Table 943.00 Fig. 943.00A Fig. 943.00B 
943.00 Table: Synergetics Quanta Module Hierarchy 
950.01 The regular tetrahedron will not associate with other regular tetrahedra to fill allspace. (See Sec. 780.10 for a conceptual definition of allspace.) If we try to fill allspace with tetrahedra, we are frustrated because the tetrahedron will not fill all the voids above the triangularbased grid pattern. (See Illus. 950.31.) If we take an equilateral triangle and bisect its edges and interconnect the midpoints, we will have a "chessboard" of four equiangular triangles. If we then put three tetrahedra chessmen on the three corner triangles of the original triangle, and put a fourth tetrahedron chessman in the center triangle, we find that there is not enough room for other regular tetrahedra to be inserted in the toosteep valleys Lying between the peaks of the tetrahedra. 
950.10 SelfPacking AllspaceFilling Geometries 
Fig. 950.12 
950.12
There are eight familiar selfpacking allspacefillers:

950.20 Cubical Coordination 
950.21 Because the cube is the basic, primenumberthreeelucidating volume, and because the cube's prime volume is three, if we assess space volumetrically in terms of the cube as volumetric unity, we will exploit three times as much space as would be required by the tetrahedron employed as volumetric unity. Employing the extreme, minimum, limit case, ergo the prime structural system of Universe, the tetrahedron (see Sec. 610.20), as prime measure of efficiency in allspace filling, the arithmeticalgeometrical volume assessment of relative space occupancy of the whole hierarchy of geometrical phenomena evaluated in terms of cubes is threefold inefficient, for we are always dealing with physical experience and structural systems whose edges consist of events whose actions, reactions, and resultants comprise one basic energy vector. The cube, therefore, requires threefold the energy to structure it as compared with the tetrahedron. We thus understand why nature uses the tetrahedron as the prime unit of energy, as its energy quantum, because it is three times as efficient in every energetic aspect as its nearest symmetrical, volumetric competitor, the cube. All the physicists' experiments show that nature always employs the most energyeconomical strategies. 
950.30 Tetrahedron and Octahedron as Complementary Allspace Fillers: A and B Quanta Modules 
950.31 We may ask: What can we do to negotiate allspace filling with tetrahedra? In an isotropic vector matrix, it will be discovered that there are only two polyhedra described internally by the configuration of the interacting lines: the regular tetrahedron and the regular octahedron. (See Illus. 950.31.) 
Next Section: 951.00 