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. [1]
Two-dimensional analogues to the stereohedra are called planigons. Higher dimensional polytopes can also be stereohedra, while they would more accurately be called stereotopes.
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.
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cube | hexagonal prism | rhombic dodecahedron | elongated dodecahedron | truncated octahedron |
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 , , and symmetry, represented by Coxeter-Dynkin diagrams: ,
and
. is a half symmetry of , and 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.
Faces | 4 | 5 | 6 | 8 | 12 | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Type | Tetrahedra | Square pyramid | Triangular bipyramid | 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) | C1 1 | C1v 2 | D2d 4 | C1v 2 | C1v 2 | C4v 8 | C2v 4 | C2v 4 | C3v 6 | Oh 48 | D3d 12 | D4h 16 | Oh 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.
Faces | 8 | 10 | 12 | |
---|---|---|---|---|
Symmetry (order) | D2d (8) | D4h (16) | ||
Images | ![]() | ![]() | ![]() | ![]() |
Cell | Gyrobifastigium | Elongated gyrobifastigium | Ten of diamonds | Elongated square bipyramid |