A parallelohedron is a space-filling polyhedron that fills space using an infinite number of similarly situated copies (Tutton 1964, pp. 567 and 723; Coxeter 1973, pp. 29-30). There are exactly five "primary"' parallelohedra: the cube, hexagonal prism, elongated dodecahedron, rhombic dodecahedron, and truncated octahedron (Coxeter 1973, p. 29).
The generalization of the cube to a parallelepiped constructed from three line segments that are not all parallel to a common plane is also a parallelohedron.
Three-dimensional parallelohedra are stereohedra, zonohedra, as well as plesiohedra which are space-filling by translation only.
In higher dimensions, a parallelohedron is a convex polytope whose translates tile Euclidean
space. The regular 24-cell is a four-dimensional
example. Novaga and Song (2026) proved that it uniquely minimizes boundary surface
area among four-dimensional parallelohedra of a prescribed volume,
up to similarity. For volume 
, the minimum boundary surface
area is 
.
