# Dodecagonal-cuboctahedral duoprism

Dodecagonal-cuboctahedral duoprism
Rank5
TypeUniform
Notation
Bowers style acronymTwaco
Coxeter diagramx12o o4x3o ()
Elements
Tera12 cuboctahedral prisms, 8 triangular-dodecagonal duoprisms, 6 square-dodecagonal duoprisms
Cells96 triangular prisms, 72 squares, 12 cuboctahedra, 24 dodecagonal prisms
Faces96 triangles, 72+288 squares, 12 dodecagons
Edges144+288
Vertices144
Vertex figureRectangular scalene, edge lengths 1, 2, 1, 2 (base rectangle), 2+3 (top), 2 (side edges)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {3+{\sqrt {3}}}}\approx 2.17533}$
Hypervolume${\displaystyle 5\left(2{\sqrt {2}}+{\sqrt {6}}\right)\approx 26.38958}$
Diteral anglesCope–co–cope: 150°
Titwadip–twip–sitwadip: ${\displaystyle \arccos \left(-{\frac {\sqrt {3}}{3}}\right)\approx 125.26439^{\circ }}$
Central density1
Number of external pieces26
Level of complexity20
Related polytopes
ArmyTwaco
RegimentTwaco
DualDodecagonal-rhombic dodecahedral duotegum
ConjugateDodecagrammic-cuboctahedral duoprism
Abstract & topological properties
Flag count23040
Euler characteristic2
OrientableYes
Properties
SymmetryB3×I2(12), order 1152
ConvexYes
NatureTame

The dodecagonal-cuboctahedral duoprism or twaco is a convex uniform duoprism that consists of 12 cuboctahedral prisms, 6 square-dodecagonal duoprisms, and 8 triangular-dodecagonal duoprisms. Each vertex joins 2 cuboctahedral prisms, 2 triangular-dodecagonal duoprisms, and 2 square-dodecagonal duoprisms.

## Vertex coordinates

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

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

## Representations

A dodecagonal-cuboctahedral duoprism has the following Coxeter diagrams:

• x12o o4x3o () (full symmetry)
• x6x o4x3o () (dodecagons as dihexagons)
• x12o x3o3x ()
• x6x x3o3x () (dodecagons as dihexagons; cuboctahedron as rhombitetratetrahedron)