Pentagonal-cubic duoprism

Pentagonal-cubic duoprism
	(OFF file)
Rank	5
Type	Uniform
Notation
Bowers style acronym	Pecube
Coxeter diagram	x5o x4o3o
Elements
Tera	5 tesseracts, 6 square-pentagonal duoprisms
Cells	5+30 cubes, 12 pentagonal prisms
Faces	30+60 squares, 8 pentagons
Edges	40+60
Vertices	40
Vertex figure	Triangular scalene, edge lengths (1+√5)/2 (top), √2 (base triangle and sides)
Measures (edge length 1)
Circumradius
Hypervolume
Diteral angles	Tes–cube–tes: 108°
	Tes–cube–squipdip: 90°
	Squipdip–pip–squipdip: 90°
Height	1
Central density	1
Number of external pieces	11
Level of complexity	10
Related polytopes
Army	Pecube
Regiment	Pecube
Dual	Pentagonal-octahedral duotegum
Conjugate	Pentagrammic-cubic duoprism
Abstract & topological properties
Euler characteristic	2
Orientable	Yes
Properties
Symmetry	B3×H2, order 480
Convex	Yes
Nature	Tame

The pentagonal-cubic duoprism or pecube, also known as a square-pentagonal duoprismatic prism, is a convex uniform duoprism that consists of 5 tesseracts and 6 square-pentagonal duoprisms. Each vertex joins 2 tesseracts and 3 square-pentagonal duoprisms. It is a duoprism based on a square and a pentagonal prism, which makes it a convex segmentoteron.

Vertex coordinates[edit | edit source]

The vertices of a pentagonal-cubic duoprism of edge length 1 are given by:

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

A pentagonal-cubic duoprism has the following Coxeter diagrams: