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arXiv · 2608.19424

A rock-paper-scissors Mandelbrot set

Charlotte Aten
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Abstract

The titular object of this paper is an analogue of the Mandelbrot set over the 3-dimensional real algebra whose multiplication is the bilinear extension of the rock-paper-scissors operation. In order to study the dynamics of the mappings $x\mapsto x^2+c$ in this setting, a notion of holomorphy for functions on general finite-dimensional real algebras is introduced. Under a mild assumption it is shown that for such algebras holomorphy is always equivalent to solving a finite system of linear first-order PDEs generalizing the Cauchy-Riemann equations. Code is provided for generating animations of these fractals and for exploring them in a video game format.

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Charlotte Aten. 2026-08-19. A rock-paper-scissors Mandelbrot set. https://arxiv.org/abs/2608.19424

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