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. + Parallelohedracubehexagonal prismrhombic dodecahedronelongated dodecahedrontruncated 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 cellsFaces456812TypeTetrahedraSquare pyramidTriangular bipyramidCubeOctahedronRhombic dodecahedronImages1/48 (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)C11C1v2D2d4C1v2C1v2C4v8C2v4C2v4C3v6Oh48D3d12D4h16Oh48HoneycombEighth pyramidilleTriangular pyramidilleOblate tetrahedrilleHalf pyramidilleSquare quarter pyramidillePyramidilleHalf oblate octahedrilleQuarter oblate octahedrilleQuarter cubilleCubilleOblate cubilleOblate octahedrilleDodecahedrille Other convex polyhedra that are stereohedra but not parallelohedra nor plesiohedra include the gyrobifastigium. + OthersFaces81012Symmetry(order)D2d (8)D4h (16)ImagesCellGyrobifastigiumElongatedgyrobifastigiumTen of diamondsElongatedsquare bipyramid
References
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