Grand antiprism

The grand antiprism, the pentagonal double antiprismoid, or gap, is a convex uniform polychoron that consists of 300 tetrahedra and 20 pentagonal antiprisms. 12 tetrahedra and 2 pentagonal antiprisms join at each vertex. It may be constructed by taking two orthogonal rings of 10 pentagonal antiprisms each, and connecting them by tetrahedra. Alternatively, one may construct the grand antiprism as a faceting of the hexacosichoron, specifically by removing two orthogonal rings of 10 vertices. The resulting diminishings intersect, thus leading to pentagonal antiprisms instead of icosahedra as cells. It is the first in an infinite family of isogonal pentagonal antiprismatic swirlchora.

Despite the name, the grand antiprism is neither a stellation nor an antiprism in any common sense of the word. It is, however, related to the duoantiprisms, being the convex hull of the compound of two orthogonal pentagonal-pentagonal duoantiprisms formed from pentagons with a size ratio of $$1:\tfrac{1+\sqrt5}{2}$$. As such, it is also the fourth member of the double antiprismoids and the only convex uniform one, formed from alternating the decagonal ditetragoltriate and then filling the gaps with tetrahedra.

One unusual property of the grand antiprism is that it contains the vertices of a small prismatodecachoron of edge length $$\tfrac{1+\sqrt{5}}{2}$$.

Its vertex figure is topologically equivalent to the Johnson solid sphenocorona, but with the edge between the two tetragonal faces made longer. This vertex figure can be formed by deleting two adjacent vertices from the regular icosahedron.

Vertex coordinates
The vertices of a grand antiprism of edge length 1 are given by:


 * $$±\left(±\frac{3+\sqrt5}{4},\,0,\,\frac{1+\sqrt5}{4},\,-\frac12\right),$$
 * $$±\left(±\frac12,\,0,\,\frac{3+\sqrt5}{4},\,-\frac{1+\sqrt5}{4}\right),$$
 * $$±\left(0,\,±\frac{3+\sqrt5}{4},\,\frac12,\,\frac{1+\sqrt5}{4}\right),$$
 * $$±\left(0,\,±\frac12,\,\frac{1+\sqrt5}{4},\,\frac{3+\sqrt5}{4}\right),$$
 * $$\left(0,\,0,\,±\frac{1+\sqrt5}{2},\,0\right),$$
 * $$\left(0,\,0,\,0,\,±\frac{1+\sqrt5}{2}\right),$$
 * $$\left(±\frac{1+\sqrt5}{4},\,±\frac{1+\sqrt5}{4},\,±\frac{1+\sqrt5}{4},\,±\frac{1+\sqrt5}{4}\right),$$
 * $$\left(±\frac{1+\sqrt5}{4},\,0,\,±\frac12,\,±\frac{3+\sqrt5}{4}\right),$$
 * $$\left(0,\,±\frac{1+\sqrt5}{4},\,±\frac{3+\sqrt5}{4},\,±\frac12\right),$$
 * $$\left(±\frac{1+\sqrt5}{4},\,±\frac12,\,±\frac{3+\sqrt5}{4},\,0\right),$$
 * $$\left(±\frac{3+\sqrt5}{4},\,±\frac{1+\sqrt5}{4},\,±\frac12,\,0\right),$$
 * $$\left(±\frac12,\,±\frac{3+\sqrt5}{4},\,±\frac{1+\sqrt5}{4},\,0\right),$$
 * $$\left(±\frac{3+\sqrt5}{4},\,±\frac12,\,0,\,±\frac{1+\sqrt5}{4}\right),$$
 * $$\left(±\frac12,\,±\frac{1+\sqrt5}{4},\,0,\,±\frac{3+\sqrt5}{4}\right),$$
 * $$\left(±\frac{1+\sqrt5}{4},\,±\frac{3+\sqrt5}{4},\,0,\,±\frac12\right).$$

These coordinates are formed by removing 20 vertices, in 2 rings of 10, from a regular hexacosichoron.

Variations
The grand antiprism is part of a continuum of more general isogonal variations that can also be called a pentagonal double antiprismoid or a decafold pentaantiprismatoswirlchoron. It has one degree of variation. These polychora generally have 20 pentagonal antiprisms in two rings, 100 tetragonal disphenoids, and 200 sphenoids, while maintaining the grand antiprism's symmetry.

Variants with lesser symmetry can have two types of pentagonal antiprisms per ring and either one or two types of pentagonal gyroprisms per ring.