Iterate one line of arithmetic, x → x² − r, and ask where it settles. For small r it lands on a single value; raise r and that value splits in two, then four, then eight, in a period-doubling cascade that piles up into chaos. Drag a box to zoom: the tree is a fractal, the same forks all the way down. Ported from my undergrad computational-physics coursework.
Feigenbaum's constant from the splitting points
Zoom into any fork and you find the whole tree again: branchings within branchings, self-similar without end.
Why it repeatsDeep in the chaotic smear, sharp white gaps open up. The system suddenly settles into a clean period-3 or -5 cycle, then shatters again.
The period-3 windowEach doubling is squeezed into a window ~4.7× narrower than the one before, so infinitely many forks fit before r ≈ 1.401.
The accumulation point