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.
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).
Goldberg added three infinite families of space-filling tetrahedra to five individual cases in the earlier literature (Goldberg 1974a).
His lists of space-filling pentahedra
were not claimed to be complete, and their numbered types denote construction methods
rather than mutually exclusive geometric forms (Goldberg 1972, 1974b).
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. The Schmitt-Conway
biprism gives nonperiodic tilings when
the layer rotation angle is an irrational
number times and mirror images are excluded (Baake and Frettlöh 2005).
Schmitt (2016) gives a summary of space-filling polyhedra via an investigation of which Dirichlet-Voronoi stereohedra the tetragonal,
trigonal, hexagonal, and cubic groups can generate.
and mirror images are excluded (Baake and Frettlöh 2005).
