Four affine transformations applied randomly at probabilities 0.01/0.85/0.07/0.07 generate a mathematically perfect fern leaf from a single starting point. This is an iterated function system — a fractal defined not by an equation but by a set of contractions.
Each transformation maps the current point to a new location. 85 % of the time the 'blade' map slightly scales and rotates the point upward — this builds the main frond. The three remaining maps create the stem, small bottom leaflets and the overall bending. The attractor has fractal dimension ≈ 1.74.
Press Start to scatter individual dots; watch the fern materialise pixel by pixel. Adjust the colour scheme to colour by transformation index. Use the preset panel to switch to Cyclosorus, Fishbone or Maple-leaf IFS systems.
Michael Barnsley developed IFS theory in 1988 while working on image compression. The fern fits in 36 bytes of numbers — versus kilobytes as an image. His company Iterated Systems briefly held the patent for fractal image compression, used in some CD-ROM encyclopaedias of the 1990s.