Stereohedron

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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]

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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
Parallelohedron edges cube.png Parallelohedron edges hexagonal prism.png Parallelohedron edges rhombic dodecahedron.png Parallelohedron edges elongated rhombic dodecahedron.png Parallelohedron edge truncated octahedron.png
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 , , and symmetry, represented by Coxeter-Dynkin diagrams: CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png, CDel node.pngCDel 4.pngCDel node.pngCDel split1.pngCDel nodes.png and CDel branch.pngCDel 3ab.pngCDel branch.png. 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.

Catoptric cells
Faces456812
Type Tetrahedra Square pyramid Triangular bipyramid Cube Octahedron Rhombic dodecahedron
Images Eighth pyramidille cell.png
1/48 (1)
Triangular pyramidille cell1.png
1/24 (2)
Oblate tetrahedrille cell.png
1/12 (4)
Half pyramidille cell.png
1/12 (4)
Square quarter pyramidille cell.png
1/24 (2)
Cubic square pyramid.png
1/6 (8)
Half oblate octahedrille cell-cube.png
1/6 (8)
Quarter oblate octahedrille cell.png
1/12 (4)
Quarter cubille cell.png
1/4 (12)
Cubic full domain.png
1 (48)
Oblate cubille cell.png
1/2 (24)
Cubic square bipyramid.png
1/3 (16)
Dodecahedrille cell.png
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
HoneycombEighth pyramidille
CDel node f1.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node f1.pngCDel 4.pngCDel node f1.png
Triangular pyramidille
CDel node f1.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node f1.pngCDel 4.pngCDel node.png
Oblate tetrahedrille
CDel node.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node f1.pngCDel 4.pngCDel node.png
Half pyramidille
CDel node fh.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node f1.pngCDel 4.pngCDel node f1.png
Square quarter pyramidille
CDel node f1.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node f1.png
Pyramidille
CDel node f1.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
Half oblate octahedrille
CDel node fh.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node f1.pngCDel 4.pngCDel node.png
Quarter oblate octahedrille
CDel node f1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node f1.pngCDel 4.pngCDel node.png
Quarter cubille
CDel node fh.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node f1.png
Cubille
CDel node f1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
Oblate cubille
CDel node fh.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node fh.png
Oblate octahedrille
CDel node.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
Dodecahedrille
CDel node fh.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png

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

Others
Faces81012
Symmetry
(order)
D2d (8)D4h (16)
Images Gyrobifastigium.png Elongated digonal gyrobicupola2.png Ten-of-diamonds decahedron skew.png Elongated oblate octahedron.png
Cell Gyrobifastigium Elongated
gyrobifastigium
Ten of diamonds Elongated
square bipyramid

References

  1. Lekkerkerker, C. G.; Gruber, Pascale (1987). Geometry of Numbers. North-Holland. p. 166.