Experimental Computation and Visual Theorems

  • Jonathan M. Borwein
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8592)


Long before current graphic, visualisation and geometric tools were available, John E. Littlewood (1885-1977) wrote in his delightful Miscellany:

A heavy warning used to be given [by lecturers] that pictures are not rigorous; this has never had its bluff called and has permanently frightened its victims into playing for safety. Some pictures, of course, are not rigorous, but I should say most are (and I use them whenever possible myself). [p. 53]

Over the past five years, the role of visual computing in my own research has expanded dramatically. In part this was made possible by the increasing speed and storage capabilities—and the growing ease of programming—of modern multi-core computing environments.

But, at least as much, it has been driven by my group’s paying more active attention to the possibilities for graphing, animating or simulating most mathematical research activities.


visual theorems experimental mathematics randomness normality of numbers short walks planar walks fractals protein confirmation 


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  1. 1.
    Aragon, F., Borwein, J.M.: Global convergence of a non-convex Douglas-Rachford iteration. J. Global Optim. 57(3), 753–769 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Aragon, F., Bailey, D.H., Borwein, J.M., Borwein, P.B.: Walking on real numbers. Mathematical Intelligencer 35(1), 42–60 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Aragon, F., Borwein, J.M., Tam, M.: Douglas-Rachford feasibility methods for matrix completion problems. ANZIAM Journal (accepted March 2014)Google Scholar
  4. 4.
    Bailey, D.H., Borwein, J.M.: Exploratory Experimentation and Computation. Notices of the AMS 58(10), 1410–1419 (2011)zbMATHMathSciNetGoogle Scholar
  5. 5.
    Borwein, J., Devlin, K.: The Computer as Crucible: an Introduction to Experimental Mathematics. AK Peters (2008)Google Scholar
  6. 6.
    Borwein, J.M., Straub, A.: Mahler measures, short walks and logsine integrals. Theoretical Computer Science 479(1), 4–21 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Borwein, J.M., Skerritt, M., Maitland, C.: Computation of a lower bound to Giuga’s primality conjecture. Integers 13 (2013), Online September 2013 at #A67,
  8. 8.
    Borwein, J.M., Straub, A., Wan, J., Zudilin, W. (with an Appendix by Don Zagier): Densities of short uniform random walks. Can. J. Math. 64(5), 961–990 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Bornemann, F., Laurie, D., Wagon, S., Waldvogel, J.: The SIAM 100-Digit Challenge: A Study In High-accuracy Numerical Computing. SIAM, Philadelphia (2004)Google Scholar
  10. 10.
    Hanna, G., de Villiers, M. (eds.) ICMI, Proof and Proving in Mathematics Education, The 19th ICMI Study. New ICMI Study Series, vol. 15. Springer (2012)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Jonathan M. Borwein
    • 1
  1. 1.CARMAUniversity of NewcastleAustralia

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