# Dodecagonal-octahedral duoprism

Dodecagonal-octahedral duoprism
Rank5
TypeUniform
Notation
Bowers style acronymTwoct
Coxeter diagramx12o o4o3x ()
Elements
Tera12 octahedral prisms, 8 triangular-dodecagonal duoprisms
Cells96 triangular prisms, 12 octahedra, 12 dodecagonal prisms
Faces96 triangles, 144 squares, 6 dodecagons
Edges72+144
Vertices72
Vertex figureSquare scalene, edge lengths 1 (base square), 2+3 (top), 2 (sides)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {5+2{\sqrt {3}}}{2}}}\approx 2.05719}$
Hypervolume${\displaystyle 2{\sqrt {2}}+{\sqrt {6}})\approx 5.27792}$
Diteral anglesOpe–oct–ope: 150°
Titwadip–twip–titwadip: ${\displaystyle \arccos \left(-{\frac {1}{3}}\right)\approx 109.47122^{\circ }}$
${\displaystyle {\frac {\sqrt {6}}{3}}\approx 0.81650}$
Central density1
Number of external pieces20
Level of complexity10
Related polytopes
ArmyTwoct
RegimentTwoct
DualDodecagonal-cubic duotegum
ConjugateDodecagrammic-octahedral duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryB3×I2(12), order 1152
ConvexYes
NatureTame

The dodecagonal-octahedral duoprism or twoct is a convex uniform duoprism that consists of 12 octahedral prisms and 8 triangular-dodecagonal duoprisms. Each vertex joins 2 octahedral prisms and 4 triangular-dodecagonal duoprisms.

## Vertex coordinates

The vertices of a dodecagonal-octahedral duoprism of edge length 1 are given by all permutations and sign changes of the last three coordinates of:

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

## Representations

A dodecahedral-octahedral duoprism has the following Coxeter diagrams:

• x12o o4o3x () (full symmetry)
• x6x o3x3o () (dodecagons as dihexagons)
• x12o o4o3x () (octahedra as tetratetrahedra)
• x6x o3x3o () (dodecagons as dihexagons and octahedra as tetratetrahedra)