# Pentagonal-decagonal duoprismatic prism

Pentagonal-decagonal duoprismatic prism
Rank5
TypeUniform
Notation
Bowers style acronymPeddip
Coxeter diagramx x5o x10o
Elements
Tera10 square-pentagonal duoprisms, 5 square-decagonal duoprisms, 2 pentagonal-decagonal duoprisms
Cells50 cubes, 5+10 decagonal prisms, 10+20 pentagonal prisms
Faces50+50+100 squares, 20 pentagons, 10 decagons
Edges50+100+100
Vertices100
Vertex figureDigonal disphenoidal pyramid, edge lengths (1+5)/2 (disphenoid base 1), (5+5)/2 (disphenoid base 2), 2 (remaining edges)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {45+12{\sqrt {5}}}{20}}}\approx 1.89516}$
Hypervolume${\displaystyle {\frac {50+25{\sqrt {5}}}{8}}\approx 13.23771}$
Diteral anglesSquipdip–pip–squipdip: 144°
Height1
Central density1
Number of external pieces17
Level of complexity30
Related polytopes
ArmyPeddip
RegimentPeddip
DualPentagonal-decagonal duotegmatic tegum
ConjugatePentagrammic-decagrammic duoprismatic prism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryH2×I2(10)×A1, order 400
ConvexYes
NatureTame

The pentagonal-decagonal duoprismatic prism or peddip, also known as the pentagonal-decagonal prismatic duoprism, is a convex uniform duoprism that consists of 2 pentagonal-decagonal duoprisms, 5 square-decagonal duoprisms, and 10 square-pentagonal duoprisms. Each vertex joins 2 square-pentagonal duoprisms, 2 square-decagonal duoprisms, and 1 pentagonal-decagonal duoprism. Being a prism based on an orbiform polytope, it is also a convex segmentoteron.

## Vertex coordinates

The vertices of a pentagonal-decagonal duoprismatic prism of edge length 1 are given by:

• ${\displaystyle \left(0,\,{\sqrt {\frac {5+{\sqrt {5}}}{10}}},\,0,\,\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {1}{2}}\right),}$
• ${\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,{\sqrt {\frac {5-{\sqrt {5}}}{40}}},\,0,\,\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {1}{2}}\right),}$
• ${\displaystyle \left(\pm {\frac {1}{2}},\,-{\sqrt {\frac {5+2{\sqrt {5}}}{20}}},\,0,\,\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {1}{2}}\right),}$
• ${\displaystyle \left(0,\,{\sqrt {\frac {5+{\sqrt {5}}}{10}}},\,\pm {\sqrt {\frac {5+{\sqrt {5}}}{8}}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {1}{2}}\right),}$
• ${\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,{\sqrt {\frac {5-{\sqrt {5}}}{40}}},\,\pm {\sqrt {\frac {5+{\sqrt {5}}}{8}}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {1}{2}}\right),}$
• ${\displaystyle \left(\pm {\frac {1}{2}},\,-{\sqrt {\frac {5+2{\sqrt {5}}}{20}}},\,\pm {\sqrt {\frac {5+{\sqrt {5}}}{8}}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {1}{2}}\right),}$
• ${\displaystyle \left(0,\,{\sqrt {\frac {5+{\sqrt {5}}}{10}}},\,\pm {\frac {\sqrt {5+2{\sqrt {5}}}}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}}\right),}$
• ${\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,{\sqrt {\frac {5-{\sqrt {5}}}{40}}},\,\pm {\frac {\sqrt {5+2{\sqrt {5}}}}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}}\right),}$
• ${\displaystyle \left(\pm {\frac {1}{2}},\,-{\sqrt {\frac {5+2{\sqrt {5}}}{20}}},\,\pm {\frac {\sqrt {5+2{\sqrt {5}}}}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}}\right).}$

## Representations

A pentagonal-decagonal duoprismatic prism has the following Coxeter diagrams:

• x x5o x10o (full symmetry)
• x x5o x5x (decagons as dipentagons)
• xx5oo xx10oo&#x (pentagonal-decagonal duoprism atop pentagonal-decagonal duoprism)
• xx5oo xx5xx&#x