# Pentagonal-cuboctahedral duoprism

Pentagonal-cuboctahedral duoprism
Rank5
TypeUniform
Notation
Bowers style acronymPeco
Coxeter diagramx5o o4x3o
Elements
Tera8 triangular-pentagonal duoprisms, 6 square-pentagonal duoprisms, 5 cuboctahedral prisms
Cells40 triangular prisms, 30 cubes, 24 pentagonal prisms, 5 cuboctahedra
Faces40 triangles, 30+120 squares, 12 pentagons
Edges60+120
Vertices60
Vertex figureRectangular scalene, edge lengths 1, 2, 1, 2 (base rectangle), (1+5)/2 (top), 2 (side edges)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {15+{\sqrt {5}}}{10}}}\approx 1.31286}$
Hypervolume${\displaystyle {\frac {5{\sqrt {50+20{\sqrt {5}}}}}{12}}\approx 4.05520}$
Diteral anglesTrapedip–pip–squipdip: ${\displaystyle \arccos \left(-{\frac {\sqrt {3}}{3}}\right)\approx 125.26439^{\circ }}$
Cope–co–cope: 108°
Trapedip–trip–cope: 90°
Squipdip–cube–cope: 90°
Central density1
Number of external pieces19
Level of complexity20
Related polytopes
ArmyPeco
RegimentPeco
DualPentagonal-rhombic dodecahedral duotegum
ConjugatePentagrammic-cuboctahedral duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryB3×H2, order 480
ConvexYes
NatureTame

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

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

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