Pentagonal-cubic duoprism

Pentagonal-cubic duoprism
Rank5
TypeUniform
Notation
Bowers style acronymPecube
Coxeter diagramx5o x4o3o
Elements
Tera5 tesseracts, 6 square-pentagonal duoprisms
Cells5+30 cubes, 12 pentagonal prisms
Faces30+60 squares, 8 pentagons
Edges40+60
Vertices40
Vertex figureTriangular scalene, edge lengths (1+5)/2 (top), 2 (base triangle and sides)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {25+2{\sqrt {5}}}{20}}}\approx 1.21392}$
Hypervolume${\displaystyle {\frac {\sqrt {5(5+2{\sqrt {5}})}}{4}}\approx 1.72048}$
Diteral anglesTes–cube–tes: 108°
Tes–cube–squipdip: 90°
Squipdip–pip–squipdip: 90°
Height1
Central density1
Number of external pieces11
Level of complexity10
Related polytopes
ArmyPecube
RegimentPecube
DualPentagonal-octahedral duotegum
ConjugatePentagrammic-cubic duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryB3×H2, order 480
ConvexYes
NatureTame

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

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

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

Representations

A pentagonal-cubic duoprism has the following Coxeter diagrams:

• x5o x4o3o (full symmetry)
• x x4o x5o (square-pentagonal duoprismatic prism)
• x x x x5o (pentagonal prismatic prismatic prism)