In
geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
, a space-filling polyhedron is a
polyhedron that can be used to fill all of
three-dimensional space via
translations,
rotation
Rotation or rotational/rotary motion is the circular movement of an object around a central line, known as an ''axis of rotation''. A plane figure can rotate in either a clockwise or counterclockwise sense around a perpendicular axis intersect ...
s 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,
triangular prism, and the
hexagonal prism. Any
parallelepiped tessellates
Euclidean 3-space, as do the five
parallelohedra including the cube, hexagonal prism,
truncated octahedron, and
rhombic dodecahedron. Other space-filling polyhedra include the
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
In geometry, a Platonic solid is a Convex polytope, convex, regular polyhedron in three-dimensional space, three-dimensional Euclidean space. Being a regular polyhedron means that the face (geometry), faces are congruence (geometry), congruent (id ...
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.
References
Space-Filling Polyhedron MathWorld
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:Space-filling polyhedra
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