Triangular antiwedge

The triangular antiwedge, or traw, also sometimes called the triangular gyrobicupolic ring, is a CRF segmentochoron (designated K-4.27 on Richard Klitzing's list). It consists of 1 octahedron (as a triangular antiprism), 2 triangular cupolas, and 6 square pyramids.

The triangular antiwedge occurs as a wedge of the regular icositetrachoron. In fact, the icositetrachoron can be broken into 6 triangular antiwedges, similar to how a regular hexagon can be broken into 6 equilateral triangles.

Vertex coordinates
The vertices of a triangular antiwedge with edge length 1 are given by:
 * $$\left(±\frac12,\,-\frac{\sqrt3}{6},\,\frac{\sqrt6}{6},\,\frac{\sqrt2}{2}\right),$$
 * $$\left(0,\,\frac{\sqrt3}{3},\,\frac{\sqrt6}{6},\,\frac{\sqrt2}{2}\right),$$
 * $$\left(±\frac12,\,\frac{\sqrt3}{6},\,-\frac{\sqrt6}{6},\,\frac{\sqrt2}{2}\right),$$
 * $$\left(0,\,-\frac{\sqrt3}{3},\,-\frac{\sqrt6}{6},\,\frac{\sqrt2}{2}\right),$$
 * $$\left(±\frac12,\,±\frac{\sqrt3}{2},\,0,\,0\right),$$
 * $$\left(±1,\,0,\,0,\,0\right).$$

Representations
A triangular antiwedge has the following Coxeter diagrams:


 * os2xo6os&#x (full symmetry)
 * xxo3oxx&#x (A2 symmetry only, seen with triangle atop gyro triangular cupola)