Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 15 (2019), 097, 21 pages      arXiv:1912.05740      https://doi.org/10.3842/SIGMA.2019.097
Contribution to the Special Issue on Algebra, Topology, and Dynamics in Interaction in honor of Dmitry Fuchs

Fun Problems in Geometry and Beyond

Boris Khesin a and Serge Tabachnikov b
a) Department of Mathematics, University of Toronto, Toronto, ON M5S 2E4, Canada
b) Department of Mathematics, Pennsylvania State University, University Park, PA 16802, USA

Received November 17, 2019; Published online December 11, 2019

Abstract
We discuss fun problems, vaguely related to notions and theorems of a course in differential geometry. This paper can be regarded as a weekend ''treasure chest'' supplementing the course weekday lecture notes. The problems and solutions are not original, while their relation to the course might be so.

Key words: clocks; spot it!; hunters; parking; frames; tangents; algebra; geometry.

pdf (1931 kb)   tex (1746 kb)  

References

  1. Akopyan A., Avvakumov S., Any cyclic quadrilateral can be inscribed in any closed convex smooth curve, Forum Math. Sigma 6 (2018), e7, 9 pages, arXiv:1712.10205.
  2. Akopyan A.V., Bobenko A.I., Incircular nets and confocal conics, Trans. Amer. Math. Soc. 370 (2018), 2825-2854, arXiv:1602.04637.
  3. Arnold V.I., Arnold's problems, Springer-Verlag, Berlin, 2004.
  4. Arnold V.I., Lobachevsky triangle altitudes theorem as the Jacobi identity in the Lie algebra of quadratic forms on symplectic plane, J. Geom. Phys. 53 (2005), 421-427.
  5. Arnold V.I., Mathematical understanding of nature. Essays on amazing physical phenomena and their understanding by mathematicians, Amer. Math. Soc., Providence, RI, 2014.
  6. Arnold V.I., Problems for children from 5 to 15, 2014, available at https://imaginary.org/background-material/school-taskbook-from-5-to-15.
  7. Arnold V.I., Gusein-Zade S.M., Varchenko A.N., Singularities of differentiable maps, Vol. I. The classification of critical points, caustics and wave fronts, Monographs in Mathematics, Vol. 82, Birkhäuser Boston, Inc., Boston, MA, 1985.
  8. Baritompa B., Löwen R., Polster B., Ross M., Mathematical table-turning revisited, Math. Intelligencer 29 (2007), 49-58.
  9. Demaine E.D., Demaine M.L., Minsky Y.N., Mitchell J.S.B., Rivest R.L., Patraşcu M., Picture-hanging puzzles, Theory Comput. Syst. 54 (2014), 531-550, arXiv:1203.3602.
  10. Fenn R., The table theorem, Bull. London Math. Soc. 2 (1970), 73-76.
  11. Fuchs D., Tabachnikov S., Mathematical omnibus. Thirty lectures on classic mathematics, Amer. Math. Soc., Providence, RI, 2007.
  12. Fuchs D., Tabachnikov S., Self-dual polygons and self-dual curves, Funct. Anal. Other Math. 2 (2009), 203-220, arXiv:0707.1048.
  13. Glutsyuk A., Izmestiev I., Tabachnikov S., Four equivalent properties of integrable billiards, Israel J. Math., to appear, arXiv:1909.09028.
  14. Gray A., Tubes, Addison-Wesley Publishing Company, Advanced Book Program, Redwood City, CA, 1990.
  15. Halpern B., Weaver C., Inverting a cylinder through isometric immersions and isometric embeddings, Trans. Amer. Math. Soc. 230 (1977), 41-70.
  16. Izmestiev I., Tabachnikov S., Ivory's theorem revisited, J. Integrable Syst. 2 (2017), xyx006, 36 pages, arXiv:1610.01384.
  17. Izosimov A., The pentagram map, Poncelet polygons, and commuting difference operators, arXiv:1906.10749.
  18. Khovanova T., Martin Gardner's mistake, College Math. J. 43 (2012), 20-24, arXiv:1102.0173, see also http://blog.tanyakhovanova.com.
  19. Kronheimer E.H., Kronheimer P.B., The tripos problem, J. London Math. Soc. 24 (1981), 182-192.
  20. Livesay G.R., On a theorem of F.J. Dyson, Ann. of Math. 59 (1954), 227-229.
  21. Matschke B., A survey on the square peg problem, Notices Amer. Math. Soc. 61 (2014), 346-352.
  22. Meyerson M.D., Remarks on Fenn's ''the table theorem'' and Zaks' ''the chair theorem'', Pacific J. Math. 110 (1984), 167-169.
  23. Michor P.W., Topics in differential geometry, Graduate Studies in Mathematics, Vol. 93, Amer. Math. Soc., Providence, RI, 2008.
  24. Raphaël E., di Meglio J.-M., Berger M., Calabi E., Convex particles at interfaces, J. Phys. I France 2 (1992), 571-579.
  25. Santaló L.A., Integral geometry and geometric probability, Encyclopedia of Mathematics and its Applications, Vol. 1, Addison-Wesley Publishing Co., Reading, Mass. - London - Amsterdam, 1976.
  26. Schwartz R., The pentagram map, Experiment. Math. 1 (1992), 71-81.
  27. Schwartz R., The Poncelet grid, Adv. Geom. 7 (2007), 157-175.
  28. Schwartz R., Tabachnikov S., Elementary surprises in projective geometry, Math. Intelligencer 32 (2010), 31-34, arXiv:0910.1952.
  29. Tabachnikov S., Geometry and billiards, Student Mathematical Library, Vol. 30, Amer. Math. Soc., Providence, RI, 2005.
  30. Tabachnikov S., The (un)equal tangents problem, Amer. Math. Monthly 119 (2012), 398-405, arXiv:1102.5577.
  31. Tabachnikov S., Kasner meets Poncelet, Math. Intelligencer 41 (2019), 56-59, arXiv:1707.09267.
  32. Tabachnikov S., Dogru F., Dual billiards, Math. Intelligencer 27 (2005), 18-25.

Previous article  Next article  Contents of Volume 15 (2019)