# Decagonal-icosidodecahedral duoprism

Decagonal-icosidodecahedral duoprism
Rank5
TypeUniform
Notation
Bowers style acronymDaid
Coxeter diagramx10o o5x3o ()
Elements
Tera20 triangular-decagonal duoprisms, 12 pentagonal-decagonal duoprisms, 10 icosidodecahedral prisms
Cells200 triangular prisms, 120 pentagonal prisms, 60 decagonal prisms, 10 icosidodecahedra
Faces200 triangles, 600 squares, 120 pentagons, 30 decagons
Edges300+600
Vertices300
Vertex figureRectangular scalene, edge lengths 1, (1+5)/2, 1, (1+5)/2 (base rectangle), (5+5)/2 (top), 2 (side edges)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {{\sqrt {2}}+{\sqrt {10}}}{2}}\approx 2.28825}$
Hypervolume${\displaystyle 5{\frac {\sqrt {32650+14590{\sqrt {5}}}}{12}}\approx 106.45343}$
Diteral anglesIddip–id–iddip: 144°
Tradedip–dip–padedip: ${\displaystyle \arccos \left(-{\sqrt {\frac {5+2{\sqrt {5}}}{15}}}\right)\approx 142.62263^{\circ }}$
Central density1
Number of external pieces42
Level of complexity20
Related polytopes
ArmyDaid
RegimentDaid
DualDecagonal-rhombic triacontahedral duotegum
ConjugateDecagrammic-great icosidodecahedral duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryH3×I2(10), order 2400
ConvexYes
NatureTame

The decagonal-icosidodecahedral duoprism or daid is a convex uniform duoprism that consists of 10 icosidodecahedral prisms, 12 pentagonal-decagonal duoprisms, and 20 triangular-decagonal duoprisms. Each vertex joins 2 icosidodecahedral prisms, 2 triangular-decagonal duoprisms, and 2 pentagonal-decagonal duoprisms.

## Vertex coordinates

The vertices of a decagonal-icosidodecahedral duoprism of edge length 1 are given by all permutations of the last three coordinates of:

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

as well as all even permutations of the last three coordinates of:

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

## Representations

A decagonal-icosidodecahedral duoprism has the following Coxeter diagrams:

• x10o o5x3o (full symmetry)
• x5x o5x3o (decagons as dipentagons)