# Decagonal-dodecagonal duoprismatic prism

Decagonal-dodecagonal duoprismatic prism
Rank5
TypeUniform
Notation
Bowers style acronymDatwip
Coxeter diagramx x10o x12o ()
Elements
Tera12 square-decagonal duoprisms, 10 square-dodecagonal duoprisms, 2 decagonal-dodecagonal duoprisms
Cells120 cubes, 10+20 dodecagonal prisms, 12+24 decagonal prisms
Faces120+120+240 squares, 24 decagons, 20 dodecagons
Edges120+240+240
Vertices240
Vertex figureDigonal disphenoidal pyramid, edge lengths (5+5)/2 (disphenoid base 1), 2+3 (disphenoid base 2), 2 (remaining edges)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {15+4{\sqrt {3}}+2{\sqrt {5}}}}{2}}\approx 2.56906}$
Hypervolume${\displaystyle {\frac {15{\sqrt {35+20{\sqrt {3}}+14{\sqrt {5}}+8{\sqrt {15}}}}}{2}}\approx 86.14553}$
Height1
Central density1
Number of external pieces24
Level of complexity30
Related polytopes
ArmyDatwip
RegimentDatwip
DualDecagonal-dodecagonal duotegmatic tegum
ConjugatesDecagonal-dodecagrammic duoprismatic prism, Decagrammic-dodecagonal duoprismatic prism, Decagrammic-dodecagrammic duoprismatic prism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryI2(10)×I2(12)×A1, order 960
ConvexYes
NatureTame

The decagonal-dodecagonal duoprismatic prism or datwip, also known as the decagonal-dodecagonal prismatic duoprism, is a convex uniform duoprism that consists of 2 decagonal-dodecagonal duoprisms, 10 square-dodecagonal duoprisms, and 12 square-decagonal duoprisms. Each vertex joins 2 square-decagonal duoprisms, 2 square-dodecagonal duoprisms, and 1 decagonal-dodecagonal duoprism. Being a prism based on an orbiform polytope, it is also a convex segmentoteron.

This polyteron can be alternated into a pentagonal-hexagonal duoantiprismatic antiprism, although it cannot be made uniform. The dodecagons can also be alternated into long rectangles to create a pentagonal-hexagonal prismatic prismantiprismoid, which is also nonuniform.

## Vertex coordinates

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

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

## Representations

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

• x x10o x12o () (full symmetry)
• x x5x x12o ()(decagons as dipentagons)
• x x10o x6x ()(dodecagons as dihexagons)
• x x5x x6x ()
• xx10oo xx12oo&#x (decagonal-hendecagonal duoprism atop decagonal-hendecagonal duoprism)
• xx5xx xx12oo&#x
• xx10oo xx6xx&#x
• xx5xx xx6xx&#x