# Decagonal duoprism

(Redirected from Dedip)
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$\frac{\sqrt2+\sqrt{10}}{2} ≈ 2.28825$ Inradius$\frac{\sqrt{5+2\sqrt5}}{2} ≈ 1.53884$ Hypervolume$\frac{25(5+2\sqrt5)}{4}≈ 59.20085$ 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:

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