# Dodecagonal-dodecahedral duoprism

Dodecagonal-dodecahedral duoprism
Rank5
TypeUniform
Notation
Coxeter diagramx12o x5o3o ()
Elements
Tera12 pentagonal-dodecagonal duoprisms, 12 dodecahedral prisms
Cells144 pentagonal prisms, 30 dodecagonal prisms, 12 dodecahedra
Faces360 squares, 144 pentagons, 20 dodecagons
Edges240+360
Vertices240
Vertex figureTriangular scalene, edge lengths (1+5)/2 (base triangle), 2+3 (top), 2 (sides)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {25+8{\sqrt {3}}+3{\sqrt {5}}}{8}}}\approx 2.38654}$
Hypervolume${\displaystyle 3{\frac {30+15{\sqrt {3}}+14{\sqrt {5}}+7{\sqrt {15}}}{4}}\approx 85.79748}$
Diteral anglesDope–doe–dope: 150°
Pitwadip–twip–pitwadip: ${\displaystyle \arccos \left(-{\frac {\sqrt {5}}{5}}\right)\approx 116.56505^{\circ }}$
Central density1
Number of external pieces24
Level of complexity10
Related polytopes
DualDodecagonal-icosahedral duotegum
ConjugatesDodecagrammic-dodecahedral duoprism, Dodecagonal-great stellated dodecahedral duoprism, Dodecagrammic-great stellated dodecahedral duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryH3×I2(12), order 2880
ConvexYes
NatureTame

The dodecagon-dodecahedral duoprism or twadoe is a convex uniform duoprism that consists of 12 dodecahedral prisms and 12 pentagonal-dodecagonal duoprisms. Each vertex joins 2 dodecahedral prisms and 3 pentagonal-dodecagonal duoprisms.

## Vertex coordinates

The vertices of a dodecagonal-dodecahedral duoprism of edge length 1 are given by:

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

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

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

## Representations

A dodecagonal-dodecahedral duoprism has the following Coxeter diagrams:

• x12o x5o3o () (full symmetry)
• x6x x5o3o () (dodecagons as dihexagons)