Domain coloring

Domain coloring is a technique for visualizing functions of a complex variable. The term "domain coloring" was coined by Frank Farris.

Motivation

Insufficient dimensions

A real function $f:\mathbb {R} \rightarrow {}\mathbb {R}$ (for example $f(x)=x^{2}$ ) can be graphed using two Cartesian coordinates on a plane.

A graph of a complex function $g:\mathbb {C} \rightarrow {}\mathbb {C}$ of one complex variable lives in a space with two complex dimensions. Since complex plane itself is two dimensional, a graph of a complex function is an object in four real dimensions. That makes complex functions difficult to visualize in our three dimensional space. One way of depicting holomorphic functions is with a Riemann surface.

Visual Encoding of complex numbers

Given a complex number $z=re^{i\theta }$ , the phase (also known as argument) $\theta$ can be represented by hue, and the modulus $r=|z|$ is represented by either intensity or variations in intensity. The arrangement of hues is arbitrary, but often it follows the color wheel. Sometimes the phase is represented by a specific gradient rather than hue.