One World for the Platonic Solids

Only the cube fills space alone (the regular tetrahedron does not, against Aristotle: Lagarias and Zong 2012; Senechal 1981), and among pairs of regular solids only the tetrahedron with the octahedron does it (Conway and Torquato 2006; Conway, Jiao and Torquato 2011). #47 showed that, with a small kit of connecting pieces, every mix of the five Platonic solids fills space, built from four separate layers. Here the solids are allowed their own sizes, and something simpler appears: every solid fits inside a skin that turns it into a dodecahedron, and then every mix lives on one cubic grid, with no layers at all. Theorem 1. With the dodecahedron of edge 1, the cube, tetrahedron and octahedron of edge φ and the icosahedron of edge √2/φ, each of the 31 mixes of the five Platonic solids fills space in two worlds: the cubic grid of edge φ (dodecahedra, Dogstars, cubes) and one octet of edge φ (tetrahedra, octahedra, and icosahedra in Pacioli's position with six Admiral Caps each). The two meet in one flat seam, face to face, and the connectors form a kit of six. Theorem 2. Each of the five Platonic solids, and the cuboctahedron, fits inside a skin of congruent pieces that makes it exactly a dodecahedron of edge 1 (Table 1); the rhombic dodecahedron is a cube of edge φ with six square pyramids. Hence every mix of these seven solids (127 mixes) fills space in one world: the alternating cubes of the cubic grid of edge φ hold dodecahedral cells and rhombic dodecahedra, the cells between are Dogstar-type remainders, and the edging is X, XR and whole cubes. No seam remains. Theorem 3 (the flat Karma layer). The plane perpendicular to a three-fold axis at level φ/(2√3) above a layer of cell centres cuts every Dogstar it crosses so that the part below is exactly a CLEO, its apex a dodecahedron corner. The octet of edge φ/√2 above the plane, with Pacioli's icosahedra (edge 1/φ, the mirror form) in its octahedra, puts an icosahedron face exactly on every CLEO base. Volumes and skins in exact arithmetic with √5; every filling checked by random points (all 31 and all 127 mixes). Follows #47 Every Mix of the Platonic Solids Fills Space (DICTO DOI 10.5281/zenodo.23551326). DICTO: ideas, names, constructions and direction. Computation, verification and drafting with Claude (Anthropic).