Permuting Heaven and Earth: Painted Expressions of Burnside's Theorem

James Mai and Daylene Zielinski
Bridges: Mathematical Connections in Art, Music, and Science (2004)
Pages 95–102


The creative process of the artist, James Mai, and an exploration of the mathematics within Permutations: Astral and Permutations: Earthly (hereafter referred to as Astral and Earthly) are the subjects of this paper. While the paintings lend themselves to mathematical interpretations, the artist's conception and execution were not strictly mathematical; considerations of visual aesthetics and metaphoric references were integrated with logical relationships. The creative process was often a circuitous one as the artist was challenged to understand the complete range of visually distinct figures through diagramming alone, without the benefit of mathematical abstraction. A mathematical analysis using Burnside's Theorem provided by Daylene Zielinski lays out an orderly approach that can be implemented in this type of artistic investigation. This useful theorem can tell the artist, in advance, how many distinct forms he will discover. It is important to note, however, that Burnside's Theorem says nothing about what these forms will look like or how to find them all; it predicts only the final count of visually distinct figures which can be created under a set of rules determined by the artist.