Square-truncated dodecahedral duoprism

Square-truncated dodecahedral duoprism
(OFF file)
Rank	5
Type	Uniform
Notation
Bowers style acronym	Squatid
Coxeter diagram	x4o x5x3o ()
Elements
Tera	4 truncated dodecahedral prisms, 12 square-decagonal duoprisms, 20 triangular-square duoprisms
Cells	4 truncated dodecahedra, 48 decagonal prisms, 30+60 cubes, 80 triangular prisms
Faces	48 decagons, 60+120+240 squares, 80 triangles
Edges	120+240+240
Vertices	240
Vertex figure	Isosceles triangular scalene, edge lengths 1, √(5+√5)/2, √(5+√5)/2 (base triangle), √2 (top and side edges)
Measures (edge length 1)
Circumradius	${\sqrt {\frac {41+15{\sqrt {5}}}{8}}}\approx 3.05248$
Hypervolume	$5{\frac {99+47{\sqrt {5}}}{12}}\approx 85.03966$
Diteral angles	Tiddip–tid–tiddip: 90°
	Tisdip–trip–tiddip: 90°
	Squadedip–dip–tiddip: 90°
	Tisdip–cube–squadedip: $\arccos \left(-{\sqrt {\frac {5+2{\sqrt {5}}}{15}}}\right)\approx 142.62263^{\circ }$
	Squadedip–cube–squadedip: $\arccos \left(-{\frac {\sqrt {5}}{5}}\right)\approx 116.56505^{\circ }$
Height	Tiddip atop tiddip: 1
Central density	1
Related polytopes
Army	Squatid
Regiment	Squatid
Dual	Square-triakis icosahedral duotegum
Conjugate	Square-quasitruncated great stellated dodecahedral duoprism
Abstract & topological properties
Flag count	28800
Euler characteristic	2
Orientable	Yes
Properties
Symmetry	H3×B2, order 960
Convex	Yes
Nature	Tame

The square-truncated dodecahedral duoprism or squatid is a convex uniform duoprism that consists of 4 truncated dodecahedral prisms, 12 square-decagonal duoprisms and 20 triangular-square duoprisms. Each vertex joins 2 truncated dodecahedral prisms, 1 triangular-square duoprism, and 2 square-decagonal duoprisms. It is a duoprism based on a square and a truncated dodecahedron, which makes it a convex segmentoteron.

Vertex coordinates

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

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

Representations

A square-truncated dodecahedral duoprism has the following Coxeter diagrams:

• x4o x5x3o (full symmetry)
• x x x5x3o (truncated dodecahedral prismatic prism)

External links

• Klitzing, Richard. "squatid".