For a framework of e rods and v vertices where the rods and vertices of the framework are the edges and vertices of a convex polyhedron, such polyhedron has (2 + e - v) faces by Euler's relationship (click here for proof.) Then, it follows from Cauchy's rigidity theorem (see below) that if the faces of the above polyhedron are all triangular, i.e., if 3v - 6 = e, then the convex polyhedral structure is rigid and not suspectible to small perturbations.
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Illustrated by Jonathan Shum
Center for Intelligent Machines
McGill University, Montreal, Canada.