Pentagonal-dodecagrammic duoprism

Pentagonal-dodecagrammic duoprism
(OFF file)
Rank	4
Type	Uniform
Notation
Coxeter diagram	x5o x12/5o ()
Elements
Cells	12 pentagonal prisms, 5 dodecagrammic prisms
Faces	60 squares, 12 pentagons, 5 dodecagrams
Edges	60+60
Vertices	60
Vertex figure	Digonal disphenoid, edge lengths (1+√5)/2 (base 1), (√6–√2)/2 (base 2), √2 (sides)
Measures (edge length 1)
Circumradius	${\sqrt {\frac {25-10{\sqrt {3}}+{\sqrt {5}}}{10}}}\approx 0.99577$
Hypervolume	${\frac {3{\sqrt {175-100{\sqrt {3}}+70{\sqrt {5}}-40{\sqrt {15}}}}}{4}}\approx 1.38300$
Dichoral angles	Stwip–12/5–stwip: 108°
	Pip–4–stwip: 90°
	Pip–5–pip: 30°
Central density	5
Number of external pieces	29
Level of complexity	12
Related polytopes
Army	Semi-uniform pitwadip
Dual	Pentagonal-dodecagrammic duotegum
Conjugates	Pentagonal-dodecagonal 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
Convex	No
Nature	Tame

The pentagonal-dodecagrammic duoprism, also known the 5-12/5 duoprism, is a uniform duoprism that consists of 12 pentagonal prisms and 5 dodecagrammic prisms, with two of each at each vertex.

Vertex coordinates

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

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

Representations

A pentagonal-dodecagrammic duoprism has the following Coxeter diagrams:

• x5o x12/5o () (full symmetry)
• x5o x6/5x () (H₂×G₂ symmetry, dodecagrams as dihexagrams)

External links

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

• Klitzing, Richard. "nd-mb-dip".