W'

W'  is a nonconvex polyhedron with all regular faces, and the first known example of a 6-5-4 acrohedron. It was discovered by John Horton Conway and given its name by Bonnie Stewart.

It has some coplanar faces, which are resolvable by blending with four tetrahedra, resulting in a 28-faced 6-5-4 acrohedron. Stewart names this polyhedron W' ' . W' ' is weakly quasi-convex.

W' is non-self-intersecting. Richard Klitzing found a self-intersecting 6-5-4 acrohedron with only 17 faces.

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

Related polytopes
W' has the same face counts, symmetry and volume as the triangular hebesphenorotunda, a Johnson solid.