# Pentagonal-decagonal duoprism

(Redirected from 5-10 duoprism)
Pentagonal-decagonal duoprism
Rank4
TypeUniform
SpaceSpherical
Info
Coxeter diagramx5o x10o
SymmetryH2×I2(10), order 200
Elements
Vertex figureDigonal disphenoid, edge lengths (1+5)/2 (base 1), (5+5)/2 (base 2), and 2 (sides)
Cells10 pentagonal prisms, 5 decagonal prisms
Faces50 squares, 10 pentagons, 5 decagons
Edges50+50
Vertices50
Measures (edge length 1)
Circumradius$\sqrt{\frac{10+3\sqrt5}{5}} ≈ 1.82802$ Hypervolume$\frac{25(2+\sqrt5)}{8} ≈ 13.23771$ Dichoral anglesPip–5–pip: 144°
Dip–5–dip: 108°
Dip–4–pip: 90°
Central density1
Euler characteristic0
Number of pieces15
Level of complexity6
Related polytopes
DualPentagonal-decagonal duotegum
ConjugatesPentagonal-decagrammic duoprism, Pentagrammic-decagonal duoprism, Pentagrammic-decagrammic duoprism
Properties
ConvexYes
OrientableYes
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:

• $\left(0,\,\sqrt{\frac{5+\sqrt5}{10}},\,0,\,±\frac{1+\sqrt5}{2}\right),$ • $\left(0,\,\sqrt{\frac{5+\sqrt5}{10}},\,±\sqrt{\frac{5+\sqrt5}{8}},\,±\frac{3+\sqrt5}{4}\right),$ • $\left(0,\,\sqrt{\frac{5+\sqrt5}{10}},\,±\frac{\sqrt{5+2\sqrt5}}{2},\,±\frac12\right),$ • $\left(±\frac{1+\sqrt5}{4},\,\sqrt{\frac{5-\sqrt5}{40}},\,0,\,±\frac{1+\sqrt5}{2}\right),$ • $\left(±\frac{1+\sqrt5}{4},\,\sqrt{\frac{5-\sqrt5}{40}},\,±\sqrt{\frac{5+\sqrt5}{8}},\,±\frac{3+\sqrt5}{4}\right),$ • $\left(±\frac{1+\sqrt5}{4},\,\sqrt{\frac{5-\sqrt5}{40}},\,±\frac{\sqrt{5+2\sqrt5}}{2},\,±\frac12\right),$ • $\left(±\frac12,\,-\sqrt{\frac{5+2\sqrt5}{20}},\,0,\,±\frac{1+\sqrt5}{2}\right),$ • $\left(±\frac12,\,-\sqrt{\frac{5+2\sqrt5}{20}},\,±\sqrt{\frac{5+\sqrt5}{8}},\,±\frac{3+\sqrt5}{4}\right),$ • $\left(±\frac12,\,-\sqrt{\frac{5+2\sqrt5}{20}},\,±\frac{\sqrt{5+2\sqrt5}}{2},\,±\frac12\right).$ ## Representations

A pentagonal-decagonal duoprism has the following Coxeter diagrams:

• x5o x10o (full symmetry)
• x5x x5o (edcagons as dipentagons, pentagon duoprism symmetry)
• ofx xxx10ooo&#xt (decagonal axial)
• ofx xxx5xxx&#xt (dipentagonal axial symmetry)