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 , , 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.

Catoptric cells
Faces 456812
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.

Others
Faces 81012
Symmetry
(order)
D2d (8) D4h (16)
Images
Cell Gyrobifastigium Elongated
gyrobifastigium
Ten of diamonds Elongated
square bipyramid

Aperiodic stereohedra

The Schmitt–Conway–Danzer tile, a convex polyhedron that tiles space, is not a stereohedron because all of its tilings are aperiodic.

References

  • Ivanov, A. B. (2001) [1994], "Stereohedron", Encyclopedia of Mathematics, EMS Press
  • B. N. Delone, N. N. Sandakova, Theory of stereohedra Trudy Mat. Inst. Steklov., 64 (1961) pp. 28–51 (Russian)
  • Goldberg, Michael Three Infinite Families of Tetrahedral Space-Fillers Journal of Combinatorial Theory A, 16, pp. 348–354, 1974.
  • Goldberg, Michael The space-filling pentahedra, Journal of Combinatorial Theory, Series A Volume 13, Issue 3, November 1972, Pages 437-443 PDF
  • Goldberg, Michael The Space-filling Pentahedra II, Journal of Combinatorial Theory 17 (1974), 375–378. PDF
  • Goldberg, Michael On the space-filling hexahedra Geom. Dedicata, June 1977, Volume 6, Issue 1, pp 99–108 PDF
  • Goldberg, Michael On the space-filling heptahedra Geometriae Dedicata, June 1978, Volume 7, Issue 2, pp 175–184 PDF
  • Goldberg, Michael Convex Polyhedral Space-Fillers of More than Twelve Faces. Geom. Dedicata 8, 491-500, 1979.
  • Goldberg, Michael On the space-filling octahedra, Geometriae Dedicata, January 1981, Volume 10, Issue 1, pp 323–335 PDF
  • Goldberg, Michael On the Space-filling Decahedra. Structural Topology, 1982, num. Type 10-II PDF
  • Goldberg, Michael On the space-filling enneahedra Geometriae Dedicata, June 1982, Volume 12, Issue 3, pp 297–306 PDF
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