Newton's laboratory
z → z − a·f(z) ⁄ f′(z), for an f of your own
Newton's method on any function you like. Drag the roots of a polynomial about and watch their territories shift, add poles, try sines and exponentials with infinitely many roots, or type a formula.
Hénon's attractor
x → 1 − a·x² + y, y → b·x
Two numbers, bent and squeezed by one rule, over and over. Whatever the start, the points gather on the same curve; look closer and the curve is two curves, then four, without end.
Chaos game
Throw a die, jump half-way, mark the spot
Pick a corner of a triangle at random and jump half-way to it, thousands of times. Chance draws a perfect lattice of triangles within triangles. Change the rules and it draws a fern, or a coastline whose length has no answer.
Flatland
A world of two dimensions, seen in one
A plane whose inhabitants are triangles, squares and circles. From above you see them whole; they see each other only as a line, brighter and dimmer. Then a sphere passes through, and a circle appears from nowhere.