Recursion & Self-Similarity
Before meeting the famous fractals, learn the two ideas that make most of them possible: repeat a rule, and watch structure reappear across scales.
One rule, repeated
A fractal often begins with an almost embarrassingly small instruction. Do this; then do the same thing to the pieces you just made. That is recursion: a construction refers back to itself. The complexity is not stored in a long recipe; it emerges from repetition.
The Cantor set says “remove the middle third.” The Koch curve says “replace one segment by four smaller segments.” An L-system rewrites symbols. The Mandelbrot set repeats . Different machinery, same deep habit: iterate a simple rule and study what survives.
Exact and statistical self-similarity
FRACTAL MICROSCOPE
Zooming is not decoration. It is an experiment.
Move the magnification slider. The point is not merely to see more detail, but to ask what repeats, what changes, and what quantity scales predictably.
A mathematical Sierpiński triangle contains smaller copies that are exact replicas. A fern in nature is different: fronds resemble the whole plant, but not pixel-for-pixel. It is better described as approximately or statistically self-similar over a finite range of scales.
“Fractal” does not simply mean “an object that looks the same when zoomed.” Some important fractals are self-similar only statistically, and Julia/Mandelbrot boundaries exhibit richer forms of scale recurrence rather than simple copy-and-paste similarity.
The three questions to carry through the fair
For every exhibit, ask: What is the rule? What quantity changes under repetition? What remains invariant? Those questions turn a beautiful image into mathematics.
Next we need one more strange idea: if a fractal is more substantial than a line but less substantial than a plane, can its dimension lie between 1 and 2?