahedron (Fig. 14-19).
For the purposes of this chapter, we want to understand the basics of how foldable great circles work; further experimentation with construction paper is left to curious (and ambitious) readers. We proceed to look at Fuller's interpretation of the significance of this behavior in terms of physical reality.
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