Pentagonal-cuboctahedral duoprism

Pentagonal-cuboctahedral duoprism
	(OFF file)
Rank	5
Type	Uniform
Notation
Bowers style acronym	Peco
Coxeter diagram	x5o o4x3o
Elements
Tera	8 triangular-pentagonal duoprisms, 6 square-pentagonal duoprisms, 5 cuboctahedral prisms
Cells	40 triangular prisms, 30 cubes, 24 pentagonal prisms, 5 cuboctahedra
Faces	40 triangles, 30+120 squares, 12 pentagons
Edges	60+120
Vertices	60
Vertex figure	Rectangular scalene, edge lengths 1, √2, 1, √2 (base rectangle), (1+√5)/2 (top), √2 (side edges)
Measures (edge length 1)
Circumradius
Hypervolume
Diteral angles	Trapedip–pip–squipdip:
	Cope–co–cope: 108°
	Trapedip–trip–cope: 90°
	Squipdip–cube–cope: 90°
Central density	1
Number of external pieces	19
Level of complexity	20
Related polytopes
Army	Peco
Regiment	Peco
Dual	Pentagonal-rhombic dodecahedral duotegum
Conjugate	Pentagrammic-cuboctahedral duoprism
Abstract & topological properties
Euler characteristic	2
Orientable	Yes
Properties
Symmetry	B3×H2, order 480
Convex	Yes
Nature	Tame

The pentagonal-cuboctahedral duoprism or peco is a convex uniform duoprism that consists of 5 cuboctahedral prisms, 6 square-pentagonal duoprisms, and 8 triangular-pentagonal duoprisms. Each vertex joins 2 cuboctahedral prisms, 2 triangular-pentagonal duoprisms, and 2 square-pentagonal duoprisms.

Vertex coordinates[edit | edit source]

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

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