# Decagonal duoprism

Decagonal duoprism
Rank4
TypeUniform
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${\displaystyle {\frac {{\sqrt {2}}+{\sqrt {10}}}{2}}\approx 2.28825}$
Inradius${\displaystyle {\frac {\sqrt {5+2{\sqrt {5}}}}{2}}\approx 1.53884}$
Hypervolume${\displaystyle {\frac {25(5+2{\sqrt {5}})}{4}}\approx 59.20085}$
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:

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

## 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)