On Packing Thirteen Points in an Equilateral Triangle

The conversation of how to maximize the minimum distance between points - or, equivalently, pack congruent circles- in an equilateral triangle began by Oler in the 1960s. In a 1993 paper, Melissen proved the optimal placements of 4 through 12 points in an equilateral triangle using only partitions and direct applications of Dirichlet’s pigeon-hole principle. In the same paper, he proposed his conjectured optimal arrangements for 13, 14, 17, and 19 points in an equilateral triangle. In 1997, Payan proved Melissen’s conjecture for the arrangement of fourteen points; and, in September 2020, Joos proved Melissen’s conjecture for the optimal arrangement of thirteen points. These proofs completed the optimal arrangements of up to and including fifteen points in an equilateral triangle. Unlike Melissen’s proofs, however, Joos’s proof for the optimal arrangement of thirteen points in an equilateral triangle requires continuous functions and calculus. I propose that it is possible to continue Melissen’s line of reasoning, and complete an entirely discrete proof of Joos’s Theorem for the optimal arrangement of thirteen points in an equilateral triangle. In this paper, we make progress towards such a proof. We prove discretely that if either of two points is fixed, Joos’s Theorem optimally places the remaining twelve. KEYWORDS: optimization; packing; equilateral triangle; distance; circles; points; thirteen; maximize