Dodecagonal-truncated dodecahedral duoprism

Dodecagonal-truncated dodecahedral duoprism
Rank5
TypeUniform
Notation
Bowers style acronymTwatid
Coxeter diagramx12o x5x3o ()
Elements
Tera20 triangular-dodecagonal duoprisms, 12 decagonal-dodecagonal duoprisms, 12 truncated dodecahedral prisms
Cells240 triangular prisms, 144 decagonal prisms, 30+60 dodecagonal prisms, 12 truncated dodecahedra
Faces240 triangles, 360+720 squares, 144 decagons, 60 dodecagons
Edges360+720+720
Vertices720
Vertex figureDigonal disphenoidal pyramid, edge lengths 1, (5+5)/2, (5+5)/2 (base triangle), 2+3 (top), 2 (side edges)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {53+8{\sqrt {3}}+15{\sqrt {5}}}{8}}}\approx 3.54255}$
Hypervolume${\displaystyle 5{\frac {198+99{\sqrt {3}}+94{\sqrt {5}}+47{\sqrt {15}}}{4}}\approx 952.11705}$
Diteral anglesTiddip–tid–tiddip: 150°
Titwadip–twip–datwadip: ${\displaystyle \arccos \left(-{\sqrt {\frac {5+2{\sqrt {5}}}{15}}}\right)\approx 142.62263^{\circ }}$
Datwadip–twip–datwadip: ${\displaystyle \arccos \left(-{\frac {\sqrt {5}}{5}}\right)\approx 116.56505^{\circ }}$
Central density1
Number of external pieces44
Level of complexity30
Related polytopes
ArmyTwatid
RegimentTwatid
DualDodecagonal-triakis icosahedral duotegum
ConjugatesDodecagrammic-truncated dodecahedral duoprism, Dodecagonal-quasitruncated great stellated dodecahedral duoprism, Dodecagrammic-quasitruncated great stellated dodecahedral duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryH3×I2(12), order 2880
ConvexYes
NatureTame

The dodecagonal-truncated dodecahedral duoprism or twatid is a convex uniform duoprism that consists of 12 truncated dodecahedral prisms, 12 decagonal-dodecagonal duoprisms and 20 triangular-dodecagonal duoprisms. Each vertex joins 2 truncated dodecahedral prisms, 1 triangular-dodecagonal duoprism, and 2 decagonal-dodecagonal duoprisms.

Vertex coordinates

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

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

Representations

A dodecagonal-truncated dodecahedral duoprism has the following Coxeter diagrams:

• x12o x5x3o () (full symmetry)
• x6x x5x3o () (dodecagons as dihexagons)