Enumerating Diamond Cuts

Abstract

We present an algorithm for enumerating all possible faceting arrangements of dihedrally symmetric diamond cuts. We first separate the question into enumerating crowns and pavilions. In each case, the method involves enumerating all plausible graphs within a fundamental domain, and then checking for planarity and triconnectivity using the Tutte spring embedding. Such graphs can be lifted to three dimensional convex crowns or pavilions via the Maxwell-Cremona correspondence.