In mathematics, affine transforms are the ones you can do to rigid bodies – rotations and translations – plus uniform scalings.

Michael Barnsley showed that a set of affine transformations will in general converge to a fractal limit, as long as they are all contractive, i.e. with a scaling factor less than 1. That is to say, applying a series of transforms, each randomly chosen from this set, to any point in space will eventually yield a point as close as you like to the limit set. He called such a set, together with a set of probabilities for choosing its members, an Iterated Function System, or IFS.

Subsequent research has shown that similar results hold for non-affine transforms, as long as they are *on average* contractive.

Barnsley showed how complex shapes, such as fern leaves, could be encoded in just a few simple transformations.

In fact, he showed, any image at all could be encoded in this way, in a process called Fractal Image Compression.

Fractal Image Compression allows very high rates of scale-free compression – images can be decompressed at any size without pixellation.

Affinity is the name of my new app for iOS and android, which applies IFS transforms to photos, videos and camera feeds, with beautiful results.