# Hexagonal-great rhombicuboctahedral duoprism

Hexagonal-great rhombicuboctahedral duoprism
Rank5
TypeUniform
Notation
Bowers style acronymHagirco
Coxeter diagramx6o x4x3x
Elements
Tera12 square-hexagonal duoprisms, 8 hexagonal duoprisms, 6 hexagonal-octagonal duoprisms, 6 great rhombicuboctahedral prisms
Cells96 cubes, 24+24+24+48 hexagonal prisms, 36 octagonal prisms, 6 great rhombicuboctahedra
Faces72+144+144+144 squares, 48+48 hexagons, 36 octagons
Edges144+144+144+288
Vertices288
Vertex figureMirror-symmetric pentachoron, edge lengths 2, 3, 2+2 (base triangle), 3 (top edge), 2 (side edges)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {17+6{\sqrt {2}}}}{2}}\approx 2.52415}$
Hypervolume${\displaystyle 3(11{\sqrt {3}}+7{\sqrt {6}})\approx 108.59696}$
Diteral anglesShiddip–hip–hiddip: ${\displaystyle \arccos \left(-{\frac {\sqrt {6}}{3}}\right)\approx 144.73561^{\circ }}$
Shiddip–hip–hodip: 135°
Hiddip–hip–hodip: ${\displaystyle \arccos \left(-{\frac {\sqrt {3}}{3}}\right)\approx 125.26439^{\circ }}$
Gircope–girco–gircope: 120°
Shiddip–cube–gircope: 90°
Hiddip–hip–gircope: 90°
Hodip–op–gircope: 90°
Central density1
Number of external pieces32
Level of complexity60
Related polytopes
ArmyHagirco
RegimentHagirco
DualHexagonal-disdyakis dodecahedral duotegum
ConjugateHexagonal-quasitruncated cuboctahedral duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryB3×G2, order 576
ConvexYes
NatureTame

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

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

## Vertex coordinates

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

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

## Representations

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

• x6o x4x3x (full symmetry)
• x3x x4x3x (hexagons as ditrigons)