Triangular duoprism

Triangular duoprism
(OFF file)
Rank	4
Type	Uniform
Notation
Bowers style acronym	Triddip
Coxeter diagram	x3o x3o ()
Tapertopic notation	1111
Elements
Cells	6 triangular prisms
Faces	6 triangles, 9 squares
Edges	18
Vertices	9
Vertex figure	Tetragonal disphenoid, edge lengths 1 (base) and √2 (sides)
Measures (edge length 1)
Circumradius	${\frac {\sqrt {6}}{3}}\approx 0.81650$
Inradius	${\frac {\sqrt {3}}{6}}\approx 0.28868$
Hypervolume	${\frac {3}{16}}=0.1875$
Dichoral angles	Trip–4–trip: 90°
	Trip–3–trip: 60°
Height	${\frac {\sqrt {3}}{2}}\approx 0.86603$
Central density	1
Number of external pieces	6
Level of complexity	3
Related polytopes
Army	Triddip
Regiment	Triddip
Dual	Triangular duotegum
Conjugate	None
Abstract & topological properties
Flag count	216
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	A2≀S2, order 72
Flag orbits	3
Convex	Yes
Nature	Tame

The triangular duoprism or triddip, also known as the triangular-triangular duoprism, the 3 duoprism or the 3-3 duoprism, is a noble uniform duoprism that consists of 6 triangular prisms, with 4 meeting at each vertex. It is the simplest possible duoprism (excluding the degenerate dichora) and is also the 6-2 gyrochoron. It is the first in an infinite family of isogonal triangular dihedral swirlchora and also the first in an infinite family of isochoric triangular hosohedral swirlchora.

It is also a convex segmentochoron (designated K-4.10 on Richard Klitzing's list), as it is a triangle atop a triangular prism.

Gallery

• 

  Segmentochoron display, triangle atop triangular prism

• 

  Wireframe, cell, net

Vertex coordinates

Coordinates for the vertices of a triangular duoprism of edge length 1, centered at the origin, are given by:

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

Simpler coordinates can be given in 6-dimensions, as all permutations of

• $\left({\dfrac {\sqrt {2}}{2}},0,0,{\dfrac {\sqrt {2}}{2}},0,0\right)$,

where the sum of the first 3 coordinates is equal to the sum of the last 3.

Representations

A triangular duoprism has the following Coxeter diagrams:

• x3o x3o () (full symmetry)
• ox xx3oo&#x (axial, triangle atop triangular prism)
• xxoo xoox&#xr (axial, vertex first)
• xxx3ooo&#x (A₂ axial)

Related polychora

Isogonal derivatives

Substitution by vertices of these following elements will produce these convex isogonal polychora:

• Triangular prism (6): Triangular duotegum
• Triangle (6): Triangular duotegum
• Square (9): Triangular duoprism
• Edge (18): Rectified triangular duoprism

External links

• Bowers, Jonathan. "Category A: Duoprisms".

• Bowers, Jonathan. "Four Dimensional Dice Up To Twenty Sides".

• Klitzing, Richard. "triddip".
• Quickfur. "The 3,3-Duoprism".

• Wikipedia contributors. "3-3 duoprism".
• Hi.gher.Space Wiki Contributors. "Duotrianglinder".