An Invitation to Polynomiography via Exponential Series

Abstract

The subject of Polynomiography deals with algorithmic visualization of polynomial equations, having many applications in STEM and art, see [Kal04]. Here we consider the polynomiography of the partial sums of the exponential series. While the exponential function is taught in standard calculus courses, it is unlikely that properties of zeros of its partial sums are considered in such courses, let alone their visualization as science or art. The Monthly article Zemyan discusses some mathematical properties of these zeros. Here we exhibit some fractal and non-fractal polynomiographs of the partial sums while also presenting a brief introduction of the underlying concepts. Polynomiography establishes a different kind of appreciation of the significance of polynomials in STEM, as well as in art. It helps in the teaching of various topics at diverse levels. It also leads to new discoveries on polynomials and inspires new applications. We also present a link for the educator to get access to a demo polynomiography software together with a module that helps teach basic topics to middle and high school students, as well as undergraduates.