# Pentagonal-dodecagonal duoprismatic prism

Pentagonal-dodecagonal duoprismatic prism
Rank5
TypeUniform
Notation
Bowers style acronymPetwip
Coxeter diagramx x5o x12o
Elements
Tera12 square-pentagonal duoprisms, 5 square-dodecagonal duoprisms, 2 pentagonal-dodecagonal duoprisms
Cells60 cubes, 5+10 dodecagonal prisms, 12+24 pentagonal prisms
Faces60+60+120 squares, 24 pentagons, 10 dodecagons
Edges60+120+120
Vertices120
Vertex figureDigonal disphenoidal pyramid, edge lengths (1+5)/2 (disphenoid base 1), 2+3 (disphenoid base 2), 2 (remaining edges)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {55+2{\sqrt {5}}+20{\sqrt {3}}}{20}}}\approx 2.16925}$
Hypervolume${\displaystyle {\frac {3{\sqrt {175+100{\sqrt {3}}+70{\sqrt {5}}+40{\sqrt {15}}}}}{4}}\approx 19.26272}$
Diteral anglesSquipdip–pip–squipdip: 150°
Height1
Central density1
Number of external pieces19
Level of complexity30
Related polytopes
ArmyPetwip
RegimentPetwip
DualPentagonal-dodecagonal duotegmatic tegum
ConjugatesPentagonal-dodecagrammic duoprismatic prism, Pentagrammic-dodecagonal duoprismatic prism, Pentagrammic-dodecagrammic duoprismatic prism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryH2×I2(12)×A1, order 480
ConvexYes
NatureTame

The pentagonal-dodecagonal duoprismatic prism or petwip, also known as the pentagonal-dodecagonal prismatic duoprism, is a convex uniform duoprism that consists of 2 pentagonal-dodecagonal duoprisms, 5 square-dodecagonal duoprisms, and 12 square-pentagonal duoprisms. Each vertex joins 2 square-pentagonal duoprisms, 2 square-dodecagonal duoprisms, and 1 pentagonal-dodecagonal duoprism. Being a prism based on an orbiform polytope, it is also a convex segmentoteron.

## Vertex coordinates

The vertices of a pentagonal-dodecagonal duoprismatic prism of edge length 1 are given by all permutations of the third and fourth coordinates of:

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

## Representations

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

• x x5o x12o (full symmetry)
• x x5o x6x (dodecagons as dihexagons)
• xx5oo xx12oo&#x (pentagonal-dodecagonal duoprism atop pentagonal-dodecagonal duoprism)
• xx5oo xx6xx&#x