Decagonal-cuboctahedral duoprism

Decagonal-cuboctahedral duoprism
	(OFF file)
Rank	5
Type	Uniform
Notation
Bowers style acronym	Daco
Coxeter diagram	x10o o4x3o ()
Elements
Tera	10 cuboctahedral prisms, 8 triangular-decagonal duoprisms, 6 square-decagonal duoprisms
Cells	80 triangular prisms, 60 cubes, 10 cuboctahedra, 24 decagonal prisms
Faces	80 triangles, 60+240 squares, 12 decagons
Edges	120+240
Vertices	120
Vertex figure	Rectangular scalene, edge lengths 1, √2, 1, √2 (base rectangle), √(5+√5)/2 (top), √2 (side edges)
Measures (edge length 1)
Circumradius
Hypervolume
Diteral angles	Cope–co–cope: 144°
	Tradedip–dip–squadedip:
	Tradedip–trip–cope: 90°
	Squadedip–cube–cope: 90°
Central density	1
Number of external pieces	24
Level of complexity	20
Related polytopes
Army	Daco
Regiment	Daco
Dual	Decagonal-rhombic dodecahedral duotegum
Conjugate	Decagrammic-cuboctahedral duoprism
Abstract & topological properties
Euler characteristic	2
Orientable	Yes
Properties
Symmetry	B3×I2(10), order 960
Convex	Yes
Nature	Tame

The decagonal-cuboctahedral duoprism or daco is a convex uniform duoprism that consists of 10 cuboctahedral prisms, 6 square-decagonal duoprisms, and 8 triangular-decagonal duoprisms. Each vertex joins 2 cuboctahedral prisms, 2 triangular-decagonal duoprisms, and 2 square-decagonal duoprisms.

Vertex coordinates[edit | edit source]

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

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

A decagonal-cuboctahedral duoprism has the following Coxeter diagrams: