The Barnsley Fern

The Barnsley Fern is a wonderful example of complex natural phenomena explained by simple mathematical structures. A point is placed in the center of the plane below, and repeatedly, one of four affine transformations is applied to it. An affine transformation is a simple function that depends only on the point’s position, but incredibly, when we color the places the point vists most often on its journey, these four transformations produce the platonic ideal of a fern. The coefficients and frequency of each transformation may have been carefully engineered, but it’s no less remarkable that something so beautiful and complex can arise from such a simple set of functions.