# Dodecagonal-great rhombicuboctahedral duoprism

Dodecagonal-great rhombicuboctahedral duoprism
Rank5
TypeUniform
Notation
Bowers style acronymTwagirco
Coxeter diagramx12o x4x3x ()
Elements
Tera12 square-dodecagonal duoprisms, 8 hexagonal-dodecagonal duoprisms, 6 octagonal-dodecagonal duoprisms, 12 great rhombicuboctahedral prisms
Cells144 cubes, 96 hexagonal prisms, 72 octagonal prisms, 24+24+24 dodecagonal prisms, 12 great rhombicuboctahedra
Faces144+288+288+288 squares, 96 hexagons, 72 octagons, 48 dodecagons
Edges288+288+288+576
Vertices576
Vertex figureMirror-symmetric pentachoron, edge lengths 2, 3, 2+2 (base triangle), 2+3 (top edge), 2 (side edges)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {21+6{\sqrt {2}}+4{\sqrt {3}}}}{2}}\approx 3.01718}$
Hypervolume${\displaystyle 6(22+14{\sqrt {2}}+11{\sqrt {3}}+7{\sqrt {6}})\approx 467.98786}$
Diteral anglesGircope–girco–gircope: 150°
Sitwadip–twip–hitwadip: ${\displaystyle \arccos \left(-{\frac {\sqrt {6}}{3}}\right)\approx 144.73561^{\circ }}$
Hitwadip–twip–otwadip: ${\displaystyle \arccos \left(-{\frac {\sqrt {3}}{3}}\right)\approx 125.26439^{\circ }}$
Central density1
Number of external pieces38
Level of complexity60
Related polytopes
ArmyTwagirco
RegimentTwagirco
DualDodecagonal-disdyakis dodecahedral duotegum
ConjugatesDodecagrammic-great rhombicuboctahedral duoprism, Dodecagonal-quasitruncated cuboctahedral duoprism, Dodecagrammic-quasitruncated cuboctahedral duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryB3×I2(12), order 1152
ConvexYes
NatureTame

The dodecagonal-great rhombicuboctahedral duoprism or twagirco is a convex uniform duoprism that consists of 12 great rhombicuboctahedral prisms, 6 octagonal-dodecagonal duoprisms, 8 hexagonal-dodecagonal duoprisms, and 12 square-dodecagonal duoprisms. Each vertex joins 2 great rhombicuboctahedral prisms, 1 square-dodecagonal duoprism, 1 hexagonal-dodecagonal duoprism, and 1 octagonal-dodecagonal duoprism.

This polyteron can be alternated into a hexagonal-snub cubic duoantiprism, although it cannot be made uniform. The dodecagons can also be edge-snubbed to create a snub cubic-hexagonal prismantiprismoid or the great rhombicuboctahedra to create a hexagonal-pyritohedral prismantiprismoid, which are also both nonuniform.

## Vertex coordinates

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

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

## Representations

A dodecagonal-great rhombicuboctahedral duoprism has the following Coxeter diagrams:

• x12o x4x3x () (full symmetry)
• x6x x4x3x () (dodecagons as dihexagons)