G3

G3 is a non-convex regular faced polyhedron. It was the smallest known 5-4-3 acrohedron until the discovery of m*, which has one face fewer. It was named by Bonnie Stewart in Adventures Among the Toroids, although Stewart was not searching for acrohedra in particular.

Vertex coordinates
The vertex coordinates of a G3 with unit edge length are given by:
 * $$\left(\pm\frac{1+\sqrt5}{4},\,\pm\frac{1+\sqrt5}{4},\,\frac{1+\sqrt5}{4}\right),$$
 * $$\left(\pm\frac12,\,0,\,\frac{3+\sqrt5}{4}\right),$$
 * $$\left(0,\,\pm\frac{3+\sqrt5}{4},\,\frac12\right),$$
 * $$\left(\frac{3+\sqrt5}{4},\,\pm\frac12,\,0\right),$$
 * $$\left(\frac{\sqrt5-1}{4},\,\pm\frac12,\,0\right),$$
 * $$\left(-\frac12,\,0,\,\frac{\sqrt5-1}{4}\right).$$