Grand antiprism

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.

The 20 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.

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