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Document Type

Article

Abstract

Teachers, students, and artists in the United States and abroad who have encountered polynomiography, in lectures, demos, or software, consistently appreciate its educational value and artistic potential. While dedicated polynomiography programs require upkeep as systems evolve, AI chatbots now offer a practical, accessible alternative. This article invites readers to explore polynomiography with ChatGPT, broadening access beyond specialized tools. While the approach will not match the full range or polish of advanced software, it provides a powerful and flexible entry point with many possibilities.

At its core, polynomiography transforms polynomial equations, each encoding a finite set of points in the complex plane, into colorful, visually striking images that can be read for their mathematical or algorithmic structure, or simply appreciated as fine art. One may specify a polynomial explicitly or designate an arbitrary set of roots, then apply or design iterative root-finding methods coupled with creative coloring and layering schemes. Metaphorically, it is a game of hide-and-seek with a constellation of dots: the polynomiographer hides points on the canvas via an equation and searches for them with an algorithm, painting as the search proceeds. Because the space of polynomials and algorithms is effectively unbounded, so are the educational and artistic outcomes. This article shows how to produce both fractal and non-fractal polynomiographs simply by prompting ChatGPT, with no code required, while also offering optional Python workflows for reproducibility, high resolution, batch processing, and classroom use.

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