Pentagonal-dodecagonal duoprism

Pentagonal-dodecagonal duoprism
(OFF file)
Rank	4
Type	Uniform
Notation
Bowers style acronym	Pitwadip
Coxeter diagram	x5o x12o ()
Elements
Cells	12 pentagonal prisms, 5 dodecagonal prisms
Faces	60 squares, 12 pentagons, 5 dodecagons
Edges	60+60
Vertices	60
Vertex figure	Digonal disphenoid, edge lengths (1+√5)/2 (base 1), (√2+√6)/2 (base 2), and √2 (sides)
Measures (edge length 1)
Circumradius	${\sqrt {\frac {25+10{\sqrt {3}}+{\sqrt {5}}}{10}}}\approx 2.11084$
Hypervolume	${\frac {3{\sqrt {175+100{\sqrt {3}}+70{\sqrt {5}}+40{\sqrt {15}}}}}{4}}\approx 19.26273$
Dichoral angles	Pip–5–pip: 150°
	Twip–12–twip: 108°
	Twip–4–pip: 90°
Central density	1
Number of external pieces	17
Level of complexity	6
Related polytopes
Army	Pitwadip
Regiment	Pitwadip
Dual	Pentagonal-dodecagonal duotegum
Conjugates	Pentagonal-dodecagrammic duoprism, Pentagrammic-dodecagonal duoprism, Pentagrammic-dodecagrammic duoprism
Abstract & topological properties
Flag count	1440
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	H2×I2(12), order 240
Flag orbits	6
Convex	Yes
Nature	Tame

The pentagonal-dodecagonal duoprism or pitwadip, also known as the 5-12 duoprism, is a uniform duoprism that consists of 5 dodecagonal prisms and 12 pentagonal prisms, with two of each joining at each vertex.

Vertex coordinates

The coordinates of a pentagonal-dodecagonal duoprism, centered at the origin and with unit edge length, are given by:

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

Representations

A pentagonal-dodecagonal duoprism has the following Coxeter diagrams:

• x5o x12o () (full symmetry)
• x5o x6x () (H₂×G₂, dodecagons as dihexagons)

External links

• Bowers, Jonathan. "Category A: Duoprisms".