# Dodecagonal-small rhombicosidodecahedral duoprism

Dodecagonal-small rhombicosidodecahedral duoprism
Rank5
TypeUniform
Notation
Bowers style acronymTwasrid
Coxeter diagramx12o x5o3x ()
Elements
Tera20 triangular-dodecagonal duoprisms, 30 square-dodecagonal duoprisms, 12 pentagonal-dodecagonal duoprisms, 12 small rhombicosidodecahedral prisms
Cells240 triangular prisms, 360 cubes, 144 pentagonal prisms, 60+60 dodecagonal prisms, 12 small rhombicosidodecahedra
Faces240 triangles, 360+720+720 squares, 144 pentagons, 60 dodecagons
Edges720+720+720
Vertices720
Vertex figureIsosceles-trapezoidal scalene, edge lengths 1, 2, (1+5)/2, 2 (base trapezoid), 2+3 (top), 2 (side edges)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {19+4{\sqrt {3}}+4{\sqrt {5}}}}{2}}\approx 2.95265}$
Hypervolume${\displaystyle 120+60{\sqrt {3}}+58{\sqrt {5}}+29{\sqrt {15}}\approx 465.93151}$
Diteral anglesTitwadip–twip–sitwadip: ${\displaystyle \arccos \left(-{\frac {{\sqrt {3}}+{\sqrt {15}}}{6}}\right)\approx 159.09484^{\circ }}$
Sriddip–srid–sriddip: 150°
Sitwadip–twip–pitwadip: ${\displaystyle \arccos \left(-{\sqrt {\frac {5+{\sqrt {5}}}{10}}}\right)\approx 148.28253^{\circ }}$
Central density1
Number of external pieces74
Level of complexity40
Related polytopes
ArmyTwasrid
RegimentTwasrid
DualDodecagonal-deltoidal hexecontahedral duotegum
ConjugatesDodecagrammic-small rhombicosidodecahedral duoprism, Dodecagonal-quasirhombicosidodecahedral duoprism, Dodecagrammic-quasirhombicosidodecahedral duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryH3×I2(12), order 2880
ConvexYes
NatureTame

The dodecagonal-small rhombicosidodecahedral duoprism or twasrid is a convex uniform duoprism that consists of 12 small rhombicosidodecahedral prisms, 12 pentagonal-dodecagonal duoprisms, 30 square-dodecagonal duoprisms, and 20 triangular-dodecagonal duoprisms. Each vertex joins 2 small rhombicosidodecahedral prisms, 1 triangular-dodecagonal duoprism, 2 square-dodecagonal duoprisms, and 1 pentagonal-dodecagonal duoprism.

## Vertex coordinates

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

• ${\displaystyle \left(\pm {\frac {1+{\sqrt {3}}}{2}},\,\pm {\frac {1+{\sqrt {3}}}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {2+{\sqrt {5}}}{2}}\right),}$
• ${\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {2+{\sqrt {3}}}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {2+{\sqrt {5}}}{2}}\right),}$
• ${\displaystyle \left(\pm {\frac {2+{\sqrt {3}}}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {2+{\sqrt {5}}}{2}}\right),}$

as well as all even permutations of the last three coordinates of:

• ${\displaystyle \left(\pm {\frac {1+{\sqrt {3}}}{2}},\,\pm {\frac {1+{\sqrt {3}}}{2}},\,0,\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {5+{\sqrt {5}}}{4}}\right),}$
• ${\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {2+{\sqrt {3}}}{2}},\,0,\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {5+{\sqrt {5}}}{4}}\right),}$
• ${\displaystyle \left(\pm {\frac {2+{\sqrt {3}}}{2}},\,\pm {\frac {1}{2}},\,0,\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {5+{\sqrt {5}}}{4}}\right),}$
• ${\displaystyle \left(\pm {\frac {1+{\sqrt {3}}}{2}},\,\pm {\frac {1+{\sqrt {3}}}{2}},\,\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {3+{\sqrt {5}}}{4}}\right),}$
• ${\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {2+{\sqrt {3}}}{2}},\,\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {3+{\sqrt {5}}}{4}}\right),}$
• ${\displaystyle \left(\pm {\frac {2+{\sqrt {3}}}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {3+{\sqrt {5}}}{4}}\right).}$

## Representations

A dodecagonal-small rhombicosidodecahedral duoprism has the following Coxeter diagrams:

• x12o x5o3x () (full symmetry)
• x6x x5o3x () (H3×G2 symmetry, dodecagons as dihexagons)