Decagonal-octahedral duoprism

Decagonal-octahedral duoprism
(OFF file)
Rank	5
Type	Uniform
Notation
Bowers style acronym	Doct
Coxeter diagram	x10o o4o3x ()
Elements
Tera	10 octahedral prisms, 8 triangular-decagonal duoprisms
Cells	80 triangular prisms, 10 octahedra, 12 decagonal prisms
Faces	80 triangles, 120 squares, 6 decagons
Edges	60+120
Vertices	60
Vertex figure	Square scalene, edge lengths 1 (base square), √(5+√5)/2 (top), √2 (sides)
Measures (edge length 1)
Circumradius	${\sqrt {\frac {4+{\sqrt {5}}}{2}}}\approx 1.76580$
Hypervolume	${\frac {5{\sqrt {10+4{\sqrt {5}})}}}{6}}\approx 3.62708$
Diteral angles	Ope–oct–ope: 144°
	Tradedip–dip–tradedip: $\arccos \left(-{\frac {1}{3}}\right)\approx 109.47122^{\circ }$
	Tradedip–trip–ope: 90°
Height	${\frac {\sqrt {6}}{3}}\approx 0.81650$
Central density	1
Number of external pieces	=18
Level of complexity	10
Related polytopes
Army	Doct
Regiment	Doct
Dual	Decagonal-cubic duotegum
Conjugate	Decagrammic-octahedral duoprism
Abstract & topological properties
Euler characteristic	2
Orientable	Yes
Properties
Symmetry	B3×I2(10), order 960
Convex	Yes
Nature	Tame

The decagonal-octahedral duoprism or doct is a convex uniform duoprism that consists of 10 octahedral prisms and 8 triangular-decagonal duoprisms. Each vertex joins 2 octahedral prisms and 4 triangular-decagonal duoprisms.

Vertex coordinates

The vertices of a decagonal-octahedral duoprism of edge length 1 are given by all permutations and sign changes of the last three coordinates of:

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

Representations

A triangular-octahedral duoprism has the following Coxeter diagrams:

• x10o o4o3x (full symmetry)
• x5x o4o3x (decagons as dipentagons)
• x10o o3x3o (octahedra as tetratetrahedra)
• x5x o3x3o (decagons as dipentagons and octahedra as tetratetrahedra)