If a polygon can tile the plane, its
prism is space-filling; examples include the
cube,
triangular prism, and the
hexagonal prism. Any
parallelepiped tessellates
Euclidean 3-space, as do the five
parallelohedra (the cube, hexagonal prism,
truncated octahedron,
elongated dodecahedron, and
rhombic dodecahedron). Other space-filling polyhedra include the
square pyramid,
plesiohedra, and
stereohedra, polyhedra whose tilings have symmetries taking every tile to every other tile, including the
gyrobifastigium, the
triakis truncated tetrahedron, and the
trapezo-rhombic dodecahedron. The cube is the only
Platonic solid that can fill space, although a tiling that combines tetrahedra and octahedra (the
tetrahedral-octahedral honeycomb) is possible. Although the regular tetrahedron cannot fill space, other tetrahedra can, including the
Goursat tetrahedra derived from the cube, and the
Hill tetrahedra. == Characteristics ==