Rectified small prismatotetracontoctachoron

Rectified small prismatotetracontoctachoron
	(OFF file)
Rank	4
Type	Isogonal
Notation
Bowers style acronym	Respic
Coxeter diagram	uo3ox4ox3ou&#zq
Elements
Cells	192 rectified triangular prisms, 144 square antiprisms, 48 cuboctahedra
Faces	1152 isosceles triangles, 384 triangles, 288+288 squares
Edges	1152+1152
Vertices	576
Vertex figure	Isosceles trapezoidal prism
Measures (short edge length 1)
Edge lengths	Edges of triangles (576): 1
	Lacing edges (576):
Circumradius
Central density	1
Related polytopes
Army	Respic
Regiment	Respic
Dual	Joined square-antitegmatic hecatontetracontatetrachoron
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	F4×2, order 2304
Convex	Yes
Nature	Tame

The rectified small prismatotetracontoctachoron or respic is a convex isogonal polychoron that consists of 48 cuboctahedra, 192 rectified triangular prisms and 144 square antiprisms. 1 cuboctahedron, 3 rectified triangular prisms, and 2 square antiprisms join at each vertex. It can be formed by rectifying the small prismatotetracontoctachoron.

It can also be formed as the convex hull of 2 oppositely oriented semi-uniform variants of the small rhombated icositetrachoron, where the edges of the cuboctahedra are half the length of the other edges. It is one of five polychora (including two transitional cases) formed in such a way, and is the transitional point between the small birhombatotetracontoctachoron and medial birhombatotetracontoctachoron.

The ratio between the longest and shortest edges is 1: ${\sqrt {2}}$ ≈ 1:1.41421.

Vertex coordinates[edit | edit source]

The vertices of a rectified small prismatotetracontoctachoron with triangles of edge length 1, centered at the origin, are given by:

$\left(0,\,\pm {\frac {\sqrt {2}}{2}},\,\pm {\frac {\sqrt {2}}{2}},\,\pm (2+{\sqrt {2}})\right),$
$\left(\pm 1,\,\pm {\frac {2+{\sqrt {2}}}{2}},\,\pm {\frac {2+{\sqrt {2}}}{2}},\,\pm (1+{\sqrt {2}})\right),$
$\left(\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1+2{\sqrt {2}}}{2}},\,\pm {\frac {3+2{\sqrt {2}}}{2}}\right),$
$\left(0,\,\pm 1,\,\pm (1+{\sqrt {2}}),\,\pm (1+{\sqrt {2}})\right).$