Symmetric yet organic: Fourier series as an artist’s tool

The domain-colouring algorithm for visualizing complex-valued functions, when used with complex Fourier series, offers novel techniques to produce mathematical art. Using a photograph instead of the usual colour wheel of the algorithm, the artist can incorporate colours and textures in designs that are symmetric, yet organic. Techniques are described in detail, including methods to create rosettes, friezes and wallpaper patterns, with various types of colour symmetry. The techniques permit new approaches to pattern metamorphosis, as seen in the work of M.C. Escher.