986.420 Min-max Limit Hierarchy of Pre-time-size Allspace-fillers
986.421 Of all the allspace-filling module components, the simplest are the three-quanta-module Mites, consisting of two A Quanta Modules (one A positive and one A negative) and of one B Quanta Module (which may be either positive or negative). Thus a Mite can be positive or negative, depending on the sign of its B Quanta Module. The Mites are not only themselves tetrahedra (the minimum-sided polyhedra), but they are also the simplest minimum-limit case of allspace-filling polyhedra of Universe, since they consist of two energy-conserving A Quanta Modules and one equivolume energy dispersing B Quanta Module. The energy conservation of the A Quanta Module is provided geometrically by its tetrahedral form, four different right-triangled facets being all foldable from one unique flat-out whole triangle (Fig. 913.01), which triangle’s boundary edges have reflective properties that bounce around internally to those triangles to produce similar smaller triangles: Ergo, the A Quanta Module acts as a local energy holder. The B Quanta Module is not foldable out of one whole triangle, and energies bouncing around within it tend to escape. The B Quanta Module acts as a local energy dispenser. (See Fig. 986.421.)
Figure 23: Fig. 986.421 A and B Quanta Modules: The top drawings present plane nets for the modules with edge lengths of the A Modules ratioed to the tetra edge and edge lengths of the B Modules ratioed to the octa edge. The middle drawings illustrate the angles and foldability. The bottom drawings show the folded assembly and their relation to each other. Tetra edge = octa edge. (Compare Figs. 913.01 and 916.01.)Figure 23: Fig. 986.421 A and B Quanta Modules: The top drawings present plane nets for the modules with edge lengths of the A Modules ratioed to the tetra edge and edge lengths of the B Modules ratioed to the octa edge. The middle drawings illustrate the angles and foldability. The bottom drawings show the folded assembly and their relation to each other. Tetra edge = octa edge. (Compare Figs. 913.01 and 916.01.)
986.422 Mite: The simplest allspace-filler is the Mite (see Secs. 953 and 986.418). The positive Mite consists of 1 A + mod, 1 A - mod, and 1 B + mod; the negative Mite consists of 1 A +mod, 1 A mod, and 1 B-mod. Sum-total number of modules
MITE( (See color plate 17.)
image
986.423 Around the four corners of the tetrahedral Mites are three right triangles. Two of them are similar right triangles with differently angled acute corners, and the third right triangle around that omni-right-angled corner is an isosceles.
986.424 The tetrahedral Mites may be inter-edge-bonded to fill allspace, but only because the spaces between them are inadvertent capturings of Mite shaped vacancies. Positive Mite inter-edge assemblies produce negative Mite vacancies, and vice versa. The minimum-limit case always provides inadvertent entry into the Negative Universe. Sum-total number of modules is1\sfrac{1}{2}
986.425 Mites can also fill allspace by inter-face-bonding one positive and one negative Mite to produce the Syte. This trivalent interface bonding requires twice as many Mites as are needed for bivalent inter-edge-bonding. Total number of modules is
986.426 Syte: The next simplest allspace-filler is the Syte. (See Sec. 953.40.) Each Syte consists of one of only three alternate ways of face-bonding two Mites to form an allspace-filling polyhedron, consisting of 2 A + mods, 2 A-mods, 1 B+mod, and 1 B-mod. Sum-total number of modules
986.427 Two of the three alternate ways of combining two Mites produce tetrahedral Sytes of one kind:
Figure 24: BITE (See color plate 17) & RITE (See color plate 19)Figure 24: BITE (See color plate 17) & RITE (See color plate 19)
while the third alternate method of combining will produce a hexahedral Syte.
Figure 25: LITE (See color plate 18)Figure 25: LITE (See color plate 18)Figure 26: Fig. 986.427 Bite, Rite, Lite:
986.428 Kite: The next simplest allspace-filler is the Kite. Kites are pentahedra or half-octahedra or half-Couplers, each consisting of one of the only two alternate ways of combining two Sytes to produce two differently shaped pentahedra, the Kate and the Kat, each of 4 A + mods, 4 A mods, 2 B + mods, and 2 B - mods. Sum-total number of modules
986.429 Two Sytes combine to produce two Kites as
KATE (See color plate 20) KAT (See color plate 21)
986.430 Octet: The next simplest allspace-filler is the Octet, a hexahedron consisting of three Sytesergo, 6 A + mods, 6 A - mods, 3 B + mods, and 3 B - mods. Sum-total number of modules
Figure 28: Fig. 986.40 OCTET (See color plate 22)Figure 28: Fig. 986.40 OCTET (See color plate 22)
986.431 Coupler: The next simplest allspace-filler is the Coupler, the asymmetric octahedron. (See Secs. 954.20–.70.) The Coupler consists of two Kites ergo, 8 A + mods, 8 A - mods, 4 B + mods, and 4 B - mods. Sum-total number of modules
Figure 29: Fig. 986.431 COUPLER (See color plate 23)Figure 29: Fig. 986.431 COUPLER (See color plate 23)
986.432 Cube: The next simplest allspace-filler is the Cube, consisting of four Octetsergo, 24 A + mods, 24 A - mods, 12 B + mods, and 12 B - mods. Sum-total number of modules
Figure 30: Fig. 986.432 CUBE (See color plate 24)Figure 30: Fig. 986.432 CUBE (See color plate 24)
986.433 Rhombic Dodecahedron: The next and last of the hierarchy of primitive allspace-fillers is the rhombic dodecahedron. The rhombic dodecahedron is the domain of a sphere (see Sec. 981.13). The rhombic dodecahedron consists of 12 Kitesergo, 48/ + mods, 48/ - mods, 24 B + mods, and 24 B mods. Sum-total number of modules
Figure 31: Fig. 986.433 RHOMBIC DODECAHEDRON (See color plate 25)Figure 31: Fig. 986.433 RHOMBIC DODECAHEDRON (See color plate 25)
986.434 This is the limit set of simplest allspace-fillers associable within one nuclear domain of closest-packed spheres and their respective interstitial spaces. There are other allspace-fillers that occur in time-size multiplications of nuclear domains, as for instance the tetrakaidecahedron. (Compare Sec. 950.12.)