# Dodecagonal-icosidodecahedral duoprism

Dodecagonal-icosidodecahedral duoprism
Rank5
TypeUniform
Notation
Bowers style acronymTwid
Coxeter diagramx12o o5x3o ()
Elements
Tera20 triangular-dodecagonal duoprisms, 12 pentagonal-dodecagonal duoprisms, 12 icosidodecahedral prisms
Cells240 triangular prisms, 144 pentagonal prisms, 60 dodecagonal prisms, 12 icosidodecahedra
Faces240 triangles, 720 squares, 144 pentagons, 30 dodecagons
Edges360+720
Vertices360
Vertex figureRectangular scalene, edge lengths 1, (1+5)/2, 1, (1+5)/2 (base rectangle), 2+3 (top), 2 (side edges)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {7+2{\sqrt {3}}+{\sqrt {5}}}{2}}}\approx 2.51994}$
Hypervolume${\displaystyle {\frac {90+45{\sqrt {3}}+34{\sqrt {5}}+17{\sqrt {15}}}{2}}\approx 154.90466}$
Diteral anglesIddip–id–iddip: 150°
Titwadip–twip–pitwadip: ${\displaystyle \arccos \left(-{\sqrt {\frac {5+2{\sqrt {5}}}{15}}}\right)\approx 142.62263^{\circ }}$
Central density1
Number of external pieces44
Level of complexity20
Related polytopes
ArmyTwid
RegimentTwid
DualDodecagonal-rhombic triacontahedral duotegum
ConjugatesDodecagrammic-icosidodecahedral duoprism, Dodecagonal-great icosidodecahedral duoprism, Dodecagrammic-great icosidodecahedral duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryH3×I2(12), order 2880
ConvexYes
NatureTame

The dodecagonal-icosidodecahedral duoprism or twid is a convex uniform duoprism that consists of 12 icosidodecahedral prisms, 12 pentagonal-dodecagonal duoprisms, and 20 triangular-dodecagonal duoprisms. Each vertex joins 2 icosidodecahedral prisms, 2 triangular-dodecagonal duoprisms, and 2 pentagonal-dodecagonal duoprisms.

## Vertex coordinates

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

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

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

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

## Representations

A dodecagonal-icosidodecahedral duoprism has the following Coxeter diagrams:

• x12o o5x3o () (full symmetry)
• x6x o5x3o () (H3×G2 symmetry, dodecagons as dihexagons)