Face to Face: A New Law of Aristotle's Pyramids

Cut a Platonic solid from its centre to each face and it falls into congruent face pyramids; the icosahedron's is the Cleo-Icos (DICTO's name). Paper #44 found which sets of the five fill space when they may touch in any way. DICTO proposed the face-to-face case, where pieces meet whole face to whole face, and a search for new deltahedra among his pieces. Main theorem (new). In every face-to-face tiling of space by face pyramids, the pyramids round each apex make up their whole solid, so the tiling is a face-to-face tiling by whole Platonic solids, each cut from its centre: either the cubic tiling or a tiling by regular tetrahedra and octahedra with two of each round every edge, such as the octet. The dodecahedron's pyramid and the Cleo-Icos occur in no such tiling, and two different pyramids meet only base to base. With #44 this completes the answer for Aristotle's pyramids, loose or face to face; face to face, the pyramids can only rebuild their own solids. The proof needs no computer. Search (negative). Gluing DICTO's pieces face to face, up to six pieces and 5,143,972 different clusters, gives only two solids with equilateral faces: the regular tetrahedron and the pentagonal bipyramid J13, from four Mosseri–Sadoc tiles t1 and two Hound Tooth (t2); the J13 is two pentagonal pyramids t1 + t2 + t1 (Koca, Koc, Koca and Al-Siyabi 2020) base to base, so neither is new. The search is a screening, complete to two pieces and restricted beyond. Extends Chapter 3 of the book Alicia in Wonderland. View in app: Kaleidohedra, Aristotle's Pyramid. Computation and proof with Claude (Anthropic). More at dictospheres.com.