OmnitruncationIn geometry, an omnitruncation of a convex polytope is a simple polytope of the same dimension, having a vertex for each flag of the original polytope and a facet for each face of any dimension of the original polytope. Omnitruncation is the dual operation to barycentric subdivision. Because the barycentric subdivision of any polytope can be realized as another polytope, the same is true for the omnitruncation of any polytope.
Uniform tilings in hyperbolic planeIn hyperbolic geometry, a uniform hyperbolic tiling (or regular, quasiregular or semiregular hyperbolic tiling) is an edge-to-edge filling of the hyperbolic plane which has regular polygons as faces and is vertex-transitive (transitive on its vertices, isogonal, i.e. there is an isometry mapping any vertex onto any other). It follows that all vertices are congruent, and the tiling has a high degree of rotational and translational symmetry.
Truncation (geometry)In geometry, a truncation is an operation in any dimension that cuts polytope vertices, creating a new facet in place of each vertex. The term originates from Kepler's names for the Archimedean solids. In general any polyhedron (or polytope) can also be truncated with a degree of freedom as to how deep the cut is, as shown in Conway polyhedron notation truncation operation. A special kind of truncation, usually implied, is a uniform truncation, a truncation operator applied to a regular polyhedron (or regular polytope) which creates a resulting uniform polyhedron (uniform polytope) with equal edge lengths.
Runcinated tesseractsIn four-dimensional geometry, a runcinated tesseract (or runcinated 16-cell) is a convex uniform 4-polytope, being a runcination (a 3rd order truncation) of the regular tesseract. There are 4 variations of runcinations of the tesseract including with permutations truncations and cantellations. The runcinated tesseract or (small) disprismatotesseractihexadecachoron has 16 tetrahedra, 32 cubes, and 32 triangular prisms. Each vertex is shared by 4 cubes, 3 triangular prisms and one tetrahedron.
Honeycomb (geometry)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. Honeycombs are usually constructed in ordinary Euclidean ("flat") space. They may also be constructed in non-Euclidean spaces, such as hyperbolic honeycombs.
Uniform 4-polytopeIn geometry, a uniform 4-polytope (or uniform polychoron) is a 4-dimensional polytope which is vertex-transitive and whose cells are uniform polyhedra, and faces are regular polygons. There are 47 non-prismatic convex uniform 4-polytopes. There are two infinite sets of convex prismatic forms, along with 17 cases arising as prisms of the convex uniform polyhedra. There are also an unknown number of non-convex star forms.
ParallelohedronIn geometry, a parallelohedron is a polyhedron that can be translated without rotations in 3-dimensional Euclidean space to fill space with a honeycomb in which all copies of the polyhedron meet face-to-face. There are five types of parallelohedron, first identified by Evgraf Fedorov in 1885 in his studies of crystallographic systems: the cube, hexagonal prism, rhombic dodecahedron, elongated dodecahedron, and truncated octahedron. Every parallelohedron is a zonohedron, a centrally symmetric polyhedron with centrally symmetric faces.
Uniform polyhedronIn geometry, a uniform polyhedron has regular polygons as faces and is vertex-transitive (i.e., there is an isometry mapping any vertex onto any other). It follows that all vertices are congruent. Uniform polyhedra may be regular (if also face- and edge-transitive), quasi-regular (if also edge-transitive but not face-transitive), or semi-regular (if neither edge- nor face-transitive). The faces and vertices need not be convex, so many of the uniform polyhedra are also star polyhedra.
Spherical polyhedronIn geometry, a spherical polyhedron or spherical tiling is a tiling of the sphere in which the surface is divided or partitioned by great arcs into bounded regions called spherical polygons. Much of the theory of symmetrical polyhedra is most conveniently derived in this way. The most familiar spherical polyhedron is the soccer ball, thought of as a spherical truncated icosahedron. The next most popular spherical polyhedron is the beach ball, thought of as a hosohedron.
ZonohedronIn geometry, a zonohedron is a convex polyhedron that is centrally symmetric, every face of which is a polygon that is centrally symmetric (a zonogon). 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. Fedorov, a Russian crystallographer. More generally, in any dimension, the Minkowski sum of line segments forms a polytope known as a zonotope.
