A space-filling polyhedron is a polyhedron which can be used to generate a tessellation of space. Although even Aristotle himself proclaimed in his work On the Heavens that the tetrahedron fills space, it in fact does not. Several space-filling polyhedra are illustrated above.
Having Dehn invariant 0 is a necessary but not sufficient condition for a polyhedron to be space-filling.
The cube is the only Platonic solid possessing this property (Gardner 1984, pp. 183-184). However, a combination of tetrahedra and octahedra do fill space (Steinhaus 1999, p. 210; Wells 1991, p. 232). In addition, octahedra, truncated octahedron, and cubes, combined in the ratio 1:1:3, can also fill space (Wells 1991, p. 235). In 1914, Föppl discovered a space-filling compound of tetrahedra and truncated tetrahedra (Wells 1991, p. 234).
There are only five space-filling convex polyhedra with regular faces: the triangular prism, hexagonal prism, cube, truncated octahedron (Steinhaus 1999, pp. 185-190; Wells 1991, pp. 233-234), and gyrobifastigium (Johnson 2000). The rhombic dodecahedron (Steinhaus 1999, pp. 185-190; Wells 1991, pp. 233-234) and elongated dodecahedron, and trapezo-rhombic dodecahedron appearing in sphere packing are also space-fillers (Steinhaus 1999, pp. 203-207), as is any non-self-intersecting quadrilateral prism. The cube, hexagonal prism, rhombic dodecahedron, elongated dodecahedron, and truncated octahedron are all "primary" parallelohedra (Coxeter 1973, p. 29).
A stereohedron is a convex polyhedron that is isohedrally space-filling, meaning the symmetries of a tiling of copies of a stereohedron take any copy to any other copy. A plesiohedron is a space-filling polyhedron which has special symmetries that take any copy of the plesiohedron in the space-filling honeycomb to any other.
In the period 1974-1980, Michael Goldberg attempted to exhaustively catalog space-filling polyhedra. According to Goldberg, there are 27 distinct space-filling hexahedra, covering all of the 7 hexahedra except the pentagonal pyramid. Of the 34 heptahedra, 16 are space-fillers, which can fill space in at least 56 distinct ways. Octahedra can fill space in at least 49 different ways. In pre-1980 papers, there are forty 11-hedra, sixteen dodecahedra, four 13-hedra, eight 14-hedra, no 15-hedra, one 16-hedron originally discovered by Föppl (Grünbaum and Shephard 1980; Wells 1991, p. 234), two 17-hedra, one 18-hedron, six icosahedra, two 21-hedra, five 22-hedra, two 23-hedra, one 24-hedron, and a believed maximal 26-hedron. In 1980, P. Engel (Wells 1991, pp. 234-235) then found a total of 172 more space-fillers of 17 to 38 faces, and more space-fillers have been found subsequently. P. Schmitt discovered a nonconvex aperiodic polyhedral space-filler around 1990, and a convex polyhedron known as the Schmitt-Conway biprism which fills space only aperiodically was found by J. H. Conway in 1993 (Eppstein).
Schmitt (2016) gives a summary of space-filling polyhedra via an investigation of which Dirichlet-Voronoi stereohedraStereohedron the tetragonal, trigonal, hexagonal, and cubic groups can generate.
