Barnsley Fern — Chaos Game

Same trick as the Sierpinski triangle — pick a random rule, apply it, plot a dot — but here the four rules are weighted affine transforms that sculpt a lifelike fern.

Points: 0
Speed: 1/frame

Manual speed 1

How it works: same chaos-game idea as the Sierpinski triangle, but instead of 3 equally-likely "jump toward a corner" rules, the fern uses 4 weighted affine transforms — each a matrix multiply plus offset,

(x,y) → (ax+by+e,
  cx+dy+f)

. One rule (picked 85% of the time) draws a leaflet, shrinking and rotating it slightly, which is what builds the self-similar frond pattern; two more rules (7% each) generate the left- and right-curving leaflets branching off the stem; the last rule (just 1% of the time) draws the short base stem segment that anchors the whole plant. Every single dot only depends on the previous one and a coin flip, yet because the transforms were hand-tuned by Michael Barnsley to mimic how real ferns grow — each frond a shrunk, rotated copy of the whole — the same recognizable fern emerges no matter the random seed.