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

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.