Space-filling polyhedron
Template:Infobox 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.
Families and their examples
If a polygon can tile the plane, its prism is space-filling; examples include the cube,Template:R triangular prism,Template:R and the hexagonal prism.Template:R Any parallelepiped tessellates Euclidean 3-space, as do the five parallelohedra (the cube, hexagonal prism, truncated octahedron, elongated dodecahedron, and rhombic dodecahedron).Template:R Other space-filling polyhedra include the square pyramid,Template:R plesiohedra,Template:R and stereohedra, polyhedra whose tilings have symmetries taking every tile to every other tile, including the gyrobifastigium,Template:R the triakis truncated tetrahedron,Template:R and the trapezo-rhombic dodecahedron.Template:R
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.Template:R Although the regular tetrahedron cannot fill space, other tetrahedra can, including the Goursat tetrahedra derived from the cube,Template:Cn and the Hill tetrahedra.Template:R
Characteristics
Every polyhedron with Dehn invariant of zero that can space-fill. Relatedly, two different polyhedra can have Dehn invariants of zero, which may be able to tile each other.Template:R
See also
- Hilbert's eighteenth problem, one of Hilbert's problems, consisting of three subproblems, one of which is a three-dimensional anisohedral tiling.
- Tridecahedron, a thirteen-sided polyhedron
References
- Space-Filling Polyhedron, MathWorld
- Template:Cite book