Phase Portraits of Hyperbolic Geometry

  • Scott B. LindstromEmail author
  • Paul Vrbik

Sometimes it is easier to see than to say. —Jonathan M. Borwein

Borwein was quite fond of saying this, and it seems fitting that his first posthumously published book was Tools and Mathematics: Instruments for Learning with Luc Trouche and John Monaghan. In his chapter [ 3], he included, along with his own commentary, a quotation that he particularly liked:
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) [5, p. 53].

The popularity of Arnold and Rogness’s video Möbius Transformations Revealed [ 2] serves to highlight that not only may pictures be rigorous, they may appeal to a much broader audience than...



The Maple code used to generate all of the phase portraits in this paper is available at


This project is dedicated to the memory of Jonathan M. Borwein, our adviser, mentor, and friend.


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    D. N. Arnold and J. Rogness. Möbius transformations revealed. Notices of the AMS 55 (2008).Google Scholar
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    D. N. Arnold and J. Rogness. Moebius Transformations Revealed [Video file]. Retrieved from, June 3, 2007.
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    J. M. Borwein. The Life of Modern Homo Habilis Mathematicus: Experimental Computation and Visual Theorems. In Tools and Mathematics, pp. 23–90, Mathematics Education Library 347. Springer, 2016.Google Scholar
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    J. M. Borwein and A. Straub. Moment function of a 4-step planar random walk. Complex Beauties 2016 (2016 Calendar). Available at
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    J. E. Littlewood. A Mathematician’s Miscellany. London: Methuen, 1953. Republished in Béla Bollobás, ed., Littlewood’s Miscellany, Cambridge University Press, 1986.Google Scholar
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    T. Needham. Visual Complex Analysis. Oxford University Press, 1997.Google Scholar
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    E. Wegert. Visual Complex Functions: An Introduction with Phase Portraits. Springer, 2012.Google Scholar
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    E. Wegert and G. Semmler. Phase Plots of Complex Functions: A Journey in Illustration. Notices of the American Mathematical Society 58(6) (2011), 768–780.MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.CARMAUniversity of NewcastleNewcastleAustralia
  2. 2.University of Toronto MississaugaMississaugaCanada

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