Square-small rhombicosidodecahedral duoprism

Square-small rhombicosidodecahedral duoprism
(OFF file)
Rank	5
Type	Uniform
Notation
Bowers style acronym	Squasrid
Coxeter diagram	x4o x5o3x ()
Elements
Tera	20 triangular-square duoprisms, 30 tesseracts, 12 square-pentagonal duoprisms, 4 small rhombicosidodecahedral prisms
Cells	80 triangular prisms, 60+60+120 cubes, 48 pentagonal prisms, 4 small rhombicosidodecahedra
Faces	80 triangles, 60+120+240+240 squares, 48 pentagons
Edges	240+240+240
Vertices	240
Vertex figure	Isosceles-trapezoidal scalene, edge lengths 1, √2, (1+√5)/2, √2 (base trapezoid), √2 (top and side edges)
Measures (edge length 1)
Circumradius	${\frac {\sqrt {13+4{\sqrt {5}}}}{2}}\approx 2.34224$
Hypervolume	${\frac {60+29{\sqrt {5}}}{3}}\approx 41.61532$
Diteral angles	Tisdip–cube–tes: $\arccos \left(-{\frac {{\sqrt {3}}+{\sqrt {15}}}{6}}\right)\approx 159.09484^{\circ }$
	Tes–cube–squipdip: $\arccos \left(-{\sqrt {\frac {5+{\sqrt {5}}}{10}}}\right)\approx 148.28253^{\circ }$
	Sriddip–srid–sriddip: 90°
	Tisdip–trip–sriddip: 90°
	Tes–cube–sriddip: 90°
	Squipdip–pip–sriddip: 90°
Height	1
Central density	1
Number of external pieces	66
Level of complexity	40
Related polytopes
Army	Squasrid
Regiment	Squasrid
Dual	Square-deltoidal hexecontahedral duotegum
Conjugate	Square-quasirhombicosidodecahedral duoprism
Abstract & topological properties
Flag count	38400
Euler characteristic	2
Orientable	Yes
Properties
Symmetry	H3×B2, order 960
Convex	Yes
Nature	Tame

The square-small rhombicosidodecahedral duoprism or squasrid is a convex uniform duoprism that consists of 4 small rhombicosidodecahedral prisms, 12 square-pentagonal duoprisms, 30 tesseracts, and 20 triangular-square duoprisms. Each vertex joins 2 small rhombicosidodecahedral prisms, 1 triangular-square duoprism, 2 tesseracts, and 1 square-pentagonal duoprism. It is a duoprism based on a square and a small rhombicosidodecahedron, which makes it a convex segmentoteron.

Vertex coordinates

The vertices of a square-small rhombicosidodecahedral duoprism of edge length 1 are given by all permutations of the last three coordinates of:

• $\left(\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {2+{\sqrt {5}}}{2}}\right),$

as well as all even permutations of the last three coordinates of:

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

Representations

A square-small rhombicosidodecahedral duoprism has the following Coxeter diagrams:

• x4o x5o3x (full symmetry)
• x x x5o3x (small rhombicosidodecahedral prismatic prism)

External links

• Klitzing, Richard. "squasrid".