Far Eastern Mathematical Journal

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Covering of a rectangle with squares from both sides

Dmitriev M. D., Ozhegov F. Yu.

2023, issue 1, P. 16-26 DOI: https://doi.org/10.47910/FEMJ202303

Abstract

The paper provides an elementary proof of Kenyon's theorem that periodic tiling of a plane by squares with periods $(1,0)$ and $(0,\lambda)$ is possible only if $\lambda$$=$$p$$\pm$$\sqrt{q^2 - r^2}$ for some rational $p\geq q\geq r\geq 0$. A similar new result is proved about covering of a rectangle with squares from both sides in one layer. The paper also proves a necessary and sufficient condition for covering with equal squares.

Keywords: periodic tilings, square, rectangle, plane.

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References

[1] E. Chien, F. Luo, “Rectangle tilings of closed surfaces from discrete harmonic 1-chains”, https://pdfs.semanticscholar.org/47a9/0f45fb72847f8caac69c924fd30c7aa74901.pdf. [2] R. Kenyon, “Tiling with squares and square-tileable surfaces”, preprint, https://www.math.brown.edu/ rkenyon/papers/squares.ps.Z. [3] F. A. Sharov, “Prjamougol'nik iz kvadratov”, Kvant, 2019, No 3, 10–14. [4] M. Prasolov, M. Skopenkov, “Tiling by rectangles and alternating current”, Journal of Combinatorial Theory, Series A, 118:3 (2011), 920–937, arXiv: http://arxiv.org/abs/1002.1356. [5] S. Shirokovskikh, “Transformations of 2-port networks and tiling by rectangles”, preprint, arXiv: https://arxiv.org/abs/2110.02558. [6] A. A. Balakin, “Okleivanie tetrajedra kvadratami”, Matematicheskoe prosveshhenie, 3:23 (2018), 134–144. [7] M. Petkova, “Salfetki ”Kvanta“ i teorema Pifagora”, Kvant, 2012, No 3, 26–27.

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