Classification of crescent configurations

Let $n$ points be in crescent configurations in $mathbb{R}^d$ if they lie in general position in $mathbb{R}^d$ and determine $n-1$ distinct distances, such that for every $1 leq i leq n-1$ there is a distance that occurs exactly $i$ times. Since Erdősu0027 conjecture in 1989 on the existence of $N$ sufficiently large such that no crescent configurations exist on $N$ or more points, he, Pomerance, and Palasti have given constructions for $n$ up to $8$ but nothing is yet known for $n geq 9$. Most recently, Burt et. al. had proven that a crescent configuration on $n$ points exists in $mathbb{R}^{n-2}$ for $n geq 3$. In this paper, we study the classification of these configurations on $4$ and $5$ points through graph isomorphism and rigidity. Our techniques, which can be generalized to higher dimensions, offer a new viewpoint on the problem through the lens of distance geometry and provide a systematic way to construct crescent configurations.