Dodecagonal-truncated icosahedral duoprism

Dodecagonal-truncated icosahedral duoprism
(OFF file)
Rank	5
Type	Uniform
Notation
Bowers style acronym	Twati
Coxeter diagram	x12o o5x3x ()
Elements
Tera	12 pentagonal-dodecagonal duoprisms, 20 hexagonal-dodecagonal duoprisms, 12 truncated icosahedral prisms
Cells	144 pentagonal prisms, 240 hexagonal prisms, 30+60 dodecagonal prisms, 12 truncated icosahedra
Faces	360+720 squares, 144 pentagons, 240 hexagons, 60 dodecagons
Edges	360+720+720
Vertices	720
Vertex figure	Digonal disphenoidal pyramid, edge lengths (1+√5)/2, √3, √3 (base triangle), √2+√3 (top), √2 (side edges)
Measures (edge length 1)
Circumradius	${\sqrt {\frac {45+8{\sqrt {3}}+9{\sqrt {5}}}{8}}}\approx 3.14207$
Hypervolume	$3{\frac {250+125{\sqrt {3}}+86{\sqrt {5}}+43{\sqrt {15}}}{4}}\approx 619.00986$
Diteral angles	Tipe–ti–tipe: 150°
	Pitwadip–twip–hitwadip: $\arccos \left(-{\sqrt {\frac {5+2{\sqrt {5}}}{15}}}\right)\approx 142.62263^{\circ }$
	Hitwadip–twip–hitwadip: $\arccos \left(-{\frac {\sqrt {5}}{3}}\right)\approx 138.18968^{\circ }$
	Pitwadip–pip–tipe: 90°
	Hitwadip–hip–tipe: 90°
Central density	1
Number of external pieces	44
Level of complexity	30
Related polytopes
Army	Twati
Regiment	Twati
Dual	Dodecagonal-pentakis dodecahedral duotegum
Conjugates	Dodecagrammic-truncated icosahedral duoprism, Dodecagonal-truncated great icosahedral duoprism, Dodecagrammic-truncated great icosahedral duoprism
Abstract & topological properties
Euler characteristic	2
Orientable	Yes
Properties
Symmetry	H3×I2(12), order 2880
Convex	Yes
Nature	Tame

The dodecagonal-truncated icosahedral duoprism or twati is a convex uniform duoprism that consists of 12 truncated icosahedral prisms, 20 hexagonal-dodecagonal duoprisms, and 12 pentagonal-dodecagonal duoprisms. Each vertex joins 2 truncated icosahedral prisms, 1 pentagonal-dodecagonal duoprism, and 2 hexagonal-dodecagonal duoprisms.

Vertex coordinates

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

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

Representations

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

• x12o o5x3x () (full symmetry)
• x6x o5x3x () (dodecagons as dihexagons)