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The Art of Modern Homo Habilis Mathematicus, or: What Would Jon Borwein Do?

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Abstract

Jonathan Borwein was a founder and early champion of the field of experimental mathematics. His high-profile accomplishments and extensive writing on the role of computational discovery have served to establish experimental mathematics as a field in its own right. In the wake of his passing, the question “What would Jon do?” has served as a frequent catalyst in the investigations of the present author. The present work draws the inspiration for its name from Borwein’s first posthumous book chapter, and it is intended as a spiritual progeny to all of Borwein’s expositions on experimental mathematics. Borwein was well known as an ardent supporter of the use of visualization. This work samples the myriad of ways in which artistic methods are instrumental to experimental mathematics, along with the ways that experimental mathematics gives birth to unexpected art. The examples are problems encountered by the present author and solved with techniques motivated by the strategies and interests of Borwein. These motivations are highlighted. Particular emphasis is placed on tools, strategies, and the broader arc that research follows: from low-dimensional, specific, and visible, to high-dimensional and general. Topics include dynamical systems, geometry, optimization, error bounds, random walks, special functions, and number theory.

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Acknowledgments

The following persons were collaborators on one or more of the present author’s projects that are described in detail above: Heinz H. Bauschke, Jonathan M. Borwein, Regina S. Burachik, Minh N. Dao, Bruno F. Lourenco, Ting Kei Pong, Brailey Sims, Anna Schneider, Matthew P. Skerritt, and Paul Vrbik. The cartoon drawn by Simon Roy of Jon and Veselin Jungic in Fig. 3 appears courtesy of Veselin Jungic. The image Walk on Pi in Fig. 24 appears courtesy of Francisco Aragón Artacho, Peter Borwein, and David H. Bailey. Armin Straub, James Wan, and Wadim Zudilin furnished valuable information about the origins of Fig. 23. The author is grateful to all of the above persons for their contributions, and especially grateful to Jonathan Borwein.

Image Sources

A small version of Fig. 1 appeared in Borwein et al. (2018) and a large version in its online appendix (Borwein et al. 2017). Figure 2 is original to this work, but is implicitly contained in Fig. 18, which appeared in Borwein et al. (2017). The photo of Jonathan Borwein in Fig. 3 was used in the presentation for Borwein and Lindstrom (2016); the cartoon was drawn by Simon Roy and included in Jungic (2016)’s remembrance of Borwein on the Jon Borwein memorial webpage managed by David Bailey. The cartoon appears courtesy of Veselin Jungic.

Figure 4 is original to this work. Figure 5 (left) and (center) are from Lindstrom and Vrbik (2019), while the image at right is original to this work. Figure 6 (left) is from Lindstrom and Vrbik (2019) while the right image is original to this work. The images in Figs. 7 and 8 are from Lindstrom and Vrbik (2019)). The images in Figs. 9, 10, and 11 are original to this work and are closely related mathematically to images in Lindstrom and Vrbik (2019). The images in Fig. 12 are from Lindstrom (2019).

Figure 13 (right) is original to this work; a slightly different version of the left image appeared in Lindstrom (2019) and originally in Díaz Millán et al. (2020). Figure 14 is original to this work. Figure 15 is original to this work, though different images of this Lyapunov function have appeared in the various related references. Figure 16 is original to this work. Figure 17 is from Borwein et al. (2018). Figure 18 is from Borwein et al. (2017), and cropped versions of these images appeared in Borwein et al. (2018). Figure 19 is from Borwein et al. (2017) and Lindstrom (2019), and is also the official artwork of the Australian Mathematical Society special interest group Mathematics of Computation and Optimization (MoCaO). Figure 20 is from Borwein et al. (2018), and a larger version is in Borwein et al. (2017). Figure 21 is original to this work and is similar to images in Lindstrom et al. (2017). Figure 22 is original to this work and based on a similar image in Lindstrom (2020).

Figure 23 first appeared in Borwein and Straub (2016) and has also been featured in Borwein and Lindstrom (2016). Related and cropped variants have appeared in other places, as described in the text above. Figure 24 is originally from Aragón Artacho et al. (2013) and has been featured in many articles in the popular press. It is also featured online in Walking on Real Numbers: A Multiple Media Mathematics Project (Aragón Artacho et al. 2014).

Figure 25 is original to this work. Figure 26 is from Bauschke and Lindstrom (2020). Figure 27 is from Burachik et al. (2019a). Figure 28 is from Burachik et al. (2021). Figure 29 is from Lindstrom et al. (2020). Figures 30 and 31 are original to this work, because that analysis was omitted from Lindstrom et al. (2020) in lieu of the shortest path to the solution.

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Lindstrom, S.B. (2021). The Art of Modern Homo Habilis Mathematicus, or: What Would Jon Borwein Do?. In: Sriraman, B. (eds) Handbook of the Mathematics of the Arts and Sciences. Springer, Cham. https://doi.org/10.1007/978-3-319-57072-3_133

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