Dodecagonal-pentagonal antiprismatic duoprism

Dodecagonal-pentagonal antiprismatic duoprism
(OFF file)
Rank	5
Type	Uniform
Notation
Bowers style acronym	Twapap
Coxeter diagram	x12o s2s10o ()
Elements
Tera	12 pentagonal antiprismatic prisms, 10 triangular-dodecagonal duoprisms, 2 pentagonal-dodecagonal duoprisms
Cells	120 triangular prisms, 24 pentagonal prisms, 12 pentagonal antiprisms, 10+10 dodecagonal prisms
Faces	120 triangles, 120+120 squares, 24 pentagons, 10 dodecagons
Edges	120+120+120
Vertices	120
Vertex figure	Isosceles-trapezoidal scalene, edge lengths 1, 1, 1, (1+√5)/2 (base trapezoid), √2+√3 (top), √2 (side edges)
Measures (edge length 1)
Circumradius	${\sqrt {\frac {21+8{\sqrt {3}}+{\sqrt {5}}}{8}}}\approx 2.15327$
Hypervolume	${\frac {10+5{\sqrt {2}}+4{\sqrt {5}}+2{\sqrt {10}}}{2}}\approx 17.67525$
Diteral angles	Pappip–pap–pappip: 150°
	Titwadip–twip–titwadip: = $\arccos \left(-{\frac {\sqrt {5}}{3}}\right)\approx 138.18969^{\circ }$
	Titwadip–twip–pitwadip: = $\arccos \left(-{\sqrt {\frac {5-2{\sqrt {5}}}{15}}}\right)\approx 100.81232^{\circ }$
	Titwadip–trip–pappip: 90°
	Pitwadip–pip–pappip: 90°
Height	${\sqrt {\frac {5+{\sqrt {5}}}{10}}}\approx 0.85065$
Central density	1
Number of external pieces	24
Level of complexity	40
Related polytopes
Army	Twapap
Regiment	Twapap
Dual	Dodecagonal-pentagonal antitegmatic duotegum
Conjugates	Dodecagrammic-pentagonal antiprismatic duoprism, dodecagonal-pentagrammic retroprismatic duoprism, dodecagrammic-pentagrammic retroprismatic duoprism
Abstract & topological properties
Euler characteristic	2
Orientable	Yes
Properties
Symmetry	I2(12)×I2(10)×A1+, order 480
Convex	Yes
Nature	Tame

The dodecagonal-pentagonal antiprismatic duoprism or twapap is a convex uniform duoprism that consists of 12 pentagonal antiprismatic prisms, 2 pentagonal-dodecagonal duoprisms, and 10 triangular-dodecagonal duoprisms. Each vertex joins 2 pentagonal antiprismatic prisms, 3 triangular-dodecagonal duoprisms, and 1 pentagonal-dodecagonal duoprism.

Vertex coordinates

The vertices of a dodecagonal-pentagonal antiprismatic duoprism of edge length 1 are given by all central inversions of the last three coordinates of:

• $\left(\pm {\frac {1+{\sqrt {3}}}{2}},\,\pm {\frac {1+{\sqrt {3}}}{2}},\,0,\,{\sqrt {\frac {5+{\sqrt {5}}}{10}}},\,{\sqrt {\frac {5+{\sqrt {5}}}{40}}}\right),$
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {2+{\sqrt {3}}}{2}},\,0,\,{\sqrt {\frac {5+{\sqrt {5}}}{10}}},\,{\sqrt {\frac {5+{\sqrt {5}}}{40}}}\right),$
• $\left(\pm {\frac {2+{\sqrt {3}}}{2}},\,\pm {\frac {1}{2}},\,0,\,{\sqrt {\frac {5+{\sqrt {5}}}{10}}},\,{\sqrt {\frac {5+{\sqrt {5}}}{40}}}\right),$
• $\left(\pm {\frac {1+{\sqrt {3}}}{2}},\,\pm {\frac {1+{\sqrt {3}}}{2}},\,\pm {\frac {1+{\sqrt {5}}}{4}},\,{\sqrt {\frac {5-{\sqrt {5}}}{40}}},\,{\sqrt {\frac {5+{\sqrt {5}}}{40}}}\right),$
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {2+{\sqrt {3}}}{2}},\,\pm {\frac {1+{\sqrt {5}}}{4}},\,{\sqrt {\frac {5-{\sqrt {5}}}{40}}},\,{\sqrt {\frac {5+{\sqrt {5}}}{40}}}\right),$
• $\left(\pm {\frac {2+{\sqrt {3}}}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1+{\sqrt {5}}}{4}},\,{\sqrt {\frac {5-{\sqrt {5}}}{40}}},\,{\sqrt {\frac {5+{\sqrt {5}}}{40}}}\right),$
• $\left(\pm {\frac {1+{\sqrt {3}}}{2}},\,\pm {\frac {1+{\sqrt {3}}}{2}},\,\pm {\frac {1}{2}},\,-{\sqrt {\frac {5-2{\sqrt {5}}}{20}}},\,{\sqrt {\frac {5+{\sqrt {5}}}{40}}}\right),$
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {2+{\sqrt {3}}}{2}},\,\pm {\frac {1}{2}},\,-{\sqrt {\frac {5-2{\sqrt {5}}}{20}}},\,{\sqrt {\frac {5+{\sqrt {5}}}{40}}}\right),$
• $\left(\pm {\frac {2+{\sqrt {3}}}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,-{\sqrt {\frac {5-2{\sqrt {5}}}{20}}},\,{\sqrt {\frac {5+{\sqrt {5}}}{40}}}\right).$

Representations

A dodecagonal-pentagonal antiprismatic duoprism has the following Coxeter diagrams:

• x12o s2s10o () (full symmetry; pentagonal antiprisms as alternated decagonal prisms)
• x12o s2s5s () (pentagonal antiprisms as alternated dipentagonal prisms)
• x6x s2s10o () (dodecagons as dihexagons)
• x6x s2s5s ()