The Barnsley fern is one of the best known images that can be produced by plotting the invariant set of a few transformations of the two dimensional plane. It is the invariant set of the following four contractions:-
S1(x,y) = (0, 0.16y)
S2(x,y) = (0.85x + 0.04y, -0.04x + 0.85y + 1.6)
S3(x,y) = (0.2x - 0.26y, 0.23x + 0.22y + 1.6)
S4(x,y) = (-0.15x + 0.28y, 0.26x + 0.24y + 0.44)
This multi-coloured version was produced by associating a colour modification with each of the four transformations used. The colour is set to dark green whenever a point is plotted as a result of the first transformation above being applied to another point in the invariant set, and faded towards a brighter shade when one of the other three transformations is used.
Roughly speaking, a contraction is a distance-reducing transformation; the definition of a contraction and of an invariant set are given on pages 113 and 114 of the set book for M835.