Dodecagonal-tetrahedral duoprism

Dodecagonal-tetrahedral duoprism
Rank5
TypeUniform
Notation
Bowers style acronymTwatet
Coxeter diagramx12o x3o3o ()
Elements
Tera12 tetrahedral prisms, 4 triangular-dodecagonal duoprisms
Cells12 tetrahedra, 48 triangular prisms, 6 dodecagonal prisms
Faces48 triangles, 72 squares, 4 dodecagons
Edges48+72
Vertices48
Vertex figureTriangular scalene, edge lengths 1 (base triangle), 2+3 (top), 2 (sides)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {19+8{\sqrt {3}}}{8}}}\approx 2.02659}$
Hypervolume${\displaystyle {\frac {2{\sqrt {2}}+{\sqrt {6}}}{4}}\approx 1.31948}$
Diteral anglesTepe–tet–tepe: 150°
Titwadip–twip–titwadip: ${\displaystyle \arccos {\left({\frac {1}{3}}\right)}\approx 70.52878^{\circ }}$
HeightsDog atop titwadip: ${\displaystyle {\frac {\sqrt {6}}{3}}\approx 0.81650}$
Twip atop perp twip: ${\displaystyle {\frac {\sqrt {2}}{2}}\approx 0.70711}$
Central density1
Number of external pieces16
Level of complexity10
Related polytopes
ArmyTwatet
RegimentTwatet
DualDodecagonal-tetrahedral duotegum
ConjugateDodecagrammic-tetrahedral duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryA3×I2(12), order 576
ConvexYes
NatureTame

The dodecagonal-tetrahedral duoprism or twatet is a convex uniform duoprism that consists of 12 tetrahedral prisms and 4 triangular-dodecagonal duoprisms. Each vertex joins 2 tetrahedral prisms and 3 triangular-dodecagonal duoprisms.

Vertex coordinates

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

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

Representations

A dodecagonal-tetrahedral duoprism has the following Coxeter diagrams:

• x12o x3o3o () (full symmetry)
• x6x x3o3o () (dodecagons as dihexagons)