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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 means of the selection of this values and the imposition of simple algebraic restrictions over the same ones is possible to create curves with different kind of symmetry and interesting geometrical properties, as you can see in the images.

The colouring of the curves is done in different steps and basically is a handmade process. At the first it is used, for convenience, the even-odd rule for distinguishing the inside and outside of each shape. Then you may fill with uniform colours, as you feel like, and at the end you apply different filters for reaching a bigger texture.

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

This idea is the one that underlies in the recent development of a real-time music visualizer, in which it has been achieved the fact that any curve is able to dance at the music’s rhythm, by connecting the changes in intensity and frequency of the sound with the changes in Fourier coefficients of the different harmonics. The colouring here is done automatically, by means of combination of various curves in each channel (RGB) and the use of transparency and blend modes.

Currently we are working in order to spread Fourier Art at 3D, searching which restrictions the coefficients must satisfy to make the generated curves keep a highly symmetry order.