# Hexagonal-truncated icosahedral duoprism

Hexagonal-truncated icosahedral duoprism
Rank5
TypeUniform
Notation
Bowers style acronymHati
Coxeter diagramx6o o5x3x
Elements
Tera12 pentagonal-hexagonal duoprisms, 20 hexagonal duoprisms, 6 truncated icosahedral prisms
Cells72 pentagonal prisms, 30+60+120 hexagonal prisms, 6 truncated icosahedral prisms
Faces180+360 squares, 72 pentagons, 60+120 hexagons
Edges180+360+360
Vertices360
Vertex figureDigonal disphenoidal pyramid, edge lengths (1+5)/2, 3, 3 (base triangle), 3 (top), 2 (side edges)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {37+9{\sqrt {5}}}{8}}}\approx 2.67219}$
Hypervolume${\displaystyle 3{\frac {125{\sqrt {3}}+43{\sqrt {15}}}{8}}\approx 143.64174}$
Diteral anglesPhiddip–pip–hiddip: ${\displaystyle \arccos \left(-{\sqrt {\frac {5+2{\sqrt {5}}}{15}}}\right)\approx 142.62263^{\circ }}$
Hiddip–pip–hiddip: ${\displaystyle \arccos \left(-{\frac {\sqrt {5}}{3}}\right)\approx 138.18968^{\circ }}$
Tipe–ti–tipe: 120°
Phiddip–pip–tipe: 90°
Hiddip–hip–tipe: 90°
Central density1
Level of complexity30
Related polytopes
ArmyHati
RegimentHati
DualHexagonal-pentakis dodecahedral duotegum
ConjugateHexagonal-truncated great icosahedral duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryH3×G2, order 1440
ConvexYes
NatureTame

The hexagonal-truncated icosahedral duoprism or hati is a convex uniform duoprism that consists of 6 truncated icosahedral prisms, 20 hexagonal duoprisms, and 12 pentagonal-hexagonal duoprisms. Each vertex joins 2 truncated icosahedral prisms, 1 pentagonal-hexagonal duoprism, and 2 hexagonal duoprisms.

## Vertex coordinates

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

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

## Representations

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

• x6o o5x3x (full symmetry)
• x3x o5x3x (hexagons as ditrigons)