BUCKYVERSE

457.00 Great Circles of Icosahedron

457.01 Three Sets of Axes of Spin: The icosahedron has three unique symmetric sets of axes of spin. It provides 20 triangular faces, 12 vertexes, and 30 edges. These three symmetrically interpatterned topological aspects faces, vertexes, and mid-edgesprovide three sets of axes of symmetric spin to generate the spherical icosahedron projection’s grid of 31 great circles.

457.02 The icosahedron has the highest number of identical and symmetric exterior triangular facets of all the symmetrical polyhedra defined by great circles.

457.10 When we interconnect the centers of area of the 20 triangular faces of the icosahedron with the centers of area of their diametrically opposite faces, we are provided with 10 axes of spin. We can spin the icosahedron on any one of these 10 axes to produce 10 equators of spin. These axes generate the set of 10 great-circle equators of the icosahedron. We may also interconnect the midpoints of the 30 edges of the icosahedron in 15 sets of diametrically opposite pairs. These axes generate the 15 great-circle equators of the icosahedron. These two sets of 10 and 15 great circles correspond to the 25 great circles of the vector equilibrium.

457.20 Six Great Circles of Icosahedron: When we interconnect the 12 vertexes of the icosahedron in pairs of diametric opposites, we are provided with six axes of spin. These axes generate the six great-circle equators of the icosahedron. The six great circles of the icosahedron go from mid-edge to mid-edge of the icosahedron’s triangular faces, and they do not go through any of its vertexes.

457.21 The icosahedron’s set of six great circles is unique among all the seven axes of symmetry (see Sec. 1040), which include both the 25 great circles of the vector equilibrium and the 31 great circles of the icosahedron. It is the only set that goes through none of the 12 vertexes of either the vector equilibrium or the icosahedron. In assiduously and most geometrically avoiding even remote contact with any of the vertexes, they represent a new behavior of great circles.

457.22 The 12 vertexes in their “in-phase” state in the vector equilibria or in their “out-of-phase” state in the icosahedra constitute all the 12 points of possible tangency of any one sphere of a closest-packed aggregate with another sphere, and therefore these 12 points are the only ones by which energy might pass to cross over into the next spheres of closest packing, thus to travel their distance from here to there. The six great circles of the icosahedron are the only ones not to go through the potential intertangency points of the closest-packed unit radius spheres, ergo energy shunted on to the six icosahedron great circles becomes locked into local holding patterns, which is not dissimilar to the electron charge behaviors.

457.30 Axes of Symmetry of Icosahedron: We have now described altogether the 10 great circles generated by the 10 axes of symmetry occurring between the centers of area of the triangular faces; plus 15 axes from the midpoints of the edges; plus six axes from the vertexes. 10+15 + 6 = 31. There is a total of 31 great circles of the icosahedron.

457.40 Spherical Polyhedra in Icosahedral System: The 31 great circles of the spherical icosahedron provide spherical edges for three other polyhedra in addition to the icosahedron: the rhombic triacontrahedron, the octahedron, and the pentagonal dodecahedron. The edges of the spherical icosahedron are shown in heavy lines in the illustration.

457.41 The spherical rhombic triacontrahedron is composed of 30 spherical rhombic diamond faces.

457.42 The spherical octahedron is composed of eight spherical triangles.

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Fig. 457.30A Axes of Rotation of Icosahedron:

  1. The rotation of the icosahedron on axes through midpoints of opposite edges define 15 great-circle planes.

  2. The rotation of the icosahedron on axes through opposite vertexes define six equatorial great-circle planes, none of which pass through any vertexes.

  3. The rotation of the icosahedron on axes through the centers of opposite faces define ten equatorial great-circle planes, which do not pass through any vertexes.

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Fig. 457.30B Projection of 31 Great-Circle Planes in Icosahedron System: The complete icosahedron system of 31 great-circle planes shown with the planar icosahedron as well as true circles on a sphere (6 + 10 + 15 = 31). The heavy lines show the edges of the original 20-faced icosahedron.

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Fig. 457.40 Definition of Spherical Polyhedra in 31-Great-Circle Icosahedron System: The 31 great circles of the spherical icosahedron provide spherical edges for three other polyhedra in addition to the icosahedron itself, whose edges are shown as heavy lines. The shading indicates a typical face, as follows:

  1. The rhombic triacontrahedron with 30 spherical rhombic faces, each consisting of four basic, least-common-denominator triangles.

  2. The octahedron with 15 basic, least-common-denominator spherical triangles.

  3. The pentagonal dodecahedron with ten basic, least-common-denominator spherical triangles.

  1. The spherical pentagonal dodecahedron is composed of 12 spherical pentagons.