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Optimal partitions of the flat torus into parts of smaller diameter

Dmitry Protasov, Alexander Tolmachev, Vsevolod Voronov

Abstract

We consider the problem of partitioning a two-dimensional flat torus $T^2$ into $m$ sets in order to minimize the maximal diameter of a part. For $m \leqslant 25$ we give numerical estimates for the maximal diameter $d_m(T^2)$ 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 $m=3$, the exact estimate is proved using elementary topological reasoning.

Optimal partitions of the flat torus into parts of smaller diameter

Abstract

We consider the problem of partitioning a two-dimensional flat torus into sets in order to minimize the maximal diameter of a part. For we give numerical estimates for the maximal diameter 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 , the exact estimate is proved using elementary topological reasoning.
Paper Structure (11 sections, 13 theorems, 50 equations, 9 figures, 3 tables)

This paper contains 11 sections, 13 theorems, 50 equations, 9 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Main result
  3. Proofs
  4. Proof of Theorem \ref{['theorem1']}
  5. Proof of Theorem \ref{['theorem2']}
  6. Verification of the estimates in Theorem \ref{['theorem3']}
  7. Upper bounds
  8. Lower and upper estimates via SAT approach
  9. Upper estimates via periodic hexagonal grids
  10. Global optimization approach
  11. Conclusion

Key Result

Theorem 1

The following inequalities are satisfied: Note that the proof of the exact estimate for $m=3$ turns out to be non-trivial. For $m \geqslant 4$, we were unable to prove that the estimates are exact.

Figures (9)

  • Figure 1: Best partitions, $m = 2, \dots, 7$
  • Figure 2: Propositions 3 and 4
  • Figure 3: Arrangement of the green-blue strip (A) and the blue-red strip (B). 1 --- red, 2 --- blue, 3 --- green.
  • Figure 4: Definition of the set $Q_i$ in Lemma 1
  • Figure 5: Colorings in 5 and 6 colors found via the kissat solver
  • ...and 4 more figures

Theorems & Definitions (20)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Theorem 4: Raikov, raikov1939addition
  • ...and 10 more