# Decagonal duoprism

Decagonal duoprism
Rank4
TypeUniform
SpaceSpherical
Notation
Bowers style acronymDedip
Coxeter diagramx10o x10o ()
Elements
Cells20 decagonal prisms
Faces100 squares, 20 decagons
Edges200
Vertices100
Vertex figureTetragonal disphenoid, edge lengths (5+5)/2 (bases) and 2 (sides)
Measures (edge length 1)
Circumradius${\displaystyle \frac{\sqrt2+\sqrt{10}}{2} ≈ 2.28825}$
Inradius${\displaystyle \frac{\sqrt{5+2\sqrt5}}{2} ≈ 1.53884}$
Hypervolume${\displaystyle \frac{25(5+2\sqrt5)}{4}≈ 59.20085}$
Dichoral anglesDip–10–dip: 144º
Dip–4–dip: 90°
Central density1
Number of external pieces20
Level of complexity3
Related polytopes
ArmyDedip
RegimentDedip
DualDecagonal duotegum
ConjugateDecagrammic duoprism
Abstract & topological properties
Euler characteristic0
OrientableYes
Properties
SymmetryI2(10)≀S2, order 800
ConvexYes
NatureTame

The decagonal duoprism or dedip, also known as the decagonal-decagonal duoprism, the 10 duoprism or the 10-10 duoprism, is a noble uniform duoprism that consists of 20 decagonal prisms, with four at each vertex. It is also the 20-9 gyrochoron. It is the first in an infinite family of isogonal decagonal dihedral swirlchora and also the first in an infinite family of isochoric decagonal hosohedral swirlchora.

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

A unit decagonal duoprism can be edge-inscribed into the small ditetrahedronary hexacosihecatonicosachoron.

## Vertex coordinates

Coordinates for the vertices of a decagonal duoprism of edge length 1, centered at the origin, are given by:

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

## Representations

A decagonal duoprism has the following Coxeter diagrams:

• x10o x10o (full symmetry)
• x5x x10o (one decagon as dipentagon)
• x5x x5x (both decagons have pentagonal symmetry)