Pentagonal-decagonal duoprism

Pentagonal-decagonal duoprism
Rank4
TypeUniform
Notation
Coxeter diagramx5o x10o ()
Elements
Cells10 pentagonal prisms, 5 decagonal prisms
Faces50 squares, 10 pentagons, 5 decagons
Edges50+50
Vertices50
Vertex figureDigonal disphenoid, edge lengths (1+5)/2 (base 1), (5+5)/2 (base 2), and 2 (sides)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {10+3{\sqrt {5}}}{5}}}\approx 1.82802}$
Hypervolume${\displaystyle {\frac {25(2+{\sqrt {5}})}{8}}\approx 13.23771}$
Dichoral anglesPip–5–pip: 144°
Dip–5–dip: 108°
Dip–4–pip: 90°
Central density1
Number of external pieces15
Level of complexity6
Related polytopes
DualPentagonal-decagonal duotegum
ConjugatePentagrammic-decagrammic duoprism
Abstract & topological properties
Flag count1200
Euler characteristic0
OrientableYes
Properties
SymmetryH2×I2(10), order 200
Flag orbits6
ConvexYes
NatureTame

The pentagonal-decagonal duoprism or padedip, also known as the 5-10 duoprism, is a uniform duoprism that consists of 5 decagonal prisms and 10 pentagonal prisms, with two of each joining at each vertex.

The convex hull of two orthogonal pentagonal-decagonal duoprisms is either the pentagonal duoexpandoprism or the pentagonal duotruncatoprism.

Vertex coordinates

The vertex coordinates of a pentagonal-decagonal duoprism, centered at the origin and with unit edge length, are given by:

• ${\displaystyle \left(0,\,{\sqrt {\frac {5+{\sqrt {5}}}{10}}},\,0,\,\pm {\frac {1+{\sqrt {5}}}{2}}\right)}$,
• ${\displaystyle \left(0,\,{\sqrt {\frac {5+{\sqrt {5}}}{10}}},\,\pm {\sqrt {\frac {5+{\sqrt {5}}}{8}}},\,\pm {\frac {3+{\sqrt {5}}}{4}}\right)}$,
• ${\displaystyle \left(0,\,{\sqrt {\frac {5+{\sqrt {5}}}{10}}},\,\pm {\frac {\sqrt {5+2{\sqrt {5}}}}{2}},\,\pm {\frac {1}{2}}\right)}$,
• ${\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,{\sqrt {\frac {5-{\sqrt {5}}}{40}}},\,0,\,\pm {\frac {1+{\sqrt {5}}}{2}}\right)}$,
• ${\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,{\sqrt {\frac {5-{\sqrt {5}}}{40}}},\,\pm {\sqrt {\frac {5+{\sqrt {5}}}{8}}},\,\pm {\frac {3+{\sqrt {5}}}{4}}\right)}$,
• ${\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,{\sqrt {\frac {5-{\sqrt {5}}}{40}}},\,\pm {\frac {\sqrt {5+2{\sqrt {5}}}}{2}},\,\pm {\frac {1}{2}}\right)}$,
• ${\displaystyle \left(\pm {\frac {1}{2}},\,-{\sqrt {\frac {5+2{\sqrt {5}}}{20}}},\,0,\,\pm {\frac {1+{\sqrt {5}}}{2}}\right)}$,
• ${\displaystyle \left(\pm {\frac {1}{2}},\,-{\sqrt {\frac {5+2{\sqrt {5}}}{20}}},\,\pm {\sqrt {\frac {5+{\sqrt {5}}}{8}}},\,\pm {\frac {3+{\sqrt {5}}}{4}}\right)}$,
• ${\displaystyle \left(\pm {\frac {1}{2}},\,-{\sqrt {\frac {5+2{\sqrt {5}}}{20}}},\,\pm {\frac {\sqrt {5+2{\sqrt {5}}}}{2}},\,\pm {\frac {1}{2}}\right)}$.

Representations

A pentagonal-decagonal duoprism has the following Coxeter diagrams:

• x5o x10o () (full symmetry)
• x5x x5o () (H2×H2, decagons as dipentagons)
• ofx xxx10ooo&#xt I2(10)×A1 axial)
• ofx xxx5xxx&#xt (H2×A1 symmetry, dipentagonal axial)