Icositruncated dodecadodecahedronIn geometry, the icositruncated dodecadodecahedron or icosidodecatruncated icosidodecahedron is a nonconvex uniform polyhedron, indexed as U45. Its convex hull is a nonuniform truncated icosidodecahedron. Cartesian coordinates for the vertices of an icositruncated dodecadodecahedron are all the even permutations of (±(2−1/τ), ±1, ±(2+τ)) (±1, ±1/τ2, ±(3τ−1)) (±2, ±2/τ, ±2τ) (±3, ±1/τ2, ±τ2) (±τ2, ±1, ±(3τ−2)) where τ = (1+)/2 is the golden ratio (sometimes written φ).
Pentagrammic prismIn geometry, the pentagrammic prism is one of an infinite set of nonconvex prisms formed by square sides and two regular star polygon caps, in this case two pentagrams. It is a special case of a right prism with a pentagram as base, which in general has rectangular non-base faces. Topologically it is the same as a convex pentagonal prism. It is the 78th model in the list of uniform polyhedra, as the first representative of uniform star prisms, along with the pentagrammic antiprism, which is the 79th model.
Octagrammic antiprismIn geometry, the octagrammic antiprism is one in an infinite set of nonconvex antiprisms formed by triangle sides and two regular star polygon caps, in this case two octagrams.
Octagrammic crossed-antiprismIn geometry, the octagrammic crossed-antiprism is one in an infinite set of nonconvex antiprisms formed by triangle sides and two regular star polygon caps, in this case two octagrams.
Projective polyhedronIn geometry, a (globally) projective polyhedron is a tessellation of the real projective plane. These are projective analogs of spherical polyhedra – tessellations of the sphere – and toroidal polyhedra – tessellations of the toroids. Projective polyhedra are also referred to as elliptic tessellations or elliptic tilings, referring to the projective plane as (projective) elliptic geometry, by analogy with spherical tiling, a synonym for "spherical polyhedron".
Hypercubic honeycombIn geometry, a hypercubic honeycomb is a family of regular honeycombs (tessellations) in n-dimensional spaces with the Schläfli symbols {4,3...3,4} and containing the symmetry of Coxeter group R_n (or B^~_n–1) for n ≥ 3. The tessellation is constructed from 4 n-hypercubes per ridge. The vertex figure is a cross-polytope {3...3,4}. The hypercubic honeycombs are self-dual. Coxeter named this family as δ_n+1 for an n-dimensional honeycomb. A Wythoff construction is a method for constructing a uniform polyhedron or plane tiling.
Great dodecahedronIn geometry, the great dodecahedron is a Kepler–Poinsot polyhedron, with Schläfli symbol {5,5/2} and Coxeter–Dynkin diagram of . It is one of four nonconvex regular polyhedra. It is composed of 12 pentagonal faces (six pairs of parallel pentagons), intersecting each other making a pentagrammic path, with five pentagons meeting at each vertex. The discovery of the great dodecahedron is sometimes credited to Louis Poinsot in 1810, though there is a drawing of something very similar to a great dodecahedron in the 1568 book Perspectiva Corporum Regularium by Wenzel Jamnitzer.
Order-2 apeirogonal tilingIn geometry, an order-2 apeirogonal tiling, apeirogonal dihedron, or infinite dihedron is a tiling of the plane consisting of two apeirogons. It may be considered an improper regular tiling of the Euclidean plane, with Schläfli symbol {∞, 2}. Two apeirogons, joined along all their edges, can completely fill the entire plane as an apeirogon is infinite in size and has an interior angle of 180°, which is half of a full 360°. The apeirogonal tiling is the arithmetic limit of the family of dihedra {p, 2}, as p tends to infinity, thereby turning the dihedron into a Euclidean tiling.
