Triangular duotegum

Triangular duotegum
(OFF file)
Rank	4
Type	Noble
Space	Spherical
Notation
Bowers style acronym	Triddit
Coxeter diagram	m3o2m3o
Elements
Cells	9 tetragonal disphenoids
Faces	18 isosceles triangles
Edges	6+9
Vertices	6
Vertex figure	Triangular tegum
Measures (based on triangles of edge length 1)
Edge lengths	Lacing (9): $\frac{\sqrt6}{3} ≈ 0.81650$
	Base (6): 1
Circumradius	$\frac{\sqrt3}{3} ≈ 0.57735$
Inradius	$\frac{\sqrt6}{12} ≈ 0.20412$
Hypervolume	$\frac{1}{32} = 0.03125$
Central density	1
Related polytopes
Army	Triddit
Regiment	Triddit
Dual	Triangular duoprism
Conjugate	None
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	A2≀S2, order 72
Convex	Yes
Nature	Tame

The triangular duotegum or triddit, also known as the triangular-triangular duotegum, the 3 duotegum, or the 3-3 duotegum, is a convex noble duotegum that consists of 9 tetragonal disphenoids and 6 vertices, with 6 cells joining at a vertex. It is the simplest possible duotegum, and is also the 6-2 step prism. It is the first in an infinite family of isogonal triangular hosohedral swirlchora and also the first in an infinite family of isochoric triangular dihedral swirlchora.

It shares the same vertex and edge configuration with the 5-dimensional hexateron. In fact, it is the simplest polytope that is not a simplex, but every pair of vertices is joined by an edge. Every n-2 step prism also has this property.

The ratio between the longest and shortest edges is 1:$\frac{\sqrt6}{2}$ ≈ 1:1.22474.

It is notable as the Birkhoff polytope B₃.

Gallery

• 

  Wireframe, cell, net

Vertex coordinates

The vertices of a triangular duotegum based on 2 unit-edge triangles, centered at the origin, are given by:

• $\left(±\frac12, -\frac{\sqrt3}{6},\,0,\,0\right),$
• $\left(0,\,\frac{\sqrt3}{3},\,0,\,0\right),$
• $\left(0,\,0,\,±\frac12,\,-\frac{\sqrt3}{6}\right),$
• $\left(0,\,0,\,0,\,\frac{\sqrt3}{3}\right).$

Due to this polytope being a Birkhoff polytope, the vertices of a triangular duotegum can also be positioned in 9D by taking all 3 × 3 permutation matrices and unraveling them in reading order:

• (1, 0, 0, 0, 1, 0, 0, 0, 1)
• (1, 0, 0, 0, 0, 1, 0, 1, 0)
• (0, 0, 1, 1, 0, 0, 0, 1, 0)
• (0, 1, 0, 1, 0, 0, 0, 0, 1)
• (0, 1, 0, 0, 0, 1, 1, 0, 0)
• (0, 0, 1, 0, 1, 0, 1, 0, 0)

The longer edge length in this case is $\sqrt{6}$, and the shorter one 2. It is centered on $\left(\frac{1}{3},\,\frac{1}{3},\,\ldots,\,\frac{1}{3}\right)$.

External links

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

• Klitzing, Richard. "triddit".

• Hi.gher.Space Wiki Contributors. "Duotrianglone".