Optimal partitions of the flat torus into parts of smaller diameter

D. S. Protasov, A. D. Tolmachev, Vsevolod A. Voronov

Abstract: We consider the problem of partitioning a two-dimensional flat torus T2<math><msup is="true"><mrow is="true"><mi is="true">T</mi></mrow><mrow is="true"><mn is="true">2</mn></mrow></msup></math> into m<math><mi is="true">m</mi></math> sets in order to minimize the maximum diameter of a part. For m⩽25<math><mrow is="true"><mi is="true">m</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">⩽</mo><mn is="true">25</mn></mrow></math> we give numerical estimates for the maximum diameter dm(T2)<math><mrow is="true"><msub is="true"><mrow is="true"><mi is="true">d</mi></mrow><mrow is="true"><mi is="true">m</mi></mrow></msub><mrow is="true"><mo is="true">(</mo><msup is="true"><mrow is="true"><mi is="true">T</mi></mrow><mrow is="true"><mn is="true">2</mn></mrow></msup><mo is="true">)</mo></mrow></mrow></math> at which the partition exists. Several approaches are proposed to obtain such estimates. In particular, we use the search for mesh partitions via the SAT solver, the global optimization approach for polygonal partitions, and the optimization of periodic hexagonal tilings. For m=3<math><mrow is="true"><mi is="true">m</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">=</mo><mn is="true">3</mn></mrow></math>, the exact estimate is proved using elementary topological reasoning.

External IDs:dblp:journals/disopt/ProtasovTV25