# Dodecagonal-truncated icosahedral duoprism

Dodecagonal-truncated icosahedral duoprism
Rank5
TypeUniform
Notation
Bowers style acronymTwati
Coxeter diagramx12o o5x3x ()
Elements
Tera12 pentagonal-dodecagonal duoprisms, 20 hexagonal-dodecagonal duoprisms, 12 truncated icosahedral prisms
Cells144 pentagonal prisms, 240 hexagonal prisms, 30+60 dodecagonal prisms, 12 truncated icosahedra
Faces360+720 squares, 144 pentagons, 240 hexagons, 60 dodecagons
Edges360+720+720
Vertices720
Vertex figureDigonal disphenoidal pyramid, edge lengths (1+5)/2, 3, 3 (base triangle), 2+3 (top), 2 (side edges)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {45+8{\sqrt {3}}+9{\sqrt {5}}}{8}}}\approx 3.14207}$
Hypervolume${\displaystyle 3{\frac {250+125{\sqrt {3}}+86{\sqrt {5}}+43{\sqrt {15}}}{4}}\approx 619.00986}$
Diteral anglesTipe–ti–tipe: 150°
Pitwadip–twip–hitwadip: ${\displaystyle \arccos \left(-{\sqrt {\frac {5+2{\sqrt {5}}}{15}}}\right)\approx 142.62263^{\circ }}$
Hitwadip–twip–hitwadip: ${\displaystyle \arccos \left(-{\frac {\sqrt {5}}{3}}\right)\approx 138.18968^{\circ }}$
Central density1
Number of external pieces44
Level of complexity30
Related polytopes
ArmyTwati
RegimentTwati
DualDodecagonal-pentakis dodecahedral duotegum
ConjugatesDodecagrammic-truncated icosahedral duoprism, Dodecagonal-truncated great icosahedral duoprism, Dodecagrammic-truncated great icosahedral duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryH3×I2(12), order 2880
ConvexYes
NatureTame

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

## Vertex coordinates

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

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

## Representations

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

• x12o o5x3x () (full symmetry)
• x6x o5x3x () (dodecagons as dihexagons)