whatOne line of arithmetic -- x becomes r x (1 - x) -- swept across every value of r and left to show its whole life at once. Each column iterates past its transient and deposits every visited x as a tick of ink: the orbit diagram, built live in front of you.

whyThe claim: order gives way to chaos by period-doubling -- one branch splits to two, two to four, faster and faster, and the ratio of successive splittings closes in on Feigenbaum's universal constant, measured here by bisection on the map itself rather than quoted. Inside the chaos, windows of order: find the wide one near r = 3.83, where the mess collapses to a beat of three. Drag a box to zoom -- the window holds the entire cascade again in miniature, and so do its windows, all the way down.

howSeeded starting points per column, several deepening passes; the HUD reports the measured period and Lyapunov exponent at the cursor r, positive for chaos, negative in the windows. A cobweb inset plays the chosen r stroke by stroke. Double-tap resets the zoom; the print recomputes the diagram fresh for the current view.