Stereohedron
# Stereohedron

In geometry and crystallography, a stereohedron is a convex polyhedron that fills space isohedrally, meaning that the symmetries of the tiling take any copy of the stereohedron to any other copy.

Two-dimensional analogues to the stereohedra are called planigons. Higher dimensional polytopes can also be stereohedra, while they would more accurately be called stereotopes.

## Plesiohedra

A subset of stereohedra are called plesiohedrons, defined as the Voronoi cells of a symmetric Delone set.

Parallelohedrons are plesiohedra which are space-filling by translation only. Edges here are colored as parallel vectors.

Parallelohedra :

| cube | hexagonal prism | rhombic dodecahedron | elongated dodecahedron | truncated octahedron |
| - | - | - | - | - |

## Other periodic stereohedra

The catoptric tessellation contain stereohedra cells. Dihedral angles are integer divisors of 180°, and are colored by their order. The first three are the fundamental domains of ${\tilde {C}}_{3}$, ${\tilde {B}}_{3}$, and ${\tilde {A}}_{3}$ symmetry, represented by Coxeter-Dynkin diagrams: , and . ${\tilde {B}}_{3}$ is a half symmetry of ${\tilde {C}}_{3}$, and ${\tilde {A}}_{3}$ is a quarter symmetry.

Any space-filling stereohedra with symmetry elements can be dissected into smaller identical cells which are also stereohedra. The name modifiers below, half, quarter, and eighth represent such dissections.

Catoptric cells :

| Faces | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 6 | 6 | 6 | 6 | 8 | 12 |
| - | - | - | - | - | - | - | - | - | - | - | - | - | - |
| Type | Tetrahedra | Tetrahedra | Tetrahedra | Tetrahedra | Square pyramid | Square pyramid | Square pyramid | Triangular bipyramid | Triangular bipyramid | Cube | Cube | Octahedron | Rhombic dodecahedron |
| Images | 1/48 (1) | 1/24 (2) | 1/12 (4) | 1/12 (4) | 1/24 (2) | 1/6 (8) | 1/6 (8) | 1/12 (4) | 1/4 (12) | 1 (48) | 1/2 (24) | 1/3 (16) | 2 (96) |
| Symmetry (order) | $C_1$ 1 | $C_{1v}$ 2 | $D_{2d}$ 4 | $C_{1v}$ 2 | $C_{1v}$ 2 | $C_{4v}$ 8 | $C_{2v}$ 4 | $C_{2v}$ 4 | $C_{3v}$ 6 | $O_h$ 48 | $D_{3d}$ 12 | $D_{4h}$ 16 | $O_h$ 48 |
| Honeycomb | Eighth pyramidille | Triangular pyramidille | Oblate tetrahedrille | Half pyramidille | Square quarter pyramidille | Pyramidille | Half oblate octahedrille | Quarter oblate octahedrille | Quarter cubille | Cubille | Oblate cubille | Oblate octahedrille | Dodecahedrille |

Other convex polyhedra that are stereohedra but not parallelohedra nor plesiohedra include the gyrobifastigium.

Others :

| Faces | 8 | 8 | 10 | 12 |
| - | - | - | - | - |
| Symmetry (order) | $D_{2d}$ (8) | $D_{2d}$ (8) | $D_{2d}$ (8) | $D_{4h}$ (16) |
| Images | | | | |
| Cell | Gyrobifastigium | Elongated gyrobifastigium | Ten of diamonds | Elongated square bipyramid |
