The law of consolidation: eight ways to grow a lattice

The preprint The Law of Consolidation of the Tetrahedral–Octahedral Lattice is on Zenodo: doi:10.5281/zenodo.23813579.

The law. A regular octahedron of edge 2a consists of six octahedra and eight tetrahedra of edge a; a tetrahedron of edge 2a, of four tetrahedra and one octahedron. Read downwards this is division. Read upwards it is consolidation: small cells assemble into larger ones, and the result is the same lattice. Earlier notes in this series used the step without stating it on its own. Here it is stated, and the results are sorted into three levels of confidence.

Proved. The law itself, and its reduction to halving two segments. And one surprising fact: a lattice of edge a admits exactly eight consolidations — eight shifts of the coarser lattice relative to it, one for each class of nodes modulo their double. So the genealogy of a cell cannot be read from the lattice alone. The eight shifts are related by translation, so without a physically marked origin they are not eight different states.

Observed numerically. Growth by lines. Draw lines through the nodes of one octahedron along its six edge directions: at the first step they meet in exactly eight new nodes — the tips of Kepler's stella octangula, whose two large tetrahedra have edge 2a. The eight split as 2 + 6: two are the stella's tetrahedra, six are corners. Repeating the step fills the node lattice in the region checked, and no intersection ever falls off the lattice.

Hypothesis. If consolidation is the primary process — the structure grew rather than was divided — then each step selected one of the eight options, and the clans introduced earlier are a record of that growth, not just a label. Throughout, there is one spatial structure described at different levels, not separate lattices for different scales.

A test idea, not yet a test. If the record has physical consequences, they might show in the octahedral (ℓ = 4) channel, for example in nuclei of spin 2 or higher. Whether it does is open.

Status: the first part is proved, the second numerically checked, the third a hypothesis. The checks script is in the Zenodo record.