Problems in arithmetic topology

Abstract

The term "fractal" was introduced by Benoit Mandelbrot in 1975 [23] to describe geometric objects exhibiting self-similarity at scale. This concept, and the examples it encompasses, has given rise to rich theories, scientific and industrial applications, and works of art that make sense of previously unwieldy fragmentations occurring in natural phenomena. We often trace the lineage of these ideas to Weierstrass' continuous-but-non-differentiable curves and related pathological functions, yet we find diverse examples in abundance in Africanist material cultures (historically and contemporaneously), such as in textiles, architecture, and religious symbols [13] . Here, we consider how understanding, teaching, and doing mathematics engaged with African and African diasporic use of fractals provides an opening for embodied, experiential, and participatory engagement.In 2012, choreographer Reggie Wilson was sourcing and developing material for a new concert dance work that would ultimately premier as Moses(es) in 2013. As Wilson described [36] : "Many of the movements and performance practices I was engaging with exhibited Africanist formal features--rhythms, brief sequences of movement, striking variations in quality and force or tempo--that did not map easily onto the schema and formalisms that characterize traditional Western concert dance. Around this same time, I encountered Ron Eglash's African Fractals which documents extensive use of fractal symmetry in African material cultures. I hypothesized that fractals were also prevalent in the performative cultures of Africa, and I invited Wolfson to help "translate" Eglash's text and to give his perspective on a selection of Africanist forms of music and dance that had most piqued my curiosity to possible usage of things fractal."Wilson brought me (Jesse) in as a "math consultant" to assist the Fist and Heel Performance Group company members as they engaged with Africanist sources and developed original movement sequences for Wilson's well-received original concert dance Moses(es). This experience confirmed the practical value of the hypothesis in the rehearsal studio: it provides tools for analyzing, playing with, reverse engineering, and generating movements in the service of creating original concert dance. The company developed a shared vocabulary for discussing and analyzing certain previously opaque features using the language of fractals, and Wilson and the dancers were able to reliably use this vocabulary to analyze and understand sourced movement in new and generative ways. The work described here grows directly out of Wilson's and the Fist and Heel's "research into performance" practice, seeking to understand the mathematical underpinnings of Wilson's hypothesis through quantitative analysis of Africanist music samples. Fractals queer our sense of spaceDimension can be a tricky concept-we often speak of degrees of freedom, independent subsets, ascending chains, or even covering combinatorics-and fractals exceed our usual intuitions. While fractals are famous...