Aristotle's Pyramid in Layers: Every Face-to-Face Tiling Is a Close-Packed Stacking

Cut every tetrahedron and octahedron of Fuller's octet truss [13] from its centre into pyramids, one on each face. Paper #44 showed that these two pyramids together fill space and that no other pair of Platonic face pyramids does, and #77 that when pieces meet whole face to whole face the pyramids always reassemble into tetrahedra and octahedra. Here we find all the ways they can then do it: the tetrahedra and octahedra always lie in flat layers, like oranges stacked in a crate. Each layer can sit on the one below in one of two ways, so the tilings are exactly the ways of stacking close-packed spheres. If faces may meet only in part, the pieces can also form cubes, and then there are far too many tilings for any list. Abstract. Let T and O be the centre pyramids of the regular tetrahedron and octahedron of edge 1 (the cone from the solid's centre over one face). Paper #77 (Face to Face) proved that in every face-to-face tiling of space by face pyramids the pyramids round each apex make up their whole solid, so a face-to-face tiling by T and O is a face-to-face tiling by regular tetrahedra and octahedra with two of each round every edge; which such tilings exist beyond the octet and its stackings it left open, as #44 (§4) did. Main theorem. The face-to-face tilings by these solids, and so by T and O, are exactly the stackings of octet layers on parallel planes, each layer placed straight or twisted on the one below, one to one with the close-packed (Barlow) stackings of equal spheres. So there are uncountably many, all made of layers, and in each the two pyramids come in exactly equal numbers. This settles the question left open in #77, and in #44 for face-to-face tilings. Without the face-to-face rule the tilings include every tiling of space by cubes of edge 1/√2 (the cube end of #51's Delian bond, and the cubes of #86 at λ = 3/8), each cube cut into 4 T and 4 O pyramids in either of two ways, and every gluing of two half-space tilings along a plane of faces; no finite list of parameters describes them. For all tilings we prove the complete list of what can surround an edge, and that every lateral edge of a T pyramid is shared, end to end, by exactly three T pyramids. The proof uses only the edge lengths, the face shapes and the dihedral angles, and every step that can be computed is checked in exact arithmetic. View in the app: https://kaleidohedra.dictospheres.com/?open=aristotle:pyramids