Decagonal-dodecagrammic duoprism

Decagonal-dodecagrammic duoprism
(OFF file)
Rank	4
Type	Uniform
Notation
Coxeter diagram	x10o x12/5o ()
Elements
Cells	12 decagonal prisms, 10 dodecagrammic prisms
Faces	120 squares, 12 decagons, 10 dodecagrams
Edges	120+120
Vertices	120
Vertex figure	Digonal disphenoid, edge lengths √(5+√5)/2 (base 1), (√6–√2)/2 (base 2), √2 (sides)
Measures (edge length 1)
Circumradius	${\sqrt {\frac {7-2{\sqrt {3}}+{\sqrt {5}}}{2}}}\approx 1.69882$
Hypervolume	${\frac {15{\sqrt {35-20{\sqrt {3}}+14{\sqrt {5}}-8{\sqrt {15}}}}}{2}}\approx 6.18497$
Dichoral angles	Stwip–12/5–stwip: 144°
	Dip–4–stwip: 90°
	Dip–10–dip: 30°
Central density	5
Number of external pieces	34
Level of complexity	12
Related polytopes
Army	Semi-uniform datwadip
Dual	Decagonal-dodecagrammic duotegum
Conjugates	Decagonal-dodecagonal duoprism, Decagrammic-dodecagonal duoprism, Decagrammic-dodecagrammic duoprism
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	I2(10)×I2(12), order 480
Convex	No
Nature	Tame

The decagonal-dodecagrammic duoprism, also known as the 10-12/5 duoprism, is a uniform duoprism that consists of 12 decagonal prisms and 10 dodecagrammic prisms, with 2 of each at each vertex.

Vertex coordinates

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

• $\left(\pm {\frac {1}{2}},\,\pm {\frac {\sqrt {5+2{\sqrt {5}}}}{2}},\,\pm {\frac {{\sqrt {3}}-1}{2}},\,\pm {\frac {{\sqrt {3}}-1}{2}}\right)$,
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {\sqrt {5+2{\sqrt {5}}}}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {2-{\sqrt {3}}}{2}}\right)$,
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {\sqrt {5+2{\sqrt {5}}}}{2}},\,\pm {\frac {2-{\sqrt {3}}}{2}},\,\pm {\frac {1}{2}}\right)$,
• $\left(\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\sqrt {\frac {5+{\sqrt {5}}}{8}}},\,\pm {\frac {{\sqrt {3}}-1}{2}},\,\pm {\frac {{\sqrt {3}}-1}{2}}\right)$,
• $\left(\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\sqrt {\frac {5+{\sqrt {5}}}{8}}},\,\pm {\frac {1}{2}},\,\pm {\frac {2-{\sqrt {3}}}{2}}\right)$,
• $\left(\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\sqrt {\frac {5+{\sqrt {5}}}{8}}},\,\pm {\frac {2-{\sqrt {3}}}{2}},\,\pm {\frac {1}{2}}\right)$,
• $\left(\pm {\frac {1+{\sqrt {5}}}{2}},\,0,\,\pm {\frac {{\sqrt {3}}-1}{2}},\,\pm {\frac {{\sqrt {3}}-1}{2}}\right)$,
• $\left(\pm {\frac {1+{\sqrt {5}}}{2}},\,0,\,\pm {\frac {1}{2}},\,\pm {\frac {2-{\sqrt {3}}}{2}}\right)$,
• $\left(\pm {\frac {1+{\sqrt {5}}}{2}},\,0,\,\pm {\frac {2-{\sqrt {3}}}{2}},\,\pm {\frac {1}{2}}\right)$.

Representations

A decagonal-dodecagrammic duoprism has the following Coxeter diagrams:

• x10o x12/5o () (full symmetry)
• x5x x12/5o () (H₂×I₂(12) symmetry, decagons as dipentagons)
• x6/5x x10o () (G₂×I₂(10) symmetry, dodecagrams as dihexagrams)
• x5x x6/5x () (H₂×G₂ symmetry)

External links

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

• Klitzing, Richard. "nd-mb-dip".