{"paper":{"title":"Packings of equal disks in a square torus","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.MG","authors_text":"Evan Solomonides, Matthew Funkhouser, Robert Connelly, Vivian Kuperberg","submitted_at":"2015-12-29T19:31:03Z","abstract_excerpt":"Packings of equal disks in the plane are known to have density at most $\\pi/\\sqrt{12}$, although this density is never achieved in the square torus, which is what we call the plane modulo the square lattice. We find packings of disks in a square torus that we conjecture to be the most dense for certain numbers of packing disks, using continued fractions to approximate $1/\\sqrt{3}$ and $2-\\sqrt{3}$. We also define a constant to measure the efficiency of a packing motived by a related constant due to Markov for continued fractions. One idea is to use the unique factorization property of Gaussian"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.08762","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}