Every Mix of the Platonic Solids Fills Space

Main theorem. For every non-empty set M of the five Platonic solids, written D, I, C, T, O (dodecahedron, icosahedron, cube, tetrahedron, octahedron: the letters of DICTO), there is a tiling of space whose tiles are copies of the solids in M, all of edge 1, each solid of M used and no other Platonic solid of edge 1, together with copies of pieces from one kit of seven connectors: the half octahedron P, the tetrahedron's face pyramid Q, the same pyramid Q′ at size φ²/√2, DICTO's Admiral Caps A, the Dogstar S, and a cube of edge φ with one of Euclid's roofs taken away (X) or added (Y). Tiles need not meet face to face. The proof stacks flat slabs, one for each solid's world: dodecahedra and Dogstars on Euclid's cubes, edged with X and Y; cubes; the octet, edged with half octahedra; and icosahedra in Pacioli's position inside octahedra of edge φ²/√2, completed by the Admiral Caps. All 31 mixes are also spot-checked numerically. Corollaries: the cube alone and the tetrahedron with the octahedron need no connector, while every mix with the dodecahedron or the icosahedron, and the tetrahedron alone, the octahedron alone, cube + tetrahedron and cube + octahedron, need one; all five solids fill space together with six connectors; the six Admiral Caps of one icosahedron and one Dogstar have the same volume, φ/2. Open: whether cube + tetrahedron + octahedron need a connector, the smallest kit, the face-to-face version and three-way periodicity. Each result opens with a plain-words explanation; figures show the seven pieces and the layer cake. The question, to connect all the Platonic solids in any combination by combining the earlier work, is DICTO's idea; computation, proofs and drafting with Claude (Anthropic). More of DICTO's geometry: https://dictospheres.com