# Hexagonal-truncated dodecahedral duoprism

Hexagonal-truncated dodecahedral duoprism
Rank5
TypeUniform
Notation
Bowers style acronymHatid
Coxeter diagramx6o x5x3o
Elements
Tera20 triangular-hexagonal duoprisms, 12 hexagonal-decagonal duoprisms, 6 truncated dodecahedral prisms
Cells120 triangular prisms, 30+60 hexagonal prisms, 72 decagonal prisms, 6 truncated dodecahedra
Faces120 triangles, 180+360 squares, 60 hexagons, 72 decagons
Edges180+360+360
Vertices360
Vertex figureDigonal disphenoidal pyramid, edge lengths 1, (5+5)/2, (5+5)/2 (base triangle), 3 (top), 2 (side edges)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {45+15{\sqrt {5}}}{8}}}\approx 3.13331}$
Hypervolume${\displaystyle 5{\frac {99{\sqrt {3}}+47{\sqrt {15}}}{8}}\approx 220.93953}$
Diteral anglesThiddip–hip–hadedip: ${\displaystyle \arccos \left(-{\sqrt {\frac {5+2{\sqrt {5}}}{15}}}\right)\approx 142.62263^{\circ }}$
Tiddip–tid–tiddip: 120°
Hadedip–hip–hadedip: ${\displaystyle \arccos \left(-{\frac {\sqrt {5}}{5}}\right)\approx 116.56505^{\circ }}$
Thiddip–trip–tiddip: 90°
Central density1
Number of external pieces38
Level of complexity30
Related polytopes
ArmyHatid
RegimentHatid
DualHexagonal-triakis icosahedral duotegum
ConjugateHexagonal-quasitruncated great stellated dodecahedral duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryH3×G2, order 1440
ConvexYes
NatureTame

The hexagonal-truncated dodecahedral duoprism or hatid is a convex uniform duoprism that consists of 6 truncated dodecahedral prisms, 12 hexagonal-decagonal duoprisms, and 20 triangular-hexagonal duoprisms. Each vertex joins 2 truncated dodecahedral prisms, 1 triangular-hexagonal duoprism, and 2 hexagonal-decagonal duoprisms.

## Vertex coordinates

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

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

## Representations

A hexagonal-truncated dodecahedral duoprism has the following Coxeter diagrams:

• x6o x5x3o (full symmetry)
• x3x x5x3o (hexagons as ditrigons)