# Icositetrachoric antiprism

Icositetrachoric antiprism
File:Icositetrachoric antiprism.png
Rank5
TypeScaliform
Notation
Bowers style acronymIcoap
Elements
Tera192 pentachora, 48 octahedral pyramids, 2 icositetrachora
Cells288+384 tetrahedra, 48 octahedra
Faces192+576 triangles
Edges144+192
Vertices48
Vertex figureOctahedron atop cube, edge length 1
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {3+{\sqrt {2}}}}{2}}\approx 1.05050}$
Height${\displaystyle {\sqrt {{\sqrt {2}}-1}}\approx 0.64359}$
Central density1
Related polytopes
DualIcositetrachoric antitegum
ConjugateIcositetrachoric antiprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
Symmetry(F4×2×A1)/2, order 2304
ConvexYes
NatureTame

The icositetrachoric antiprism or icoap, also known as the icositetrachoric alterprism, is a convex scaliform polyteron that consists of 2 icositetrachora, 48 octahedral pyramids, and 192 pentachora. Each vertex joins 1 icositetrachoron, 7 octahedral pyramids, and 18 pentachora. It can be constructed by lacing two icositetrachora in opposite orientations together.

It is also a quotient prism based on the stellated tetracontoctachoron.

## Vertex coordinates

The vertices of an icositetrachoric antiprism of edge length 1 are given by all permutations excluding the last coordinate of:

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