# Hexagonal-dodecahedral duoprism

Hexagonal-dodecahedral duoprism
Rank5
TypeUniform
Notation
Coxeter diagramx6o x5o3o
Elements
Tera12 pentagonal-hexagonal duoprisms, 6 dodecahedral prisms
Cells72 pentagonal prisms, 30 hexagonal prisms, 6 dodecahedra
Faces180 squares, 72 pentagons, 20 hexagons
Edges120+180
Vertices120
Vertex figureTriangular scalene, edge lengths (1+5)/2 (base triangle), 3 (top), 2 (sides)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {17+3{\sqrt {5}}}{8}}}\approx 1.72149}$
Hypervolume${\displaystyle 3{\frac {15{\sqrt {3}}+7{\sqrt {15}}}{8}}\approx 19.90937}$
Diteral anglesDope–doe–dope: 120°
Phiddip–hip–phiddip: ${\displaystyle \arccos \left(-{\frac {\sqrt {5}}{5}}\right)\approx 116.56505^{\circ }}$
Phiddip–pip–dope: 90°
Central density1
Number of external pieces18
Level of complexity10
Related polytopes
DualHexagonal-icosahedral duotegum
ConjugateHexagonal-great stellated dodecahedral duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryH3×G2, order 1440
ConvexYes
NatureTame

The hexagonal-dodecahedral duoprism or hadoe is a convex uniform duoprism that consists of 6 dodecahedral prisms and 12 pentagonal-hexagonal duoprisms. Each vertex joins 2 dodecahedral prisms and 3 pentagonal-hexagonal duoprisms.

## Vertex coordinates

The vertices of a hexagonal-dodecahedral duoprism of edge length 1 are given by:

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

as well as all even permutations of the last three coordinates of:

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

## Representations

A hexagonal-dodecahedral duoprism has the following Coxeter diagrams:

• x6o x5o3o (full symmetry)
• x3x x5o3o (hexagons as ditrigons)