Minimum Perimeter Developments of the Platonic Solids
A development of a convex polyhedron is a connected plane figure obtained by cutting the surface of the polyhedron and unfolding it. In this paper, we determine the length and configuration of a minimum perimeter development for each of the Platonic solids. We show that such developments are obtained by cutting the surface of the polyhedron along a Steiner minimal tree. We introduce the concept of Steiner isomorphism to develop a search algorithm for determining these Steiner minimal trees. Each of these trees is completely symmetric with respect to rotation around a fixed point.
Akiyama, J., Chen, X., Nakamura, G., & Ruiz, M. J. (2011). Minimum perimeter developments of the platonic solids. Thai Journal of Mathematics, 9(3), 461-487.