A convex polyhedron which is not equifacettable

A famous theorem due to E. Steinitz states, in one of its formulations, that every planar (or, equivalently, every spherical) graph can be realized as the graph of edges and vertices of a convex polyhedron in Euclidean 3-space (see, for example, Grünbaum [1, Section 13.1] or Ziegler [3, Chapter 4]). This representation is possible in many different ways, but in all of them the circuits that bound faces of the polyhedron are the same. Malkevitch considered the question whether, in case the graph is a triangulation of the sphere (or, equivalently, the graph of a triangle-faced convex polyhedron), one can insist that the polyhedron in Steinitz's theorem has congruent isosceles triangles as faces. He shows by an elegant example that the answer is negative, and discusses various other questions.