Icositetrachoric antiprism

Icositetrachoric antiprism
File:Icositetrachoric antiprism.png (OFF file)
Rank	5
Type	Scaliform
Notation
Bowers style acronym	Icoap
Elements
Tera	192 pentachora, 48 octahedral pyramids, 2 icositetrachora
Cells	288+384 tetrahedra, 48 octahedra
Faces	192+576 triangles
Edges	144+192
Vertices	48
Vertex figure	Octahedron atop cube, edge length 1
Measures (edge length 1)
Circumradius	${\frac {\sqrt {3+{\sqrt {2}}}}{2}}\approx 1.05050$
Height	${\sqrt {{\sqrt {2}}-1}}\approx 0.64359$
Central density	1
Related polytopes
Dual	Icositetrachoric antitegum
Conjugate	Icositetrachoric antiprism
Abstract & topological properties
Euler characteristic	2
Orientable	Yes
Properties
Symmetry	(F4×2×A1)/2, order 2304
Convex	Yes
Nature	Tame

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:

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

External links

• Klitzing, Richard. "Icoap".