# Decagonal-dodecahedral duoprism

Decagonal-dodecahedral duoprism
Rank5
TypeUniform
Notation
Coxeter diagramx10o x5o3o ()
Elements
Tera12 pentagonal-decagonal duoprisms, 10 dodecahedral prisms
Cells120 pentagonal prisms, 30 decagonal prisms, 10 dodecahedra
Faces300 squares, 120 pentagons, 20 decagons
Edges200+300
Vertices200
Vertex figureTriangular scalene, edge lengths (1+5)/2 (base triangle), (5+5)/2 (top), 2 (sides)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {21+7{\sqrt {5}}}{8}}}\approx 2.14046}$
Hypervolume${\displaystyle 5{\frac {\sqrt {4450+1990{\sqrt {5}}}}{8}}\approx 58.96164}$
Diteral anglesDope–doe–dope: 144°
Padedip–dip–padedip: ${\displaystyle \arccos \left(-{\frac {\sqrt {5}}{5}}\right)\approx 116.56505^{\circ }}$
Central density1
Number of external pieces22
Level of complexity10
Related polytopes
DualDecagonal-icosahedral duotegum
ConjugateDecagrammic-great stellated dodecahedral duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryH3×I2(10), order 2400
ConvexYes
NatureTame

The decagonal-dodecahedral duoprism or dadoe is a convex uniform duoprism that consists of 10 dodecahedral prisms and 12 pentagonal-decagonal duoprisms. Each vertex joins 2 dodecahedral prisms and 3 pentagonal-decagonal duoprisms.

## Vertex coordinates

The vertices of a decagonal-dodecahedral duoprism of edge length 1 are given by:

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

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

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

## Representations

A decagonal-dodecahedral duoprism has the following Coxeter diagrams:

• x10o x5o3o ()(full symmetry)
• x5x x5o3o () (decagons as dipentagons)