459.00 Great Circle Foldabilities of Icosahedron
459.01 The great circles of the icosahedron can be folded out of circular discs of paper by three different methods: (a) 15 multi-bow ties of four tetrahedra each; (b) six pentagonal bow ties; and (c) 10 multi-bow ties. Each method defines certain of the surface arcs and central angles of the icosahedron’s great circle system, but all three methods taken together do not define all of the surface arcs and central angles of the icosahedron’s three sets of axis of spin (see drawings section).
459.02 The 15 great circles of the icosahedron can be folded into multi-bowties of four tetrahedra each. Four times 15 equals 60, which is half the number of triangles on the sphere. Sixty additional triangles inadvertently appear, revealing the 120 identical spherical triangles which are the maximum number of like units which may be used to subdivide the sphere.
459.03 The six great circles of the icosahedron can be folded from central angles of 36 degrees each to form six pentagonal bow ties. (See illustration 458.12.)
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Fig. 458.12 Folding of Great Circles into the Icosahedron System:
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The 15 great circles of the icosahedron folded into “multi-bow-ties” consisting of four tetrahedrons each. Four times 15 equals 60, which is 16 the number of triangles on the sphere. Sixty additional triangles inadvertently appear, revealing the 120 identical (although right- and left-handed) spherical triangles, which are the maximum number of like units that may be used to subdivide the sphere.
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The six great-circle icosahedron system created from six pentagonal “bow-ties.”