Space-filling polyhedron
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| Definition | a polyhedron that fill all of three-dimensional space via translations, rotations, or reflections |
|---|---|
In geometry, a space-filling polyhedron is a polyhedron that can be used to fill all of three-dimensional space via translations, rotations and/or reflections, where filling means that; taken together, all the instances of the polyhedron constitute a partition of three-space. Any periodic tiling or honeycomb of three-space can, in fact, be generated by translating a primitive cell polyhedron.
If a polygon can tile the plane, its prism is space-filling; examples include the cube,[1] triangular prism,[2] and the hexagonal prism.[3] Any parallelepiped tessellates Euclidean 3-space, as do the five parallelohedra (the cube, hexagonal prism, truncated octahedron, elongated dodecahedron, and rhombic dodecahedron).[3] Other space-filling polyhedra include the square pyramid,[4] plesiohedra,[1] and stereohedra, polyhedra whose tilings have symmetries taking every tile to every other tile, including the gyrobifastigium,[5] the triakis truncated tetrahedron,[6] and the trapezo-rhombic dodecahedron.[7]
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.[8] Although the regular tetrahedron cannot fill space, other tetrahedra can, including the Goursat tetrahedra derived from the cube,[citation needed] and the Hill tetrahedra.[9]
