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Fourier Art is a form of computer art created by using Fourier series.

Fourier series are used to define curves and surfaces in a parametric form by expressing the coordinates of the points with different series.

As an example, a 2D curve in Cartesian coordinates will be as follows:

Where a, b, c and d are the Fourier coefficients, and T is the period of each series.

By imposing simple algebraic restrictions over this parameters it is possible to create curves with different kind of symmetry and interesting geometrical properties, as you can see in our gallery. You can also check out this random shape generator to get the main idea.

The colouring of the curves is done in different steps and is basically a handmade process. At first, the even-odd rule is used for distinguishing the inside and outside of each curve. Then, the shape is filled with plain colours, as you feel like, and, at the end, different filters are applied for reaching richer textures.

Fourier Art also allows you to create beautiful animations by smoothly changing the Fourier coefficients of a curve. This change can be uniform for all the coefficients or modeled by a specific function for each one of them, which offers a great potencial for expression.

This idea underlies in the recent development of many real-time music visualizers, in which any shape is able to dance at the music’s rhythm by connecting the changes in intensity and frequency of the sound with the changes in the Fourier coefficients of the different harmonics. The colouring here is done automatically by means of the combination of several shapes for each channel (RGBA) and the use of blend modes. Check out the video section or our YouTube channel to watch more examples.

Currently we are working on FourierArt 3D, searching for the restrictions the coefficients must satisfy to create 3D curves and surfaces with a highly symmetry order.