# Pentagonal-icosahedral duoprism

Pentagonal-icosahedral duoprism
Rank5
TypeUniform
Notation
Bowers style acronymPike
Coxeter diagramx5o o5o3x ()
Elements
Tera20 triangular-pentagonal duoprisms, 5 icosahedral prisms
Cells100 triangular prisms, 30 pentagonal prisms, 5 icosahedra
Faces100 triangles, 150 squares, 12 pentagons
Edges60+150
Vertices60
Vertex figurePentagonal scalene, edge lengths 1 (base pentagon), (1+5)/2 (top), 2 (sides)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {45+9{\sqrt {5}}}{40}}}\approx 1.27598}$
Hypervolume${\displaystyle 5{\frac {\sqrt {650+290{\sqrt {5}}}}{48}}\approx 3.75356}$
Diteral anglesTrapedip–pip–trapedip: ${\displaystyle \arccos \left(-{\frac {\sqrt {5}}{3}}\right)\approx 138.18969^{\circ }}$
Ipe–ike–ipe: 108°
Trapedip–trip–ipe: 90°
Central density1
Number of external pieces25
Level of complexity10
Related polytopes
ArmyPike
RegimentPike
DualPentagonal-dodecahedral duotegum
ConjugatePentagrammic-great icosahedral duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryH3×H2, order 1200
ConvexYes
NatureTame

The pentagonal-icosahedral duoprism or pike is a convex uniform duoprism that consists of 5 icosahedral prisms and 20 triangular-pentagonal duoprisms. Each vertex joins 2 icosahedral prisms and 5 triangular-pentagonal duoprisms.

## Vertex coordinates

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

• ${\displaystyle \left(0,\,{\sqrt {\frac {5+{\sqrt {5}}}{10}}},\,0,\,\pm {\frac {1}{2}},\,\pm {\frac {1+{\sqrt {5}}}{4}}\right)}$,
• ${\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,{\sqrt {\frac {5-{\sqrt {5}}}{40}}},\,0,\,\pm {\frac {1}{2}},\,\pm {\frac {1+{\sqrt {5}}}{4}}\right)}$,
• ${\displaystyle \left(\pm {\frac {1}{2}},\,-{\sqrt {\frac {5+2{\sqrt {5}}}{20}}},\,0,\,\pm {\frac {1}{2}},\,\pm {\frac {1+{\sqrt {5}}}{4}}\right)}$.