One iteration, zₖ₊₁ = zₖ² + c, produces entirely different objects according to what is held fixed. Fix the constant c and vary the starting point: those z whose successive images never escape to infinity form a filled Julia set - connected and continent-like for some c, a fine Cantor dust for others. Fix the origin and vary c: the values that keep the orbit bounded assemble the Mandelbrot set, whose main cardioid sprouts bulbs of higher period and whose boundary teems with infinitesimal copies of the whole. The same quadratic, confined to the real axis, is conjugate to the logistic map and therefore exhibits the infinite cascade of period-doubling bifurcations that precedes chaos. Fatou and Julia studied these iterations in 1918; computer pictures by Mandelbrot in 1980 made the sets iconic.