stereohedron

{{Short description|Convex polyhedron that fills space isohedrally}}

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.

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|+ Parallelohedra

align=center

align=center

|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: {{CDD|node|4|node|3|node|4|node}}, {{CDD|node|4|node|split1|nodes}} and {{CDD|branch|3ab|branch}}. ${\tilde{B}}_3$ 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.

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|+ Catoptric cells

!Faces

!colspan=4|4

colspan=3|5

colspan=4|6

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!Type

|colspan=4|Tetrahedra

|colspan=3|Square pyramid

|colspan=2|Triangular bipyramid

|colspan=2|Cube

|Octahedron

|Rhombic dodecahedron

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!Images

|80px
1/48 (1)

|80px
1/24 (2)

|80px
1/12 (4)

|80px
1/24 (2)

|80px
1/6 (8)

|80px
1/12 (4)

|80px
1/4 (12)

|80px
1 (48)

|60px
1/2 (24)

|80px
1/3 (16)

|60px
2 (96)

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!Symmetry
(order)

|C₁
1

|C_1v
2

|D_2d
4

|C_1v
2

|C_4v
8

|C_2v
4

|C_3v
6

|O_h
48

|D_3d
12

|D_4h
16

|O_h
48

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!Honeycomb

|Eighth pyramidille
{{CDD|node_f1|4|node_f1|3|node_f1|4|node_f1}}

|Triangular pyramidille
{{CDD|node_f1|4|node_f1|3|node_f1|4|node}}

|Oblate tetrahedrille
{{CDD|node|4|node_f1|3|node_f1|4|node}}

|Half pyramidille
{{CDD|node_fh|4|node|3|node_f1|4|node_f1}}

|Square quarter pyramidille
{{CDD|node_f1|4|node_f1|3|node|4|node_f1}}

|Pyramidille
{{CDD|node_f1|4|node_f1|3|node|4|node}}

|Half oblate octahedrille
{{CDD|node_fh|4|node|3|node_f1|4|node}}

|Quarter oblate octahedrille
{{CDD|node_f1|4|node|3|node_f1|4|node}}

|Quarter cubille
{{CDD|node_fh|4|node|3|node|4|node_f1}}

|Cubille
{{CDD|node_f1|4|node|3|node|4|node}}

|Oblate cubille
{{CDD|node_fh|4|node|3|node|4|node_fh}}

|Oblate octahedrille
{{CDD|node|4|node_f1|3|node|4|node}}

|Dodecahedrille
{{CDD|node_fh|4|node|3|node|4|node}}

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

class=wikitable \|+ Others !Faces !colspan=2\|8	10	12
align=center !Symmetry (order) \|colspan=3\|D_2d (8) \|D_4h (16)
align=center !Images \|120px \|60px \|80px \|80px
Cell \|Gyrobifastigium \|Elongated gyrobifastigium \|Ten of diamonds \|Elongated square bipyramid

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|+ Others

!Faces

!colspan=2|8

align=center

!Symmetry
(order)

|colspan=3|D_2d (8)

|D_4h (16)

align=center

!Images

Cell

|Elongated
gyrobifastigium

|Ten of diamonds

|Elongated
square bipyramid

References

{{SpringerEOM|title=Stereohedron|first=A. B.|last=Ivanov|id=Stereohedron&oldid=31579}}
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 [http://www.sciencedirect.com/science/article/pii/0097316572900775] [http://documentslide.com/documents/the-space-filling-pentahedra.html PDF]
Goldberg, Michael The Space-filling Pentahedra II, Journal of Combinatorial Theory 17 (1974), 375–378. [http://documentslide.com/documents/the-space-filling-pentahedra-ii.html PDF]
Goldberg, Michael On the space-filling hexahedra Geom. Dedicata, June 1977, Volume 6, Issue 1, pp 99–108 [https://link.springer.com/article/10.1007/BF00181585] [http://documentslide.com/documents/on-the-space-filling-hexahedra.html PDF]
Goldberg, Michael On the space-filling heptahedra Geometriae Dedicata, June 1978, Volume 7, Issue 2, pp 175–184 [https://link.springer.com/article/10.1007/BF00181630] [http://documentslide.com/documents/on-the-space-filling-heptahedra.html 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 [https://link.springer.com/article/10.1007/BF01447431] [http://documents.mx/documents/on-the-space-filling-octahedra.html PDF]
Goldberg, Michael On the Space-filling Decahedra. Structural Topology, 1982, num. Type 10-II [https://upcommons.upc.edu/handle/2099/990 PDF]
Goldberg, Michael On the space-filling enneahedra Geometriae Dedicata, June 1982, Volume 12, Issue 3, pp 297–306 [https://link.springer.com/article/10.1007/BF00147314] [http://documentslide.com/documents/on-the-space-filling-enneahedra.html PDF]

Category:Space-filling polyhedra