Dodecagonal-truncated dodecahedral duoprism

Dodecagonal-truncated dodecahedral duoprism
(OFF file)
Rank	5
Type	Uniform
Notation
Bowers style acronym	Twatid
Coxeter diagram	x12o x5x3o ()
Elements
Tera	20 triangular-dodecagonal duoprisms, 12 decagonal-dodecagonal duoprisms, 12 truncated dodecahedral prisms
Cells	240 triangular prisms, 144 decagonal prisms, 30+60 dodecagonal prisms, 12 truncated dodecahedra
Faces	240 triangles, 360+720 squares, 144 decagons, 60 dodecagons
Edges	360+720+720
Vertices	720
Vertex figure	Digonal disphenoidal pyramid, edge lengths 1, √(5+√5)/2, √(5+√5)/2 (base triangle), √2+√3 (top), √2 (side edges)
Measures (edge length 1)
Circumradius	${\sqrt {\frac {53+8{\sqrt {3}}+15{\sqrt {5}}}{8}}}\approx 3.54255$
Hypervolume	$5{\frac {198+99{\sqrt {3}}+94{\sqrt {5}}+47{\sqrt {15}}}{4}}\approx 952.11705$
Diteral angles	Tiddip–tid–tiddip: 150°
	Titwadip–twip–datwadip: $\arccos \left(-{\sqrt {\frac {5+2{\sqrt {5}}}{15}}}\right)\approx 142.62263^{\circ }$
	Datwadip–twip–datwadip: $\arccos \left(-{\frac {\sqrt {5}}{5}}\right)\approx 116.56505^{\circ }$
	Titwadip–trip–tiddip: 90°
	Datwadip–dip–tiddip: 90°
Central density	1
Number of external pieces	44
Level of complexity	30
Related polytopes
Army	Twatid
Regiment	Twatid
Dual	Dodecagonal-triakis icosahedral duotegum
Conjugates	Dodecagrammic-truncated dodecahedral duoprism, Dodecagonal-quasitruncated great stellated dodecahedral duoprism, Dodecagrammic-quasitruncated great stellated dodecahedral duoprism
Abstract & topological properties
Euler characteristic	2
Orientable	Yes
Properties
Symmetry	H3×I2(12), order 2880
Convex	Yes
Nature	Tame

The dodecagonal-truncated dodecahedral duoprism or twatid is a convex uniform duoprism that consists of 12 truncated dodecahedral prisms, 12 decagonal-dodecagonal duoprisms and 20 triangular-dodecagonal duoprisms. Each vertex joins 2 truncated dodecahedral prisms, 1 triangular-dodecagonal duoprism, and 2 decagonal-dodecagonal duoprisms.

Vertex coordinates

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

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

Representations

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

• x12o x5x3o () (full symmetry)
• x6x x5x3o () (dodecagons as dihexagons)