Gaussian periods
Suppose that $\omega, d$ and $n$ are positive integers satisfying $\omega^d \equiv 1 \mod n$. A Gaussian period is a function $\sigma^{\mathbb{Z}/n}_{\langle\omega\rangle}: \mathbb{Z}_n \to \mathbb{C}$ defined by the exponential sum of the powers of $\omega$:
\[\sigma^{\mathbb{Z}/n}_{\langle\omega\rangle}(y) = \sum_{k=1}^{d-1} e^{2\pi i \frac{\omega^k y}{n}}\]The picture above plots the image of the gaussian periods as $y$ ranges from $1$ to $n$. When “Colorize” is enabled, each point is colored according to the value of $y \mod c$.
Mathematical framework
The gaussian period relates the multiplicative and additive structure of the ring $\mathbb{Z}/n$. For a multiplicative unit $\omega \in \mathbb{Z}/n^\times$, consider the multiplicative subgroup spanned by $\omega$:
\[\langle \omega \rangle = \{\omega^k \mod n | k \in \mathbb{N}\}\]The Gaussian period is a sum of exponential functions with exponent valued in $\langle \omega \rangle$.
\[\sigma^{\mathbb{Z}/n}_{\langle\omega\rangle}(y) = \sum_{x \in \langle \omega \rangle} e^{2\pi i \frac{x y}{n}}\]We should think of $\langle \omega \rangle$ as a geometric progression in modular arithmetic, and the gaussian period as its (discrete) Fourier transform. The Fourier transform of a sequence detects additive periodicity. For instance, the Fourier transform of an arithmetic progression $x_k = a\cdot k + b \mod n$ is given by
\[\sigma^{\mathbb{Z}/n}_{ \{ x_k\} }(y) = \sum_k e^{2 \pi i \frac{(a\cdot k +b) y }{n}}\]Whenever $y \cdot a \equiv 0 \mod n$, the fourier transform is as large as possible, satsifying $\vert\sigma^{\mathbb{Z}/n}_{ { x_k} }(y)\vert = \vert{x_k}\vert$. Inverting this, if a set has a large fourier transform for some $y$, then we should imagine it as being nearly periodic. Therefore, the Gaussian periods detect additive periodicity within geometric progressions in $\mathbb{Z}/n$.
The plots above were first explored in The graphic nature of Gaussian periods, by William Duke, Stephan Ramon Garcia, Bob Lutz. I’d recommend first reading their notices survey, Gauss’s Hidden Menagerie: From Cyclotomy to Supercharecers.