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.

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 duoantiprisms formed from pentagons with a size ratio of 1:($\sqrt{5}$+1)/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 ($\sqrt{5}$+1)/2.

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


 * ±(±($\sqrt{5}$+1)/2, 0, 1, −($\sqrt{5}$–1)/2),
 * ±(±($\sqrt{5/8}$–1)/2, 0, ($\sqrt{5}$+1)/2, −1),
 * ±(0, ±($\sqrt{5}$+1)/2, ($\sqrt{5}$–1)/2, 1),
 * ±(0, ±($\sqrt{5}$–1)/2, 1, ($\sqrt{5}$+1)/2),

along with all changes of sign of:


 * (0, 0, 2, 0),
 * (0, 0, 0, 2),
 * (1, 1, 1, 1),
 * (1, 0, ($\sqrt{5}$–1)/2, ($\sqrt{5}$+1)/2),
 * (0, 1, ($\sqrt{5}$+1)/2, ($\sqrt{5}$–1)/2),
 * (1, ($\sqrt{5}$–1)/2, ($\sqrt{5}$+1)/2, 0),
 * (($\sqrt{5}$+1)/2, 1, ($\sqrt{5}$–1)/2, 0),
 * (($\sqrt{5}$–1)/2, ($\sqrt{5}$+1)/2, 1, 0),
 * (($\sqrt{5}$+1)/2, ($\sqrt{5}$–1)/2, 0, 1),
 * (($\sqrt{5}$–1)/2, 1, 0, ($\sqrt{5}$+1)/2),
 * (1, ($\sqrt{5}$+1)/2, 0, ($\sqrt{5}$–1)/2).

These coordinates are formed by removing two rings of 10 vertices from the coordinates of the regular hexacosichoron.