Holyhedron

A holyhedron is a polyhedron where every face has at least one hole. The full definition, which is rather involved, excludes faces where the hole touches the outer boundary of the face. John Horton Conway offered a prize of US$10,000 divided by the number of faces.

Circa 1997, George Hart discovered some near-examples. In 1999, Jade Vinson discovered the first known example, with 78,585,627 faces, with the additional property of being non-self-intersecting. In 2001, Grunbaum and Shepard found a self-intersecting 24-face isohedral holyhedron, but it is a compound and might not be considered a valid example.

Don Hatch found a non-self-intersecting 492-face example in 2003. In 2014, Nathan Ho discovered a 12-face self-intersecting holyhedron based on a pentagrammic pyramid.