Triangular duotegum

The triangular duotegum or triddit, also known as the triangular-triangular duotegum, the 3 duotegum, or the 3-3 duotegum, is a 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.

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 &times; 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 2, and the shorter one $$\sqrt{6}$$. It is centered on $$\left(\frac{1}{3},\,\frac{1}{3},\,\ldots,\,\frac{1}{3}\right)$$.