# Decagonal-great rhombicuboctahedral duoprism

Decagonal-great rhombicuboctahedral duoprism
Rank5
TypeUniform
Notation
Bowers style acronymDagirco
Coxeter diagramx10o x4x3x ()
Elements
Tera12 square-decagonal duoprisms, 8 hexagonal-decagonal duoprisms, 6 octagonal-decagonal duoprisms
Cells120 cubes, 80 hexagonal prisms, 60 octagonal prisms, 24+24+24 decagonal prisms, 10 great rhombicuboctahedra
Faces120+240+240+240 squares, 80 hexagons, 60 octagons, 48 decagons
Edges240+240+240+480
Vertices480
Vertex figureMirror-symmetric pentachoron, edge lengths 2, 3, 2+2 (base triangle), (5+5)/2 (top edge), 2 (side edges)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {19+6{\sqrt {2}}+2{\sqrt {5}}}}{2}}\approx 2.82654}$
Hypervolume${\displaystyle 5{\sqrt {1095+770{\sqrt {2}}+438{\sqrt {5}}+308{\sqrt {10}}}}\approx 321.61016}$
Diteral anglesSquadedip–dip–hadedip: ${\displaystyle \arccos \left(-{\frac {\sqrt {6}}{3}}\right)\approx 144.73561^{\circ }}$
Gircope–girco–gircope: 144°
Hadedip–dip–odedip: ${\displaystyle \arccos \left(-{\frac {\sqrt {3}}{3}}\right)\approx 125.26439^{\circ }}$
Odedip–op–gircope: 90°
Central density1
Number of external pieces36
Level of complexity60
Related polytopes
ArmyDagirco
RegimentDagirco
DualDecagonal-disdyakis dodecahedral duotegum
ConjugatesDecagrammic-great rhombicuboctahedral duoprism, Decagonal-quasitruncated cuboctahedral duoprism, Decagrammic-quasitruncated cuboctahedral duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryB3×I2(10), order 960
ConvexYes
NatureTame

The decagonal-great rhombicuboctahedral duoprism or dagirco is a convex uniform duoprism that consists of 10 great rhombicuboctahedral prisms, 6 octagonal-decagonal duoprisms, 8 hexagonal-decagonal duoprisms, and 12 square-decagonal duoprisms. Each vertex joins 2 great rhombicuboctahedral prisms, 1 square-decagonal duoprism, 1 hexagonal-decagonal duoprism, and 1 octagonal-decagonal duoprism.

This polyteron can be alternated into a pentagonal-snub cubic duoantiprism, although it cannot be made uniform. The great rhombicuboctahedra can be edge-snubbed to create a pentagonal-pyritohedral prismantiprismoid, which is also nonuniform.

## Vertex coordinates

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

• ${\displaystyle \left(0,\,\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {1+2{\sqrt {2}}}{2}},\,\pm {\frac {1+{\sqrt {2}}}{2}},\,\pm {\frac {1}{2}}\right),}$
• ${\displaystyle \left(\pm {\sqrt {\frac {5+{\sqrt {5}}}{8}}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {1+2{\sqrt {2}}}{2}},\,\pm {\frac {1+{\sqrt {2}}}{2}},\,\pm {\frac {1}{2}}\right),}$
• ${\displaystyle \left(\pm {\frac {\sqrt {5+2{\sqrt {5}}}}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1+2{\sqrt {2}}}{2}},\,\pm {\frac {1+{\sqrt {2}}}{2}},\,\pm {\frac {1}{2}}\right).}$

## Representations

A decagonal-great rhombicuboctahedral duoprism has the following Coxeter diagrams:

• x10o x4x3x () (full symmetry)
• x5x x4x3x ()(decagons as dipentagons)