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Locating New Polyhedral Systems The first new polyhedron consists simply of one octahedron with a tetrahedron on two opposite sides (Fig. 9-8). The result, a The next candidate, the VE, is too familiar to warrant further description at this point. Twelve cuboctahedral vertices can be located around every point in the IVM, thereby embracing eight tetrahedra and six half octahedra.
Furthermore, higher-frequency versions of any of the above polyhedra—tetrahedron, octahedron, rhombohedron, and VE—can be easily located within the matrix, thus establishing the foundation for truncated polyhedra. Subtract a half octahedron from each of the six corners of a three-frequency octahedron to yield a symmetrical "truncated octahedron" with fourteen faces: six squares and eight regular hexagons (Fig. 9-9b). A "truncated tetrahedron," with four hexagons and four triangles, is left after a single-frequency tetrahedron is removed from each corner of a three-frequency tetrahedron (Fig. 9-9a). The same procedure applies to higher-frequency versions of any of the above shapes, as well as further truncations of truncated shapes. Such transformations can be plotted indefinitely. Duality and the IVM
We now introduce a new level of flexibility with the addition of a new set of vertices—in the exact center of each tetrahedron and octahedron. These new vertices are connected to the original IVM vertices, thereby introducing radial vectors into each of the original cells. Figure 9-10 shows a single octahedron and tetrahedron with these central nodes.
Angles The central angles of a tetrahedron is approximately 109° 28', exactly equal to the octahedron's dihedral angle. No longer surprised by such relationships, we go on to look inside the octahedron and note its central angle of 90 degrees, which is the surface angle of a cube. The octahedron's three body diagonals, or six radil, thus form the Polyhedra We now observe considerable expansion of our inventory of generated shapes. Starting with the most familiar, we isolate the minimum cube. Formed by a single tetrahedron embraced by four neighboring eighth-octahedral pyramids, or
Next, we embrace a single octahedron by eight quarter tetrahedra, thereby outlining the rhombic dodecahedron, whose twelve diamond faces have obtuse angles of 109° 28' and acute angles of 70° 32'—generated by the tetrahedral central angle and two adjacent axial angles, respectively. Its eight three-valent vertices are the centers of embracing tetrahedra, while its six four-valent vertices are the original octahedron vertices (Fig. 9-12). Once again, we observe the relationship of duality between the VE and rhombic dodecahedron. The former has fourteen faces (six four-sided and eight three-sided) corresponding to the four-valent vertices and three-valent vertices of the latter. Likewise, the twelve four-valent VE vertices line up with the twelve rhombic faces. (Refer to Fig. 4-12.)
Domain The duality between VE and rhombic dodecahedron illustrates the relationship of duality and domain. Having already seen that spheres in closest packing outline the vertices of the VE, we now turn our attention to the We now have an experiential basis for the VE—rhombic-dodecahedron duality. Twelve vectors emanate from any point in the IVM, locating the Returning to the IVM, we observe that four rhombic dodecahedra, or "spherics," come together at the center of each tetrahedron, such that the tetrahedron central angle becomes the obtuse surface angle of the spheric. In the same way, eight cubes meet at the center of each octahedron, as allowed by the shared 90-degree surface and central angles, respectively. For clarity, we shall refer to the new network, interconnecting the centers of all octahedral and tetrahedral cells, as IVM', and we can draw the following conclusion. If the vertices of a given polyhedron are located in the IVM, then that system's dual will be outlined by the IVM', and vice versa. Similarly, if a polyhedron is centered on a vertex of the IVM, its dual will be centered on a vertex in IVM'. For example, we recall our first case of duality: the vertices of the octahedron's dual, the cube, are supplied by octahedron centers, which are nodes of IVM'. This discovery leads us to another assumption. As truncation of our familiar polyhedra yields shapes contained within the IVM, the dual operation, stellation, should produce polyhedra outlined by IVM'. The assumption is valid: the additional IVM' vertices provide the loci for the vertices of stellated versions of these basic shapes. Actually, this observation is not new, for we have already seen that quarter-tetrahedral pyramids affixed to octahedron faces produce Fuller's spheric, or, in other words, that a degenerately stellated octahedron becomes a rhombic dodecahedron. The three-valent vertices of this diamond faceted shape are tetrahedron centers, by definition nodes of IVM'. |

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