On the symmetry of finite sums of exponentials

TL;DR

This paper explores the geometric properties of curves formed by the sum of two complex exponentials, revealing their symmetry groups and intersection points through elementary methods, and discusses phenomena for sums of more exponentials.

Contribution

It provides a complete characterization of symmetry and intersection points for sums of two exponentials, and discusses phenomena for more than two exponentials using elementary arguments.

Findings

Determined symmetry groups of the curves.

Identified points of self-intersection.

Described phenomena for sums of more than two exponentials.

Abstract

In this note we are interested in the rich geometry of the graph of a curve $\gamma_{a,b}: [0,1] \rightarrow \mathbb{C}$ defined as \begin{equation*} \gamma_{a,b}(t) = \exp(2\pi i a t) + \exp(2\pi i b t), \end{equation*} in which $a,b$ are two different positive integers. It turns out that the sum of only two exponentials gives already rise to intriguing graphs. We determine the symmetry group and the points of self intersection of any such graph using only elementary arguments and describe various interesting phenomena that arise in the study of graphs of sums of more than two exponentials.