# Octagonal-dodecagonal duoprism

Octagonal-dodecagonal duoprism
Rank4
TypeUniform
Notation
Coxeter diagramx8o x12o ()
Elements
Cells12 octagonal prisms, 8 dodecagonal prisms
Faces96 squares, 12 octagons, 8 dodecagons
Edges96+96
Vertices96
Vertex figureDigonal disphenoid, edge lengths 2+2 (base 1), (2+6)/2 (base 2), and 2 (sides)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {6+{\sqrt {2}}+2{\sqrt {3}}}{2}}}\approx 2.33220}$
Hypervolume${\displaystyle 6(2+2{\sqrt {2}}+{\sqrt {3}}+{\sqrt {6}})\approx 54.05981}$
Dichoral anglesOp–8–op: 150°
Twip–12–twip: 135°
Op–4–twip: 90°
Central density1
Number of external pieces20
Level of complexity6
Related polytopes
DualOctagonal-dodecagonal duotegum
ConjugatesOctagonal-dodecagrammic duoprism, Octagrammic-dodecagonal duoprism, Octagrammic-dodecagrammic duoprism
Abstract & topological properties
Euler characteristic0
OrientableYes
Properties
SymmetryI2(8)×I2(12), order 384
ConvexYes
NatureTame

The octagonal-dodecagonal duoprism or otwadip, also known as the 8-12 duoprism, is a uniform duoprism that consists of 8 dodecagonal prisms and 12 octagonal prisms, with two of each joining at each vertex.

This polychoron can be alternated into a square-hexagonal duoantiprism, although it cannot be made uniform. The octagons can also be alternated into long rectangles to create a hexagonal-square prismantiprismoid or the dodecagons into long ditrigons to create a square-hexagonal prismantiprismoid, or it can be subsymmetrically faceted into a digonal-triangular tetraswirlprism, which are nonuniform.

## Vertex coordinates

The coordinates of an octagonal-dodecagonal duoprism of edge length 1, centered at the origin, are given by:

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

## Representations

An octagonal-dodecagonal duoprism has the fllowing Coxeter diagrams:

• x8o x12o (full symmetry)
• x4x x12o (octagons as ditetragons)
• x6x x8o (dodecagons as dihexagons)
• x4x x6x (both of these applied)