Clan Topology of the Tetrahedral-Octahedral Lattice: Closed Forms, the Exact 3/4 Law and the 2:1 Attractor

The tetrahedral-octahedral lattice (TOT; the alternated cubic honeycomb, octet truss) is self-similar: halving the edge splits every octahedron into 6 octahedra and 8 tetrahedra, and every tetrahedron into 4 tetrahedra and 1 octahedron. The rules themselves are classical (they underlie, for example, octahedral-tetrahedral mesh refinement). Here we follow not the number of cells but their lineage — the sequence of cell types through which a cell descends from a root cell — and call the set of cells sharing a lineage a clan. We give closed forms for all counts: starting from one octahedron, generation n contains On = (2·8ⁿ + 2ⁿ)/3 octahedra and Tn = 4(8ⁿ − 2ⁿ)/3 tetrahedra, distributed over 2ⁿ clans. The pure octahedral clan O←O←…←O holds exactly 6ⁿ cells and occupies exactly (3/4)ⁿ of the volume — a geometric law, not an asymptotic one. Weighting lineages by volume turns clan topology into a two-state Markov chain with transition probabilities 3/4, 1/4, 1/2, 1/2, whose stationary distribution is 2/3 octahedral and 1/3 tetrahedral volume — the volume form of the 2:1 tetrahedron-to-octahedron ratio — and whose entropy rate is 0.8742 bit per generation: of the 2ⁿ clans, effectively only 1.833ⁿ carry the volume. The division matrix [[6, 1], [8, 4]] has the integer eigenvalues 8 and 2; its dominant eigenvector makes T = 2·O an attractor of division from any starting cell. Book, interactive 3D model of the lattice and the open laboratory: https://theoryof.space