# Octagonal-dodecagonal duoprismatic prism

Octagonal-dodecagonal duoprismatic prism
Rank5
TypeUniform
Notation
Bowers style acronymOtwip
Coxeter diagramx x8o x12o
Elements
Tera12 square-octagonal duoprisms, 8 square-dodecagonal duoprisms, 2 octagonal-dodecagonal duoprisms
Cells96 cubes, 8+16 dodecagonal prisms, 12+24 octagonal prisms
Faces96+96+192 squares, 24 octagons, 16 dodecagons
Edges96+192+192
Vertices192
Vertex figureDigonal disphenoidal pyramid, edge lengths 2+2 (disphenoid base 1), 2+3 (disphenoid base 2), 2 (remaining edges)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {13+2{\sqrt {2}}+4{\sqrt {3}}}}{2}}\approx 2.38520}$
Hypervolume${\displaystyle 6(2+2{\sqrt {2}}+{\sqrt {3}}+{\sqrt {6}})\approx 54.05981}$
Diteral anglesSodip–op–sodip: 150°
Height1
Central density1
Number of external pieces22
Level of complexity30
Related polytopes
ArmyOtwip
RegimentOtwip
DualOctagonal-dodecagonal duotegmatic tegum
ConjugatesOctagonal-dodecagrammic duoprismatic prism, Octagrammic-dodecagonal duoprismatic prism, Octagrammic-dodecagrammic duoprismatic prism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryI2(8)×I2(12)×A1, order 768
ConvexYes
NatureTame

The octagonal-dodecagonal duoprismatic prism or otwip, also known as the octagonal-dodecagonal prismatic duoprism, is a convex uniform duoprism that consists of 2 octagonal-dodecagonal duoprisms, 8 square-dodecagonal duoprisms, and 12 square-octagonal duoprisms. Each vertex joins 2 square-octagonal duoprisms, 2 square-dodecagonal duoprisms, and 1 octagonal-dodecagonal duoprism. Being a prism based on an orbiform polytope, it is also a convex segmentoteron.

This polyteron can be alternated into a square-hexagonal duoantiprismatic antiprism, although it cannot be made uniform. The octagons can also be alternated into long rectangles to create a hexagonal-square prismatic prismantiprismoid or the dodecagons into long ditrigons to create a square-hexagonal prismatic prismantiprismoid, which are also both nonuniform.

## Vertex coordinates

The vertices of an octagonal-dodecagonal duoprismatic prism of edge length 1 are given by all permutations of the first two coordinates of:

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

## Representations

An octagonal-dodecagonal duoprismatic prism has the following Coxeter diagrams:

• x x8o x12o (full symmetry)
• x x4x x12o (octagons as ditetragons)
• x x8o x6x (dodecagons as dihexagons)
• x x4x x6x
• xx8oo xx12oo&#x (octagonal-enneagonal duoprism atop octagonal-enneagonal duoprism)
• xx4xx xx12oo&#x
• xx8oo xx6xx&#x
• xx4xx xx6xx&#x