# Pentagonal-great rhombicuboctahedral duoprism

Pentagonal-great rhombicuboctahedral duoprism
Rank5
TypeUniform
Notation
Bowers style acronymPegirco
Coxeter diagramx5o x4x3x ()
Elements
Tera12 square-pentagonal duoprisms, 8 pentagonal-hexagonal duoprisms, 6 pentagonal-octagonal duoprisms, 5 great rhombicuboctahedral prisms
Cells60 cubes, 24+24+24 pentagonal prisms, 40 hexagonal prisms, 30 octagonal prisms, 5 great rhombicuboctahedra
Faces60+120+120+120 squares, 48 pentagons, 40 hexagons, 30 octagons
Edges120+120+120+240
Vertices240
Vertex figureMirror-symmetric pentachoron, edge lengths 2, 3, 2+2 (base triangle), (1+5)/2 (top edge), 2 (side edges)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {75+30{\sqrt {2}}+2{\sqrt {5}}}{20}}}\approx 2.46879}$
Hypervolume${\displaystyle {\frac {\sqrt {5475+3850{\sqrt {2}}+2190{\sqrt {5}}+1540{\sqrt {10}}}}{2}}\approx 71.91422}$
Diteral anglesSquipdip–pip–phiddip: ${\displaystyle \arccos \left(-{\frac {\sqrt {6}}{3}}\right)\approx 144.73561^{\circ }}$
Squipdip–pip–podip: 135°
Phiddip–pip–podip: ${\displaystyle \arccos \left(-{\frac {\sqrt {3}}{3}}\right)\approx 125.26439^{\circ }}$
Gircope–girco–gircope: 108°
Squipdip–cube–gircope: 90°
Phiddip–hip–gircope: 90°
Podip–op–gircope: 90°
Central density1
Number of external pieces31
Level of complexity60
Related polytopes
ArmyPegirco
RegimentPegirco
DualPentagonal-disdyakis dodecahedral duotegum
ConjugatesPentagrammic-great rhombicuboctahedral duoprism, Pentagonal-quasitruncated cuboctahedral duoprism, Pentagrammic-quasitruncated cuboctahedral duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryB3×H2, order 480
ConvexYes
NatureTame

The pentagonal-great rhombicuboctahedral duoprism or pegirco is a convex uniform duoprism that consists of 5 great rhombicuboctahedral prisms, 6 pentagonal-octagonal duoprisms, 8 pentagonal-hexagonal duoprisms, and 8 square-pentagonal duoprisms. Each vertex joins 2 great rhombicuboctahedral prisms, 1 square-pentagonal duoprism, 1 pentagonal-hexagonal duoprism, and 1 pentagonal-octagonal duoprism.

## Vertex coordinates

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

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