A space-filling hexahedron is a hexahedron whose congruent copies can tile three-dimensional Euclidean space. The copies may be translated, rotated, or reflected. The examples below are convex polyhedra.
The illustrated examples have the same arrangements of faces and edges as a cube, hemicube, hemiobelisk, pentagonal wedge, tetragonal antiwedge, and triangular dipyramid, respectively (E. Pegg, Jr., pers. comm., Oct. 23, 2026). These represent six of the seven classes of convex hexahedron distinguished by their arrangements of faces, edges, and vertices. No pentagonal pyramid example is included. Its absence from this collection does not establish that no such space-filling polyhedron exists.
The ability to tile space depends on the actual shape, rather than only the arrangement of faces. A hexahedron with the face arrangement of a hemiobelisk and nonzero Dehn invariant cannot tile space, whereas the illustrated hexahedron of hemiobelisk type can. Likewise, a hexahedron with the same arrangement of faces as a cube need not be a regular cube.
The three illustrated dissections give further examples. Their common parent is obtained by truncating each vertex of a regular tetrahedron one quarter of the way along each incident edge. Its hexagonal faces have alternating edge lengths in the ratio 2:1. This parent admits dissections into four or twelve congruent hexahedra of hemiobelisk type, or twelve congruent hexahedra with the same arrangement of faces as a cube (E. Pegg, Jr., pers. comm., Oct. 23, 2026). Two parent solids form a repeating translation motif, giving 8, 24, or 24 congruent pieces per motif, respectively.
For example, the hexahedron of hemiobelisk type in the four-piece dissection has vertices
 |
Its volume is 
, while the parent solid has volume 
in these coordinates.
The seven-cube cluster obtained by removing one corner unit cube from a cube of edge length 2 admits a dissection
into three congruent pyramidal
frustums (Goldberg 1980, pp. 375-377). The cluster tiles
space, so each fragment is a space-filling hexahedron. One piece satisfies 
, 
, and 
. The other two are obtained by cyclic
permutations of the coordinates.
There are infinitely many mutually non-similar space-filling hexahedra, as is already demonstrated by cuboids with varying edge lengths. Thus a finite list of examples or construction methods, such as Goldberg's investigation (Goldberg 1977), should not be interpreted as a complete count of geometric forms.
