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Parallel covering a rhombus with squares

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Abstract

Suppose that \(R^{\alpha }\) is a rhombus with side length 1 and with the acute angle \(\alpha \). Let \(\{S_{i}\}\) be a collection of squares. A tight lower bound of the sum of the areas of squares from \(\{S_{i}\}\) that permit a parallel covering of \(R^{\alpha }\) is given.

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References

  1. H.L. Abbott, M. Katchalski, Covering squares with squares. Discrete Comput. Geom. 24, 151–169 (2000)

    Article  MathSciNet  Google Scholar 

  2. P. Erdős, Squares in a square. Geombinatorics IV, 110–114 (1995)

    MathSciNet  Google Scholar 

  3. G. Fejes Tóth and W. Kuperberg, Packing and covering with convex sets, in Handbook of Convex Geometry, pp. 799–860 (1993)

  4. Z. Füredi, Covring a triangle with homothetic copies. Discrete Geom. Monoger. Textb. Pure Appl. Math. 253, 434–445 (2003)

    Google Scholar 

  5. H. Groemer, Coverings and packings by sequences of convex sets. Discrete Geom. Convexity 440, 262–278 (1985)

    MathSciNet  Google Scholar 

  6. J. Januszewski, Covering a triangle with sequences of its homothetic copies. Period. Math. Hung. 36, 183–189 (1998)

    Article  MathSciNet  Google Scholar 

  7. J. Januszewski, Translative covering by sequences of homothetic copies. Acta Math. Hung. 91, 337–342 (2001)

    Article  MathSciNet  Google Scholar 

  8. J. Januszewski, Covering by sequences of squares. Stud. Sci. Math. Hung. 39, 179–188 (2002)

    MathSciNet  Google Scholar 

  9. J. Januszewski, Parallel packing and covering of an equilateral triangle with sequences of squares. Acta Math. Hung. 125, 249–260 (2009)

    Article  MathSciNet  Google Scholar 

  10. M. Lu, Z. Su, Parallel covering of isosceles triangles with squares. Acta Math. Hung. 155(2), 266–297 (2018)

    Article  ADS  MathSciNet  Google Scholar 

  11. J.W. Moon, L. Moser, Some packing and covering theorems. Colloq. Math. 17, 103–110 (1967)

    Article  MathSciNet  Google Scholar 

  12. C. Su, Z. Su, L. Yuan, Parallel packing and covering of an isosceles right triangle with sequences of squares. Ars Combin. 130, 3–16 (2017)

    MathSciNet  Google Scholar 

Download references

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant Number 12171362) and the Doctoral Research Project of Tianjin Normal University (Grant Number 52XB2112).

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Authors and Affiliations

  1. College of Mathematical Science, Tianjin Normal University, Tianjin, 300387, People’s Republic of China

    Chen-Yang Su

  2. College of Science, Tianjin University of Commerce, Tianjin, 300134, People’s Republic of China

    Xue Li

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  1. Chen-Yang Su
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  2. Xue Li
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Correspondence to Xue Li.

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Su, CY., Li, X. Parallel covering a rhombus with squares. Period Math Hung 88, 190–203 (2024). https://doi.org/10.1007/s10998-023-00547-0

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