# Dodecagonal-icosahedral duoprism

Dodecagonal-icosahedral duoprism
Rank5
TypeUniform
Notation
Bowers style acronymTwike
Coxeter diagramx12o o5o3x ()
Elements
Tera20 triangular-dodecagonal duoprisms, 12 icosahedral prisms
Cells240 triangular prisms, 30 dodecagonal prisms, 12 icosahedra
Faces240 triangles, 360 squares, 12 dodecagons
Edges144+360
Vertices144
Vertex figurePentagonal scalene, edge lengths 1 (base pentagon), 2+3 (top), 2 (sides)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {21+8{\sqrt {3}}+{\sqrt {5}}}{8}}}\approx 2.15327}$
Hypervolume${\displaystyle 5{\frac {6+3{\sqrt {3}}+2{\sqrt {5}}+{\sqrt {15}}}{4}}\approx 24.42659}$
Diteral anglesIpe–ike–ipe: 150°
Titwadip–twip–titwadip: ${\displaystyle \arccos \left(-{\frac {\sqrt {5}}{3}}\right)\approx 138.18969^{\circ }}$
Central density1
Related polytopes
ArmyTwike
RegimentTwike
DualDodecagonal-dodecahedral duotegum
ConjugatesDodecagrammic-icosahedral duoprism, Dodecagonal-great icosahedral duoprism, Dodecagrammic-great icosahedral duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryH3×I2(12), order 2880
ConvexYes
NatureTame

The dodecagonal-icosahedral duoprism or twike is a convex uniform duoprism that consists of 12 icosahedral prisms and 20 triangular-dodecagonal duoprisms. Each vertex joins 2 icosahedral prisms and 5 triangular-dodecahedral duoprisms.

## Vertex coordinates

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

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

## Representations

A dodecagonal-icosahedral duoprism has the following Coxeter diagrams:

• x12o o5o3x () (full symmetry)
• x6x o5o3x () (dodecagons as dihexagons)