W'

W'  is a convex polyhedron with all regular faces. However it is not a Johnson solid, since several of its faces are coplanar. It is notable for being the first discovered example of a 6-5-4 acrohedron. It was discovered by John Horton Conway and given its name by Bonnie Stewart.

Vertex coordinates
A W' of edge length 1 has vertices given by the following coordinates:


 * $$\left(\pm\frac{1}{2},\,-\frac{\sqrt{3}}{6},\,\frac{3\sqrt{3}+\sqrt{15}}{6}\right)$$,
 * $$\left(0,\,\frac{\sqrt{3}}{3},\,\frac{3\sqrt{3}+\sqrt{15}}{6}\right)$$,
 * $$\left(\pm1,\,\frac{\sqrt{3}}{6},\,\frac{3\sqrt{3}+\sqrt{15}}{6}\right)$$,
 * $$\left(0,\,-\frac{2\sqrt{3}}{3},\,\frac{3\sqrt{3}+\sqrt{15}}{6}\right)$$,
 * $$\left(\pm\frac12,\,\frac{2\sqrt{3}+\sqrt{15}}{6},\,\frac{\sqrt{3}+\sqrt{15}}{6}\right)$$,
 * $$\left(\pm\frac{3+\sqrt5}{4},\,-\frac{\sqrt{15}-\sqrt{3}}{12},\,\frac{\sqrt{3}+\sqrt{15}}{6}\right)$$,
 * $$\left(\pm\frac{1+\sqrt5}{4},\,-\frac{5\sqrt{3}+\sqrt{15}}{12},\,\frac{\sqrt{3}+\sqrt{15}}{6}\right)$$,
 * $$\left(\pm\frac{1}{2},\,\pm\frac{\sqrt{3}}{2},\,0\right)$$,
 * $$\left(\pm1,\,0,\,0\right)$$.

Related polytopes
W' is closely related to the triangular hebesphenorotunda, a Johnson solid. The two have the same face counts, symmetry and volume.

Its coplanar faces are resolvable by excavating them with four tetrahedra, resulting in a 28-faced 6-5-4 acrohedron. Stewart names this polyhedron W' ' . W' ' is weakly quasi-convex.

Alex Doskey found that W' can be augmented with a square pyramid to produce a 6-5-3-3 acrohedron.

W' is non-self-intersecting. Richard Klitzing found a self-intersecting 6-5-4 acrohedron with the same symmetry, that has only 17 faces.