Triangle Presentations Encoded by Perfect Difference Sets

When James Singer exhibited projective planes for all prime power orders in 1938, he realized these using the trace function of cubic extensions of a finite field and linked $\text{trace}=0$ to perfect difference sets. In 1993, Cartwright, Mantero, Steger, and Zappa found that this trace function can be used to create a triangle presentation, which determines the structure of an $\tilde{A}_2$ building. We demonstrate a new, intrinsic connection between the perfect different sets of Singer and the triangle presentations of Cartwright et al., and show that this connection improves the efficiency of algorithms that generate these triangle presentations. Moreover, we translate the panel-regular groups of Essert \cite{essert2013geometric} and Witzel \cite{witzel2017panel} using triangle presentation nomenclature. This translation creates a uniform understanding of the panel-regular groups and vertex-regular groups via triangle presentations.