# Hexagonal-decagonal duoprism

Hexagonal-decagonal duoprism
Rank4
TypeUniform
SpaceSpherical
Info
Coxeter diagramx6o x10o
SymmetryG2×I2(10), order 240
Elements
Vertex figureDigonal disphenoid, edge lengths 3 (base 1), (5+5)/2 (base 2), and 2 (sides)
Cells10 hexagonal prisms, 6 decagonal prisms
Faces60 squares, 10 hexagons, 6 decagons
Edges60+60
Vertices60
Measures (edge length 1)
Circumradius$\sqrt{\frac{5+\sqrt5}{2}} ≈ 1.90211$ Hypervolume$\frac{15\sqrt{15+6\sqrt5}}{4} ≈ 19.99014$ Dichoral anglesHip–6–hip: 144°
Dip–10–dip: 120°
Dip–4–hip: 90°
Central density1
Euler characteristic0
Number of pieces16
Level of complexity6
Related polytopes
DualHexagonal-decagonal duotegum
ConjugateHexagonal-decagrammic duoprism
Properties
ConvexYes
OrientableYes
NatureTame

The hexagonal-decagonal duoprism or hadedip, also known as the 6-10 duoprism, is a uniform duoprism that consists of 6 decagonal prisms and 10 hexagonal prisms, with two of each joining at each vertex.

This polychoron can be alternated into a triangular-pentagonal duoantiprism, although it cannot be made uniform.

## Vertex coordinates

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

• $\left(0,\,±1,\,0,\,±\frac{1+\sqrt5}{2}\right),$ • $\left(0,\,±1,\,±\sqrt{\frac{5+\sqrt5}{8}},\,±\frac{3+\sqrt5}{4}\right),$ • $\left(0,\,±1,\,±\frac{\sqrt{5+2\sqrt5}}{2},\,±\frac12\right),$ • $\left(±\frac{\sqrt3}{2},\,±\frac12,\,0,\,±\frac{1+\sqrt5}{2}\right),$ • $\left(±\frac{\sqrt3}{2},\,±\frac12,\,±\sqrt{\frac{5+\sqrt5}{8}},\,±\frac{3+\sqrt5}{4}\right),$ • $\left(±\frac{\sqrt3}{2},\,±\frac12,\,±\frac{\sqrt{5+2\sqrt5}}{2},\,±\frac12\right).$ ## Representations

A hexagonal-decagonal duoprism has the following Coxeter diagrams:

• x6o x10o (full symmetry)
• x3x x10o (hexagons as ditrigons)
• x5x x6o (decagons as dipentagons)
• x3x x5x (both applied)
• xux xxx10ooo&#xt (decagonal axial)
• xux xxx5xxx&#xt (dipentagonal axial)