# Decagonal-icosahedral duoprism

Decagonal-icosahedral duoprism
Rank5
TypeUniform
Notation
Bowers style acronymDike
Coxeter diagramx10o o5o3x ()
Elements
Tera20 triangular-decagonal duoprisms, 10 icosahedral prisms
Cells200 triangular prisms, 30 decagonal prisms, 10 icosahedra
Faces200 triangles, 300 squares, 12 decagons
Edges120+300
Vertices120
Vertex figurePentagonal scalene, edge lengths 1 (base pentagon), (5+5)/2 (top), 2 (sides)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {17+5{\sqrt {5}}}{8}}}\approx 1.87684}$
Hypervolume${\displaystyle 25{\frac {\sqrt {130+58{\sqrt {5}}}}{24}}\approx 16.78642}$
Diteral anglesIpe–ike–ipe: 144°
Tradedip–dip–tradedip: ${\displaystyle \arccos \left(-{\frac {\sqrt {5}}{3}}\right)\approx 138.18969^{\circ }}$
Central density1
Number of external pieces30
Level of complexity10
Related polytopes
ArmyDike
RegimentDike
DualDecagonal-dodecahedral duotegum
ConjugateDecagrammic-great icosahedral duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryH3×I2(10), order 2400
ConvexYes
NatureTame

The decagonal-icosahedral duoprism or dike is a convex uniform duoprism that consists of 10 icosahedral prisms and 20 triangular-decagonal duoprisms. Each vertex joins 2 icosahedral prisms and 5 triangular-decagonal 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:

• ${\displaystyle \left(0,\,\pm {\frac {1+{\sqrt {5}}}{2}},\,0,\,\pm {\frac {1}{2}},\,\pm {\frac {1+{\sqrt {5}}}{4}}\right),}$
• ${\displaystyle \left(\pm {\sqrt {\frac {5+{\sqrt {5}}}{8}}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,0,\,\pm {\frac {1}{2}},\,\pm {\frac {1+{\sqrt {5}}}{4}}\right),}$
• ${\displaystyle \left(\pm {\frac {\sqrt {5+2{\sqrt {5}}}}{2}},\,\pm {\frac {1}{2}},\,0,\,\pm {\frac {1}{2}},\,\pm {\frac {1+{\sqrt {5}}}{4}}\right).}$

## Representations

A decagonal-icosahedral duoprism has the following Coxeter diagrams:

• x10o o5o3x () (full symmetry)
• x5x o5o3x () (decagons as dipentagons)