On Polyhedral Realization with Isosceles Triangles

Answering a question posed by Joseph Malkevitch, we prove that there exists polyhedral graph, with triangular faces, such every realization of it as the graph convex polyhedron includes at least one face is scalene triangle. Our construction based on Kleetopes, and shows an integer $i$ all $i$-iterated Kleetopes have face. However, also show triangulated graphs non-convex non-self-crossing realizations in which faces are isosceles. We answer another Malkevitch observing spherical tiling Dawson (2005) leads to fourth infinite family polyhedra congruent isosceles triangles, adding three families previously known Malkevitch. bounded diameter dominating sets size.