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Pushing a Rectangle down a Path

  • Mathematical Gems and Curiosities
  • Sergei Tabachnikov, Editor
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Notes

  1. Some readers might wonder whether this notion of a square or rectangle moving along a track might be related to the bicycle with square tires [8] one can ride at Macalaster College (or MoMath in New York). I doubt that there is a deep connection, however.

References

  1. A. Akopyan and S. Avvakumov. Any cyclic quadrilateral can be inscribed in any closed convex smooth curve. arXiv:1712.10205v1 (2017).

  2. J. Aslam, S. Chen, F. Frick, S. Saloff-Coste, L. Setiabrate, and H. Thomas. Splitting loops and necklaces: variants of the square peg problem. arXiv:1806.02484 (2018).

  3. C. Hugelmeyer. Every smooth Jordan curve has an inscribed rectangle with aspect ratio equal to \(\sqrt{3}\). arXiv:1803.07417 (2018).

  4. B. Matschke. A survey on the square peg problem. Notices of the AMS 61(4) (2014), 346–351.

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  5. I. Pak. Lectures on Discrete and Polyhedral Geometry. Available at http://www.math.ucla.edu/~pak/geompol8.pdf (2018).

  6. R. E. Schwartz, A trichotomy for rectangles inscribed in Jordan loops. Preprint, 2018.

  7. T. Tao. An integration approach to the Toeplitz square peg conjecture. Forum of Mathematics, Sigma 5 (2017).

  8. P. Zorn. Riding on square wheels. The Mathematical Tourist. Available at http://mathtourist.blogspot.com/2011/05/riding-on-square-wheels.html. May 15, 2011.

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Acknowledgments

I would like to thank Peter Doyle and Sergei Tabachnikov for helpful discussions about this topic. This research was supported by N.S.F. Research Grant DMS-1204471.

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Authors and Affiliations

  1. Department of Mathematics, Brown University, Providence, USA

    Richard Evan Schwartz

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  1. Richard Evan Schwartz
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Correspondence to Richard Evan Schwartz.

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This column is a place for those bits of contagious mathematics that travel from person to person in the community, because they are so elegant, surprising, or appealing that one has an urge to pass them on.

Contributions are most welcome.

Submissions should be uploaded to http://tmin.edmgr.comor sent directly to Sergei Tabachnikov, tabachni@math.psu.edu

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Schwartz, R.E. Pushing a Rectangle down a Path. Math Intelligencer 41, 7–10 (2019). https://doi.org/10.1007/s00283-018-9819-1

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