Jessen's icosahedron

Jessen's icosahedron or Jessen's orthogonal icosahedron is a concave isogonal icosahedron with pyritohedral symmetry. It has the same face, vertex, and edge count as the regular icosahedron. All edges have a dihedral angle of 90 degrees, but the six longer edges are concave.

The vertex coordinates are given by the twelve points $$(\pm 2,\pm 1,0)$$, $$(0,\pm 2,\pm 1)$$, and $$(\pm 1,0,\pm 2)$$.

A shape similar to Jessen's icosahedron can be produced by symmetrically "indenting" six edges of a regular icosahedron, but the longer-to-shorter edge ratio of the resulting figure is different (equal to the golden ratio in this case).