Domain Coloring Visualization: Turning Complex Functions into Vivid Art

Meta Description ➜ Domain coloring maps complex numbers to colour, transforming invisible math into stunning images. Scroll down for an interactive canvas and deep-dive guide.

Hue encodes phase; brightness encodes magnitude—core idea of domain colouring.

What Is Domain Coloring?

Domain coloring is a visualisation technique that assigns colour (hue, saturation, lightness) to complex numbers so that every point z = x + iy appears as a coloured pixel. By choosing a mapping where:

  • Hue represents the argument (phase)
  • Brightness/Saturation represent the magnitude (|z|)

you get an instant, intuitive portrait of analytic behaviour: zeros blossom into black dots, poles into white flares, essential singularities into fireworks of hue.

Classic Colour Scheme (HSV Wheel)

A popular mapping sets hue = arg / 2π (so 0° → red, 120° → green, 240° → blue) and brightness = 1 ⁄ (1 + ln(1+|z|)). This keeps infinities bright while finite areas stay vivid.

Interactive Demo 🎨

z = 0.00 + 0.00i

Why Mathematicians & Artists Love It

Teaching Complex Analysis: Students “see” singularities, branch cuts, and conformal mappings.
Research Diagnostics: Quickly sanity-check analytic continuations or locate zeros numerically.
Generative Art: Entire galleries showcase domain-colored functions printed on canvas or 3-D shaders.

DIY Tips for Better Plots

  1. Adaptive Sampling: Increase resolution near critical points to avoid aliasing.
  2. Contour Overlays: Draw thin black lines where |f(z)| = 1 to emphasise modulus levels.
  3. Gamma Tweak: A slightγ-correction on brightness keeps mid-tones from washing out.

Frequently Asked Questions