# Dodecagonal-truncated tetrahedral duoprism

Dodecagonal-truncated tetrahedral duoprism
Rank5
TypeUniform
Notation
Bowers style acronymTwatut
Coxeter diagramx12o x3x3o ()
Elements
Tera4 triangular-dodecagonal duoprisms, 12 truncated tetrahedral prisms, 4 hexagonal-dodecagonal duoprisms
Cells48 triangular prisms, 48 hexagonal prisms, 12 truncated tetrahedra, 6+12 dodecagonal prisms
Faces48 triangles, 72+144 squares, 48 hexagons, 12 dodecagons
Edges72+144+144
Vertices144
Vertex figureDigonal disphenoidal pyramid, edge lengths 1, 3, 3 (base triangle), 2+3 (top), 2 (side edges)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {27+8{\sqrt {3}}}{8}}}\approx 2.25988}$
Hypervolume${\displaystyle 23{\frac {2{\sqrt {2}}+{\sqrt {6}}}{4}}\approx 30.34802}$
Diteral anglesTuttip–tut–tuttip: 150°
Titwadip-twip-hitwadip: ${\displaystyle \arccos \left(-{\frac {1}{3}}\right)\approx 109.47122^{\circ }}$
Hitwadip–twip–hitwadip: ${\displaystyle \arccos \left({\frac {1}{3}}\right)\approx 70.52877^{\circ }}$
Central density1
Number of external pieces20
Level of complexity30
Related polytopes
ArmyTwatut
RegimentTwatut
DualDodecagonal-triakis tetrahedral duotegum
ConjugateDodecagrammic-truncated tetrahedral duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryA3×I2(12), order 576
ConvexYes
NatureTame

The dodecagonal-truncated tetrahedral duoprism or twatut is a convex uniform duoprism that consists of 12 truncated tetrahedral prisms, 4 hexagonal-dodecagonal duoprisms, and 4 triangular-dodecagonal duoprisms. Each vertex joins 2 truncated tetrahedral prisms, 1 triangular-dodecagonal duoprism, and 2 hexagonal-dodecagonal duoprisms.

## Vertex coordinates

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

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

## Representations

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

• x12o x3x3o (full symmetry)
• x6x x3x3o () (A3×G2 symmetry, dodecagons as dihexagons)