# Pentagonal-small rhombicuboctahedral duoprism

Pentagonal-small rhombicuboctahedral duoprism
Rank5
TypeUniform
Notation
Bowers style acronymPesirco
Coxeter diagramx5o x4o3x
Elements
Tera8 triangular-pentagonal duoprisms, 6+12 square-pentagonal duoprisms, 5 small rhombicuboctahedral prisms
Cells40 triangular prisms, 30+60 cubes, 24+24 pentagonal prisms, 5 small rhombicuboctahedra
Faces40 triangles, 30+60+120+120 squares, 24 pentagons
Edges120+120+120
Vertices120
Vertex figureIsosceles-trapezoidal scalene, edge lengths 1, 2, 2, 2 (base trapezoid), (1+5)/2 (top), 2 (side edges)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {35+10{\sqrt {2}}+2{\sqrt {5}}}{20}}}\approx 1.63729}$
Hypervolume${\displaystyle {\frac {\sqrt {2150+1500{\sqrt {2}}+860{\sqrt {5}}+60{\sqrt {10}}}}{6}}\approx 14.99232}$
Diteral anglesTrapedip–pip–squipdip: ${\displaystyle \arccos \left(-{\frac {\sqrt {6}}{3}}\right)\approx 144.73561^{\circ }}$
Squipdip–pip–squipdip: 135°
Sircope–sirco–sircope: 108°
Trapedip–trip–sircope: 90°
Squipdip–cube–sircope: 90°
Central density1
Number of external pieces31
Level of complexity40
Related polytopes
ArmyPesirco
RegimentPesirco
DualPentagonal-deltoidal icositetrahedral duotegum
ConjugatesPentagrammic-small rhombicuboctahedral duoprism, Pentagonal-quasirhombicuboctahedral duoprism, Pentagrammic-quasirhombicuboctahedral duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryB3×H2, order 480
ConvexYes
NatureTame

The pentagonal-small rhombicuboctahedral duoprism or pesirco is a convex uniform duoprism that consists of 5 small rhombicuboctahedral prisms, 18 square-pentagonal duoprisms of two kinds, and 8 triangular-pentagonal duoprisms. Each vertex joins 2 small rhombicuboctahedral prisms, 1 triangular-pentagonal duoprism, and 3 square-pentagonal duoprisms.

## Vertex coordinates

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

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