Decagonal-dodecagonal duoprism

Decagonal-dodecagonal duoprism
(OFF file)
Rank	4
Type	Uniform
Notation
Bowers style acronym	Datwadip
Coxeter diagram	x10o x12o ()
Elements
Cells	12 decagonal prisms, 10 dodecagonal prisms
Faces	120 squares, 12 decagons, 10 dodecagons
Edges	120+120
Vertices	120
Vertex figure	Digonal disphenoid, edge lengths √(5+√5)/2 (base 1), (√2+√6)/2 (base 2), and √2 (sides)
Measures (edge length 1)
Circumradius	${\sqrt {\frac {7+2{\sqrt {3}}+{\sqrt {5}}}{2}}}\approx 2.51994$
Hypervolume	${\frac {15{\sqrt {35+20{\sqrt {3}}+14{\sqrt {5}}+8{\sqrt {15}}}}}{2}}\approx 86.14553$
Dichoral angles	Twip–12–twip: 144°
	Dip–10–dip: 150°
	Twip–4–dip: 90°
Central density	1
Number of external pieces	22
Level of complexity	6
Related polytopes
Army	Datwadip
Regiment	Datwadip
Dual	Decagonal-dodecagonal duotegum
Conjugates	Decagonal-dodecagrammic duoprism, Decagrammic-dodecagonal duoprism, Decagrammic-dodecagrammic duoprism
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	I2(10)×I2(12), order 480
Convex	Yes
Nature	Tame

The decagonal-dodecagonal duoprism or datwadip, also known as the 10-12 duoprism, is a uniform duoprism that consists of 10 dodecagonal prisms and 12 decagonal prisms, with two of each joining at each vertex.

This polychoron can be alternated into a pentagonal-hexagonal duoantiprism, although it cannot be made uniform. The dodecagons can also be alternated into long ditrigons to create a pentagonal-hexagonal prismantiprismoid, which is also nonuniform.

Vertex coordinates

The coordinates of a decagonal-dodecagonal duoprism of edge length 1, centered at the origin, are given by:

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

Representations

A decagonal-dodecagonal duoprism has the following Coxeter diagrams:

• x10o x12o (full symmetry)
• x5x x12o (decagons as dipentagons)
• x6x x10o (dodecagons as dihexagons)
• x5x x6x (both of these applied)

External links

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