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Let's begin with a representative system, the octahedron. We interconnect polar opposites, starting with vertices, followed by mid-edge points and finally face centers. Six paired vertices are connected by three mutually perpendicular lines, the familiar XYZ axes meeting at the octahedral center of gravity (Fig. 14-3a). These axes define three orthogonal great circles, which divide the sphere's surface into eight triangular areas, or octants. Three symmetrically arranged great circles will always form the edges of a spherical octahedron (Fig. 14-2, middle). That much is straightforward.
On to edges. The octahedron's twelve edges consist of six opposing pairs, the midpoints of which can be connected by six intersecting axes (Fig. 14-3b). The same number of great circles are thereby generated, delineating another facet of the octahedron's symmetry (Fig. 14-4). Unlike the previous case, the pattern made by six great circles does not look like an octahedron. Its twenty-four right isosceles triangles (1) outline the edges of both the spherical cube and rhombic dodecahedron, as well as the edges of two intersecting spherical tetrahedra (otherwise known as the "star tetrahedron"), thus highlighting the topological relationship between these four systems. This exercise demonstrates a new aspect of "intertransformability": great circles generated by a given polyhedron often delineate the spherical edges of its symmetrical cousins.
Finally, the centers of opposite faces are joined together (Fig. 14-3c). The octahedron's four pairs of triangles define four intersecting axes, and in turn four symmetrically arrayed great circles (Fig.14-5). As the spherical edges of Fuller's vector equilibrium, this pattern is particularly significant, as will be developed below.
The sets of three, six, and four great circles can all be superimposed on one sphere to exhaustively display the "unique topological aspects" of the octahedron. With a grand total of thirteen, we have located all the great circles that correspond to the octahedron's symmetry. We can go no further.
Now consider the cube. It is quickly apparent that its total great-circle pattern will be identical to that of the octahedron. The cube's eight vertices generate the same four great circles as the octahedron's faces, its six faces correspond to the three orthogonal great circles, and the twelve edge midpoints are identically situated to those of the octahedron. Such is the result of duality. The same sets of circles are generated by different elements, and the end results are equivalent.
Next, we turn to the VE. With two kinds of faces the situation might seem more complicated; however, the above procedure still applies. The VE's eight triangles define the same four axes as the faces of the octahedron, while its squares contribute three orthogonal (XYZ) axes. Seven great circles altogether are generated by the axes of the fourteen VE faces. Twelve vertices correspond to the same six great circles as those of the octahedron (or cube) edges, and finally 24 edges spin out the unfamiliar pattern of twelve great circles. With a total of 25 great circles, the topological parameters of the VE are exhausted (Fig. 14-6).
The tetrahedron presents a slightly different situation in that its vertices do not group into polar opposites, but rather are positioned directly across from the centers of faces. However, four axes of symmetry (all going through the center of gravity) can be created by connecting the vertices with their opposite faces. In having axes of rotational symmetry that connect faces and vertices, the tetrahedron is again unique; as we recall from Chapter 4, only tetrahedra have the same number of vertices as faces. (Only the tetrahedron is its own dual.) The four great circles generated by these unorthodox axes produce (once again) the spherical vector equilibrium. This is not surprising if we recall the lesson from "multiplication by division," which first uncovered this shared symmetry between the tetrahedron, octahedron, and VE: By simply interconnecting mid-edge points, all three polyhedra were found to be inherent in the topological makeup of the starting-point tetrahedron. They share the same four axes of symmetry.
The axes of symmetry associated with the tetrahedron's six edges are more orthodox: mid-edge points of opposite edges are simply joined to reestablish the XYZ axes. Familiar by now with evidence of right angles hiding within this triangular shape, we can no longer be caught off guard by this discovery. The corresponding three great circles are the edges of the spherical octahedron, once again illustrating the depth of the octahedron-tetrahedron relationship. And now all topological aspects are used up; the tetrahedron has seven great circles, the minimum number possible for symmetrical polyhedra.
Next, we look at the maximum case. It's not easy to accept the concept of an upper limit on the number of symmetrically positioned great circles that can be imposed on a sphere; common sense suggests that we should be able to keep adding new rings indefinitely. However, we recall from Chapter 3 that all systems are polyhedralthat is, everything that divides inside from outside can be described in terms of some number (four or more) of "event complexes" and their relationshipsand from Chapter 4 that the system with the greatest number of identical regular polygons and equivalent vertices is an icosahedron. This tells us that the number of great circles allowed by the topological aspects of the icosahedron is the maximum for these symmetrical patterns.
The axes defined by the icosahedron's twelve vertices introduce six great circles. We already have a pattern with six circles (octahedron-VE, Fig. 14-4); however, this is an entirely new set, "out of phase" with the earlier group, as defined by the "jitterbug" transformation (Fig. 14-7). Outlining 12 pentagons and 20 triangles (as opposed to 24 isosceles triangles), these arcs present the most symmetrical arrangement of six great circles. When a vector equilibrium contracts into an icosahedron in the jitterbug transformation, the radiusedge-length equivalence is lost, but the distances between adjacent vertices on the surface are suddenly all equal. As an icosahedron produces the most symmetrical distribution of twelve vertices on a closed system, it follows that the same is true for their corresponding six great circles.
Next, the axes of symmetry connecting the centers of icosahedron faces generate ten great circles, while the thirty edges spin out fifteen more: 6 + 10 + 15 = 31, the total number of great circles in the limit-case pattern (Fig. 14-8). These circles provide the spherical edges of a pentagonal dodecahedron as well as those of an icosahedron, and (less predictably) also include the octahedron. These relationships will be explored below.
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