# Pentagonal-hexagonal duoprism

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Pentagonal-hexagonal duoprism
Rank4
TypeUniform
Notation
Bowers style acronymPhiddip
Coxeter diagramx5o2x6o ()
Elements
Cells6 pentagonal prisms, 5 hexagonal prisms
Faces30 squares, 6 pentagons, 5 hexagons
Edges30+30
Vertices30
Vertex figureDigonal disphenoid, edge lengths (1+5)/2 (base 1), 3 (base 2), and 2 (sides)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {15+{\sqrt {5}}}{10}}}\approx 1.31286}$
Hypervolume${\displaystyle {\frac {3{\sqrt {75+30{\sqrt {5}}}}}{8}}\approx 4.46993}$
Dichoral anglesPip–5–pip: 120°
Hip–6–hip: 108°
Hip–4–pip: 90°
Central density1
Number of external pieces11
Level of complexity6
Related polytopes
ArmyPhiddip
RegimentPhiddip
DualPentagonal-hexagonal duotegum
ConjugatePentagrammic-hexagonal duoprism
Abstract & topological properties
Flag count720
Euler characteristic0
OrientableYes
Properties
SymmetryH2×G2, order 120
Flag orbits6
ConvexYes
NatureTame

The pentagonal-hexagonal duoprism or phiddip, also known as the 5-6 duoprism, is a uniform duoprism that consists of 5 hexagonal prisms and 6 pentagonal prisms, with two of each joining at each vertex.

## Vertex coordinates

Coordinates for the vertices of a pentagonal-hexagonal duoprism with edge length 1 are given by:

• ${\displaystyle \left(0,\,{\sqrt {\frac {5+{\sqrt {5}}}{10}}},\,0,\,\pm 1\right)}$,
• ${\displaystyle \left(0,\,{\sqrt {\frac {5+{\sqrt {5}}}{10}}},\,\pm {\frac {\sqrt {3}}{2}},\,\pm {\frac {1}{2}}\right)}$,
• ${\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,{\sqrt {\frac {5-{\sqrt {5}}}{40}}},\,0,\,\pm 1\right)}$,
• ${\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,{\sqrt {\frac {5-{\sqrt {5}}}{40}}},\,\pm {\frac {\sqrt {3}}{2}},\,\pm {\frac {1}{2}}\right)}$,
• ${\displaystyle \left(\pm {\frac {1}{2}},\,-{\sqrt {\frac {5+2{\sqrt {5}}}{20}}},\,0,\,\pm 1\right)}$,
• ${\displaystyle \left(\pm {\frac {1}{2}},\,-{\sqrt {\frac {5+2{\sqrt {5}}}{20}}},\,\pm {\frac {\sqrt {3}}{2}},\,\pm {\frac {1}{2}}\right)}$.

## Representations

A pentagonal-hexagonal duoprism has the following Coxeter diagrams:

• x5o2x6o () (full symmetry)
• x3x2x5o () (A2×H2 symmetry, hexagons as ditrigons)
• ofx xxx6ooo&#xt (G2×A1 axial)
• ofx xxx3xxx&#xt (A2×A1 symmetry, ditrigonal axial)
• xux xxx5ooo&#xt (H2×A1 axial)