# Grand antiprism

Grand antiprism
Rank4
TypeUniform
Notation
Bowers style acronymGap
Coxeter diagramxofo5oxof2ofxo5foox&#zx
Elements
Cells100+200 tetrahedra, 20 pentagonal antiprisms
Faces100+200+400 triangles, 20 pentagons
Edges100+200+200
Vertices100
Vertex figureSphenocorona variant, edge lengths 1 and (5+1)/2
Measures (edge length 1)
Circumradius${\displaystyle {\frac {1+{\sqrt {5}}}{2}}\approx 1.61803}$
Hypervolume${\displaystyle {\frac {5\left(55+26{\sqrt {5}}\right)}{24}}\approx 23.57037}$
Dichoral anglesTet–3–tet: ${\displaystyle \arccos \left(-{\frac {1+3{\sqrt {5}}}{8}}\right)\approx 164.47751^{\circ }}$
Pap–5–pap: 144°
Pap–3–tet: ${\displaystyle \arccos \left(-{\frac {\sqrt {10}}{4}}\right)\approx 142.23876^{\circ }}$
Central density1
Number of external pieces320
Level of complexity22
Related polytopes
ArmyGap
RegimentGap
DualPentagonal double antitegmoid
ConjugatePentagrammic double antiprismoid
Abstract & topological properties
Flag count8800
Euler characteristic0
OrientableYes
Properties
SymmetryI2(10)+≀S2×2, order 400
ConvexYes
NatureTame

The grand antiprism or gap, is a convex uniform polychoron that consists of 300 tetrahedra and 20 pentagonal antiprisms. Twelve tetrahedra and 2 pentagonal antiprisms join at each vertex. Despite its name, it is not in any sense an antiprism, although it is closely related to the duoantiprisms.

The grand antiprism can be constructed using two rings of 10 pentagonal antiprisms each, which are placed in orthogonal planes. These two rings do not touch, and the space between them is filled with tetrahedra. Its vertex figure is the convex hull of an icosahedron minus two adjacent vertices, forming a distorted sphenocorona.

The only non-Wythoffian convex uniform polychoron, it was the final member of the convex uniform polychora to be discovered (in 1965 by Conway and Guy). Much later, its construction formed the basis for the more general family of swirlchora (in which the grand antiprism is of the isogonal type).

## Properties

It is also known as the pentagonal double antiprismoid or decafold pentaantiprismatoswirlchoron.

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.

The grand antiprism is the only convex uniform polychoron where the number of edge types in its highest symmetry is greater than the number of degrees of freedom. In this case, it has three edge types, but only two degrees of freedom.

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 or two inversely oriented pentagonal antiditetragoltriates formed from pentagons with a size ratio of 1:${\displaystyle {\tfrac {1+{\sqrt {5}}}{2}}}$. 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 ${\displaystyle {\tfrac {1+{\sqrt {5}}}{2}}}$. This is due to the fact that the inscribed small prismatodecachoron, which has gyrochoric symmetry, can be thought of as a convex hull of two orthogonal stretched 10-3 step prisms.

## Vertex coordinates

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

• ${\displaystyle \pm \left(\pm {\frac {3+{\sqrt {5}}}{4}},\,0,\,{\frac {1+{\sqrt {5}}}{4}},\,-{\frac {1}{2}}\right),}$
• ${\displaystyle \pm \left(\pm {\frac {1}{2}},\,0,\,{\frac {3+{\sqrt {5}}}{4}},\,-{\frac {1+{\sqrt {5}}}{4}}\right),}$
• ${\displaystyle \pm \left(0,\,\pm {\frac {3+{\sqrt {5}}}{4}},\,{\frac {1}{2}},\,{\frac {1+{\sqrt {5}}}{4}}\right),}$
• ${\displaystyle \pm \left(0,\,\pm {\frac {1}{2}},\,{\frac {1+{\sqrt {5}}}{4}},\,{\frac {3+{\sqrt {5}}}{4}}\right),}$
• ${\displaystyle \left(0,\,0,\,\pm {\frac {1+{\sqrt {5}}}{2}},\,0\right),}$
• ${\displaystyle \left(0,\,0,\,0,\,\pm {\frac {1+{\sqrt {5}}}{2}}\right),}$
• ${\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {1+{\sqrt {5}}}{4}}\right),}$
• ${\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,0,\,\pm {\frac {1}{2}},\,\pm {\frac {3+{\sqrt {5}}}{4}}\right),}$
• ${\displaystyle \left(0,\,\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {1}{2}}\right),}$
• ${\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {1}{2}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,0\right),}$
• ${\displaystyle \left(\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {1}{2}},\,0\right),}$
• ${\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {1+{\sqrt {5}}}{4}},\,0\right),}$
• ${\displaystyle \left(\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {1}{2}},\,0,\,\pm {\frac {1+{\sqrt {5}}}{4}}\right),}$
• ${\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {1+{\sqrt {5}}}{4}},\,0,\,\pm {\frac {3+{\sqrt {5}}}{4}}\right),}$
• ${\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,0,\,\pm {\frac {1}{2}}\right).}$

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. Both variants have two degrees of freedom and one degree of variation. As a pentagonal double antiprismoid, it has 20 pentagonal antiprisms, 100 tetragonal disphenoids and 200 sphenoids, while maintaining the grand antiprism's symmetry. As a decafold pentaantiprismatoswirlchoron, with half the symmetry of the grand antiprism, its cells are 20 pentagonal gyroprisms, 100 phyllic disphenoids and 200 irregular tetrahedra; its faces are 20 pentagons, 100 isosceles triangles and 200+200+200 scalene triangles and it has 100+100+100+200 edges.