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.
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.
 |  |  |  |  |
| 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 |
In geometry, a convex uniform honeycomb is a uniform tessellation which fills three-dimensional Euclidean space with non-overlapping convex uniform polyhedral cells.
In elementary geometry, a polytope is a geometric object with flat sides (faces). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions n as an n-dimensional polytope or n-polytope. For example, a two-dimensional polygon is a 2-polytope and a three-dimensional polyhedron is a 3-polytope. In this context, "flat sides" means that the sides of a (k + 1)-polytope consist of k-polytopes that may have (k – 1)-polytopes in common.
In geometry, a zonohedron is a convex polyhedron that is centrally symmetric, every face of which is a polygon that is centrally symmetric. Any zonohedron may equivalently be described as the Minkowski sum of a set of line segments in three-dimensional space, or as a three-dimensional projection of a hypercube. Zonohedra were originally defined and studied by E. S. Fedorove, a Russian crystallographer. More generally, in any dimension, the Minkowski sum of line segments forms a polytope known as a zonotope.
In geometry, the elongated square bipyramid is the polyhedron constructed by attaching two equilateral square pyramids onto a cube's faces that are opposite each other. It can also be seen as 4 lunes linked together with squares to squares and triangles to triangles. It is also been named the pencil cube or 12-faced pencil cube due to its shape.
The cubic honeycomb or cubic cellulation is the only proper regular space-filling tessellation in Euclidean 3-space made up of cubic cells. It has 4 cubes around every edge, and 8 cubes around each vertex. Its vertex figure is a regular octahedron. It is a self-dual tessellation with Schläfli symbol {4,3,4}. John Horton Conway called this honeycomb a cubille.
In geometry, a pentagonal tiling is a tiling of the plane where each individual piece is in the shape of a pentagon.
In geometry, a honeycomb is a space filling or close packing of polyhedral or higher-dimensional cells, so that there are no gaps. It is an example of the more general mathematical tiling or tessellation in any number of dimensions. Its dimension can be clarified as n-honeycomb for a honeycomb of n-dimensional space.
In four-dimensional euclidean geometry, the tesseractic honeycomb is one of the three regular space-filling tessellations, represented by Schläfli symbol {4,3,3,4}, and constructed by a 4-dimensional packing of tesseract facets.
In geometry, the 5-cubic honeycomb or penteractic honeycomb is the only regular space-filling tessellation in Euclidean 5-space. Four 5-cubes meet at each cubic cell, and it is more explicitly called an order-4 penteractic honeycomb.
In mathematics, the John ellipsoid or Löwner–John ellipsoidE(K) associated to a convex body K in n-dimensional Euclidean space Rn can refer to the n-dimensional ellipsoid of maximal volume contained within K or the ellipsoid of minimal volume that contains K.
James W. Cannon is an American mathematician working in the areas of low-dimensional topology and geometric group theory. He was an Orson Pratt Professor of Mathematics at Brigham Young University.
In geometry, an enneahedron is a polyhedron with nine faces. There are 2606 types of convex enneahedron, each having a different pattern of vertex, edge, and face connections. None of them are regular.
In geometry, the cyclotruncated simplicial honeycomb is a dimensional infinite series of honeycombs, based on the symmetry of the affine Coxeter group. It is given a Schläfli symbol t0,1{3[n+1]}, and is represented by a Coxeter-Dynkin diagram as a cyclic graph of n+1 nodes with two adjacent nodes ringed. It is composed of n-simplex facets, along with all truncated n-simplices.
In four-dimensional Euclidean geometry, the bitruncated 16-cell honeycomb is a uniform space-filling tessellation in Euclidean 4-space.
In four-dimensional Euclidean geometry, the birectified 16-cell honeycomb is a uniform space-filling tessellation in Euclidean 4-space.
In geometry, a plesiohedron is a special kind of space-filling polyhedron, defined as the Voronoi cell of a symmetric Delone set. Three-dimensional Euclidean space can be completely filled by copies of any one of these shapes, with no overlaps. The resulting honeycomb will have symmetries that take any copy of the plesiohedron to any other copy.
A purely combinatorial approach to mirror symmetry was suggested by Victor Batyrev using the polar duality for -dimensional convex polyhedra. The most famous examples of the polar duality provide Platonic solids: e.g., the cube is dual to octahedron, the dodecahedron is dual to icosahedron. There is a natural bijection between the -dimensional faces of a -dimensional convex polyhedron and -dimensional faces of the dual polyhedron and one has . In Batyrev's combinatorial approach to mirror symmetry the polar duality is applied to special -dimensional convex lattice polytopes which are called reflexive polytopes.
In geometry, the elongated gyrobifastigium or gabled rhombohedron is a space-filling octahedron with 4 rectangles and 4 right-angled pentagonal faces.
In geometry, the ten-of-diamonds decahedron is a space-filling polyhedron with 10 faces, 2 opposite rhombi with orthogonal major axes, connected by 8 identical isosceles triangle faces. Although it is convex, it is not a Johnson solid because its faces are not composed entirely of regular polygons. Michael Goldberg named it after a playing card, as a 10-faced polyhedron with two opposite rhombic (diamond-shaped) faces. He catalogued it in a 1982 paper as 10-II, the second in a list of 26 known space-filling decahedra.
