f3201b

Fig. 1132.01B Composite of Vector Equilibrium and Icosahedron Great Circle Sets: This is a black-and- white version of color plate 32. The Basic Equilibrium 48 LCD triangle appears here shaded in the spherical grid. In this composite spherical matrix we see all the 25 primary vector equilibrium great circles and two sets skewed-positive and negative of the icosahedron 31 great circle sets. (31 ' 2 = 62. 62 + 25 = 87. But 14 of the 87 are redundant.) Four of the VE great circles are congruent with four of the icosa's 10-great circle set. Three of the VE great circles are congruent with three of the icosa's 15-great- circle set. Thus seven positive are redundant and seven negative are redundant. (87__14=73.) There are 73 great circles in the composite set. (See color plate 32.) This composite shows the vector equilibrium great circles and the icosahedron great circles in the two alternate ways of pumping the VE jitterbug pattern.
3.8 "While great circles are the shortest distance around spheres, a single great-circle band around a sphere will readily slide off....Not until we have three noncommonly polarized, great-circle bands providing omnitriangulation as in a spherical octahedron, do we have the great circles acting structurally to self-interstabilize their respective spherical postionings by finitely intertriangulating fixed points less than 180 degrees apart." (I, 706.20)

3.9 "There are only three possible cases of fundamental omnisymmetrical, omnitrianguläted, least-effort structural systems in nature: the tetrahedron, with three triangles at each vertexi the octahedron, with four triangles at each vertex; and the icosahedron, with five triangles at each vertex." (I, 532.41)

3.10 "Design strategies: All the calculations required for the design of geodesic domes may be derived from the three basic triangles of the three basic structural systems:

the 120 right spherical triangles of the icosahedron
the 48 right spherical triangles of the octahedron, and
the 24 right spherical triangles of the tetrahedron.
All the great-circle behaviors occurring around the whole sphere take place within just one of those three basic right triangles and repeat themselves in all others." (II, 795.08)