One Double Pyramid, Two Splits: Cut Across Its Square It Builds Only Cubes, Cut Along a Diagonal It Builds the Rhombic Dodecahedron's Pyramids Freely

Cut a cube's corners with a growing octahedron and a second, smaller solid appears in the gaps; together the two fill space as caesium and chloride ions do in a salt crystal. We ask whether the two pieces could fill space in any other way, as the octet's pieces can be stacked in layers. They cannot: along the whole walk the pieces lock into their crystal and nowhere else, because at every edge the angles leave only one choice and no edge can rest on a face. The pyramids of the rhombic dodecahedron are free instead: on every face of a cube grid, one of two diagonals can be chosen. Abstract. Paper #90 (Proposition 6) placed the sections Pμ = C ∩ μC* and Qμ = C ∩ (3 − μ)C* of a cube and its polar octahedron in caesium chloride (CsCl) pairs that fill space for every μ, from the cuboctahedron and octahedron through DICTO's Unborn Axes and Icosaxes, Kelvin's cell and the Golden Cut Cubes to the cube; #51 gave a second walk, T(t) and O(t), through the octet from Kepler's rhombic dodecahedron to the cube. Main theorem. (A) For every μ in (1, 3), every tiling by copies of Pμ and Qμ, face to face or not, is the CsCl tiling; (B) for every t in (0, 1/3) ∪ (1/3, 1), every tiling by T(t) and O(t) is #51's. No pair on either walk has a layer variant: #92's freedom lives only at the octet, the sliding freedom only at the cube; Kelvin's cells and Kepler's rhombic dodecahedra each tile in one way only. (C) The rhombic dodecahedron's centre pyramids are free: their face-to-face tilings correspond one to one with the choices of a diagonal on every square face of the cube lattice, so #77's law that face pyramids rebuild their solid fails for the rhombic dodecahedron. (D) The cube's centre pyramids tile face to face only as cubes (#77); otherwise their lateral edges are shared end to end by three, and no finite list describes the tilings. The proofs use the angle sums round edges and one lemma on faces; 67 exact checks.