The Unreachable Face Centre of the TOT Lattice

The tetrahedral–octahedral lattice TOT [1] (Fuller's isotropic vector matrix [3]) refines into itself by deconsolidation, the inverse of the consolidation law [2]: every edge is halved. We show that the whole refinement reduces to a single one-dimensional operation — halving a segment — applied to two lengths of the lattice: the edge a and the octahedron diagonal a√2; the triangles, tetrahedra and octahedra of the next level follow. Hence every node of every level has coordinates of the form k/2ⁿ in a basis of the original lattice. Of the four natural centres of the lattice — edge midpoint, octahedron centre, tetrahedron centre and face centre — the first three become nodes at levels 1, 1 and 2, whereas the centre G of a regular triangular face never does. We prove this geometrically (G always lies inside the medial triangle) and arithmetically (1/3 is not a dyadic fraction); in the 3-adic norm of affine coordinates the nodes stay at a fixed distance from G — a 3-adic gap in the dyadic tetrahedral–octahedral hierarchy. Finally, a regular tetrahedron of the same edge, placed inside an octahedron on one of its faces, has its apex exactly at the centre of the opposite face: the centre, unreachable for the lattice, is reached only by an object inserted from outside the framework.