The Law of Consolidation of the Tetrahedral–Octahedral Lattice: Growth by Lines, an Eightfold Choice, and Clans as a Record of Growth

The tetrahedral–octahedral lattice (TOT) is self-similar: a regular octahedron of edge 2a consists of six octahedra and eight tetrahedra of edge a, and a regular tetrahedron of edge 2a of four tetrahedra and one octahedron. Read upwards, this is the law of consolidation of the lattice, which several earlier notes of this series use without stating it on its own. We state it here and separate three levels of result. Proved: the law itself, its reduction to the halving of two segment lengths, and an eightfold choice in consolidation: a given lattice of edge a admits exactly eight consolidations, eight shifts of the coarser level relative to it, one for each class of the node lattice modulo its double. Consequently the genealogy of a cell cannot be read from the lattice alone. The eight shifts are related by the translation symmetry of the lattice, so without a physically marked origin they are not eight physically different states. Observed numerically: growth by lines. Lines drawn through the nodes of one octahedron along the six edge directions meet, at the first step, in exactly eight new nodes, the tips of Kepler's stella octangula, whose two large tetrahedra have edge 2a; repeating the step fills the node lattice inside the region checked, and no intersection falls off the lattice. Hypothesis: if consolidation is the primary process, so that the structure grew rather than was divided, each step selected one of the eight choices, and the clans are a record of this history rather than a label. Throughout, there is one spatial structure, described at different levels, not separate lattices for different scales. Whether the record has physical consequences is open.