Көпбұрыштың кеңейтілген антипризмасы

Uniform 4 polytope bounded by 320 cells
Grand antiprism(Schlegel diagram wireframe)TypeUniform 4 polytopeUniform index47Cells100+200 (3.3.3) 20 (3.3.3.5)Faces20 {5}700 {3}Edges500Vertices100Vertex figureSphenocoronaSymmetry groupIonic diminished Coxeter group [[10,2+,10]] of order 400PropertiesconvexA net showing two disjoint rings of 10 antiprisms. 200 tetrahedra (yellow) are in face contact with the antiprisms and 100 tetrahedra (red) contact only other tetrahedra. In geometry, the grand antiprism or pentagonal double antiprismoid is a uniform 4 polytope (4 dimensional uniform polytope) bounded by 320 cells: 20 pentagonal antiprisms, and 300 tetrahedra. It is an anomalous, non Wythoffian uniform 4 polytope, discovered in 1965 by Conway and Guy. Topologically, under its highest symmetry, the pentagonal antiprisms have D5d symmetry and there are two types of tetrahedra, one with S4 symmetry and one with Cs symmetry.

20 stacked pentagonal antiprisms occur in two disjoint rings of 10 antiprisms each. The antiprisms in each ring are joined to each other via their pentagonal faces. The two rings are mutually perpendicular, in a structure similar to a duoprism. The 300 tetrahedra join the two rings to each other, and are laid out in a 2 dimensional arrangement topologically equivalent to the 2 torus and the ridge of the duocylinder. These can be further divided into three sets. 100 face mate to one ring, 100 face mate to the other ring, and 100 are centered at the exact midpoint of the duocylinder and edge mate to both rings. This latter set forms a flat torus and can be "unrolled" into a flat 10×10 square array of tetrahedra that meet only at their edges and vertices. See figure below. In addition the 300 tetrahedra can be partitioned into 10 disjoint Boerdijk–Coxeter helices of 30 cells each that close back on each other. The two pentagonal antiprism tubes, plus the 10 BC helices, form an irregular discrete Hopf fibration of the grand antiprism that Hopf maps to the faces of a pentagonal antiprism. The two tubes map to the two pentagonal faces and the 10 BC helices map to the 10 triangular faces. The structure of the grand antiprism is analogous to that of the 3 dimensional antiprisms. However, the grand antiprism is the only convex uniform analogue of the antiprism in 4 dimensions (although the 16 cell may be regarded as a regular analogue of the digonal antiprism). The only nonconvex uniform 4 dimensional antiprism analogue uses pentagrammic crossed antiprisms instead of pentagonal antiprisms, and is called the pentagrammic double antiprismoid.

The vertex figure of the grand antiprism is a sphenocorona or dissected regular icosahedron: a regular icosahedron with two adjacent vertices removed. In their place 8 triangles are replaced by a pair of trapezoids, edge lengths φ, 1, 1, 1 (where φ is the golden ratio), joined together along their edge of length φ, to give a tetradecahedron whose faces are the 2 trapezoids and the 12 remaining equilateral triangles. 12 (3.3.3)2 (3.3.3.5)Dissected regular icosahedron

These are two perspective projections, projecting the polytope into a hypersphere, and applying a stereographic projection into 3 space. Wireframe, viewed down one of the pentagonal antiprism columns. with transparent triangular facesOrthographic projectionCentered on hyperplane of an antiprism in one of the two rings.3D orthographic projectionof 100 of 120 600 cell vertices and 500 edges {488 of 1/2 (3 Sqrt[5]) and 12 of 2/(3+Sqrt[5])}.