# Decagonal duoprismatic prism

Decagonal duoprismatic prism
Rank5
TypeUniform
Notation
Coxeter diagramx x10o x10o ()
Elements
Tera20 square-decagonal duoprisms, 2 decagonal duoprisms
Cells100 cubes, 20+40 decagonal prisms
Faces200+200 squares, 40 decagons
Edges100+400
Vertices200
Vertex figureTetragonal disphenoidal pyramid, edge lengths (5+5)/2 (disphenoid bases) and 2 (remaining edges)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {13+4{\sqrt {5}}}}{2}}\approx 2.34224}$
Hypervolume${\displaystyle {\frac {25(5+2{\sqrt {5}})}{4}}\approx 59.20085}$
HeightDedip atop dedip: 1
Central density1
Number of external pieces22
Level of complexity15
Related polytopes
DualDecagonal duotegmatic tegum
ConjugateDecagrammic duoprismatic prism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryI2(10)≀S2×A1, order 1600
ConvexYes
NatureTame

The decagonal duoprismatic prism or daddip, also known as the decagonal-decagonal prismatic duoprism, is a convex uniform duoprism that consists of 2 decagonal duoprisms and 20 square-decagonal duoprisms. Each vertex joins 4 square-decagonal duoprisms and 1 decagonal duoprism. Being a prism based on an orbiform polytope, it is also a convex segmentoteron.

This polyteron can be alternated into a pentagonal duoantiprismatic antiprism, although it cannot be made uniform.

## Vertex coordinates

The vertices of an decagonal duoprismatic prism of edge length 2sin(π/9) are given by:

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

## Representations

A decagonal duoprismatic prism has the following Coxeter diagrams:

• x x10o x10o () (full symmetry)
• x x5x x5x () (decagons as dipentagons)
• xx10oo xx10oo&#x (decagonal duoprism atop decagonal duoprism)
• xx5xx xx5xx&#x