Dodecagonal-tetrahedral duoprism

Dodecagonal-tetrahedral duoprism
(OFF file)
Rank	5
Type	Uniform
Notation
Bowers style acronym	Twatet
Coxeter diagram	x12o x3o3o ()
Elements
Tera	12 tetrahedral prisms, 4 triangular-dodecagonal duoprisms
Cells	12 tetrahedra, 48 triangular prisms, 6 dodecagonal prisms
Faces	48 triangles, 72 squares, 4 dodecagons
Edges	48+72
Vertices	48
Vertex figure	Triangular scalene, edge lengths 1 (base triangle), √2+√3 (top), √2 (sides)
Measures (edge length 1)
Circumradius	${\sqrt {\frac {19+8{\sqrt {3}}}{8}}}\approx 2.02659$
Hypervolume	${\frac {2{\sqrt {2}}+{\sqrt {6}}}{4}}\approx 1.31948$
Diteral angles	Tepe–tet–tepe: 150°
	Tepe–trip–titwadip: 90°
	Titwadip–twip–titwadip: $\arccos {\left({\frac {1}{3}}\right)}\approx 70.52878^{\circ }$
Heights	Dog atop titwadip: ${\frac {\sqrt {6}}{3}}\approx 0.81650$
	Twip atop perp twip: ${\frac {\sqrt {2}}{2}}\approx 0.70711$
Central density	1
Number of external pieces	16
Level of complexity	10
Related polytopes
Army	Twatet
Regiment	Twatet
Dual	Dodecagonal-tetrahedral duotegum
Conjugate	Dodecagrammic-tetrahedral duoprism
Abstract & topological properties
Euler characteristic	2
Orientable	Yes
Properties
Symmetry	A3×I2(12), order 576
Convex	Yes
Nature	Tame

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:

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

Representations

A dodecagonal-tetrahedral duoprism has the following Coxeter diagrams:

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