Two Congruence Classes of Extremal Twenty-Point Sets

Alexeev, Mixon and Parshall proved that twenty points in the plane have at most 54 unit-distance pairs and that every maximizing set has the same abstract unit-distance graph. We determine its Euclidean realization classes: there are exactly two, even when reflections are allowed. Both are explicit Minkowski sums of the unit diamond and a five-point triangular fan. Cartesian-product squares force every injective realization to have this form, and one additional edge leaves two relative rotations. A fourth distance moment separates the resulting sets. The proof uses the established abstract-graph enumeration and otherwise consists of elementary geometry and exact algebra. This is a finite congruence classification; it does not resolve the asymptotic question in Erdős Problem 668.