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Appendix D: Special Properties of the Tetrahedron (1) Minimum system: the tetrahedron is the first case of insideness and outsideness. (2) The regular tetrahedron fits inside the cube, with its edges providing the diagonals across the cube's six faces, and thereby supplying the six supporting struts needed to stabilize the otherwise unstable cube. Furthermore, two intersecting regular tetrahedra outline all eight vertices of the cube. (3) The tetrahedron is unique in being its own dual. (4) The six edges of the regular tetrahedron are parallel to the six intersecting vectors that define the vector equilibrium. (5) Similarly, the four faces of the regular tetrahedron are the same four planes of symmetry inherent in the vector equilibrium and in cubic closepacking of spheres. The tetrahedron is thus at the root of an omnisymmetrical space-filling vector matrix, or isotropic vector matrix. (6) When the volume of a tetrahedron is specified as one unit, other ordered polyhedra are found to have precise whole-number volume ratios, as opposed to the cumbersome and often irrational quantities generated by employing the cube as the unit of volume. Furthermore, the tetrahedron has the most surface area per unit of volume. (7) Of all polyhedra, the tetrahedron has the greatest resistance to an applied load. It is the only system that cannot "dimple"; reacting to an external force, a tetrahedron must either remain unchanged or turn completely "inside out." (8) The surface angles of the tetrahedron add up to 720 degrees, which is the "angular takeout" inherent in all closed systems. (9) The tetrahedron is the starting point, or "whole system," in Fuller's "Cosmic Hierarchy," and as such contains the axes of symmetry that characterize all the polyhedra of the isotropic vector matrix, or face-centered cubic symmetry in crystallography. (10) Packing spheres together requires a minimum of four balls, to produce a stable arrangment, automatically forrning a regular tetrahedron. The centers of the four spheres define the tetrahedral vertices. In Fuller's words, "four balls lock." (11) It has been demonstrated that many unstable polyhedra can be folded into tetrahedra, as in the jitterbug transformation. (12) Fuller refers to the six edges of a tetrahedron as one "quantum" of structure, because the number of edges in regular, semiregular, and high-frequency geodesic polyhedra is always a multiple of six. |

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