Phase Portraits of Hyperbolic Geometry
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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:
Sometimes it is easier to see than to say. —Jonathan M. Borwein
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...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 Maple code used to generate all of the phase portraits in this paper is available at https://ogma.newcastle.edu.au/vital/access/manager/Repository/uon:29853.
This project is dedicated to the memory of Jonathan M. Borwein, our adviser, mentor, and friend.
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