Triangular-tetrahedral duoprism

Triangular-tetrahedral duoprism
(OFF file)
Rank	5
Type	Uniform
Notation
Bowers style acronym	Tratet
Coxeter diagram	x3o x3o3o ()
Tapertopic notation	1211
Elements
Tera	3 tetrahedral prisms, 4 triangular duoprisms
Cells	3 tetrahedra, 6+12 triangular prisms
Faces	4+12 triangles, 18 squares
Edges	12+18
Vertices	12
Vertex figure	Triangular scalene, edge lengths 1 (base triangle and top edge) and √2 (sides)
Measures (edge length 1)
Circumradius	${\frac {\sqrt {102}}{12}}\approx 0.84163$
Hypervolume	${\frac {\sqrt {6}}{48}}\approx 0.051031$
Diteral angles	Tepe–trip–triddip: 90°
	triddip–trip–triddip: $\arccos {\left({\frac {1}{3}}\right)}\approx 70.52878^{\circ }$
	Tepe–tet–tepe: 60°
Heights	Tet atop tepe: ${\frac {\sqrt {3}}{2}}\approx 0.86603$
	Trig atop triddip: ${\frac {\sqrt {6}}{3}}\approx 0.81650$
	Trip atop perp trip: ${\frac {\sqrt {2}}{2}}\approx 0.70711$
Central density	1
Number of external pieces	7
Level of complexity	10
Related polytopes
Army	Tratet
Regiment	Tratet
Dual	Triangular-tetrahedral duotegum
Conjugate	None
Abstract & topological properties
Flag count	1440
Euler characteristic	2
Orientable	Yes
Properties
Symmetry	A3×A2, order 144
Convex	Yes
Nature	Tame

The triangular-tetrahedral duoprism or tratet is a convex uniform duoprism that consists of 3 tetrahedral prisms and 4 triangular duoprisms. Each vertex joins 2 tetrahedral prisms and 3 triangular duoprisms. It is a duoprism based on a triangle and a tetrahedron, and is thus also a convex segmentoteron, as a tetrahedron atop tetrahedral prism.

Vertex coordinates

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

• $\left(0,\,{\frac {\sqrt {3}}{3}},\,{\frac {\sqrt {2}}{4}},\,{\frac {\sqrt {2}}{4}},\,{\frac {\sqrt {2}}{4}}\right),$
• $\left(\pm {\frac {1}{2}},\,-{\frac {\sqrt {3}}{6}},\,{\frac {\sqrt {2}}{4}},\,{\frac {\sqrt {2}}{4}},\,{\frac {\sqrt {2}}{4}}\right).$

Representations

A triangular-tetrahedral duoprism has the following Coxeter diagrams:

• x3o x3o3o (full symmetry)
• xx3oo ox3oo&#x (A2×A2 symmetry, triangle atop triangular duoprism)
• ox xx3oo3oo&#x (A3×A1 symmetry, tetrahedron atop tetrahedral prism)
• ox xo xx3oo&#x (A2×A1×A1 symmetry, triangular prism atop orthogonal triangular prism)
• ooo3ooo3xxx&#x (A3 symmetry, tetrahedra considered different)
• oox ooo3xxx&#x (A2×A1 symmetry, tetrahedra have mirror symmetry only)
• oooo3xxxx&#x (A2 symmetry, tetrahedra have no symmetry)
• xxoo xoox3oooo&#xr (A2×A1 symmetry)

External links

• Klitzing, Richard. "tratet".