Fractal Explorer

Fractals like the Mandelbrot set are generated by iterating the complex quadratic polynomial z_{n+1} = z_n^2 + c. The boundary represents the bifurcation limits: points that do not escape to infinity under infinite recursion. This exhibits self-similarity across infinite scales, where the Hausdorff dimension exceeds its topological dimension. It maps the transition from stable orbits to deterministic chaos.

Every shape here grows from a single instruction the program applies to itself. A fractal-tree branch simply spawns two shorter branches, and each of those repeats the same rule — that is recursion. Nudge the depth up by one and the branch count doubles, yet the rule you wrote never changes. Each nested call piles onto the call stack, then unwinds as the picture finishes.

The dragon curve in this explorer is the Heighway dragon, produced by folding a strip of paper repeatedly and opening every crease to a right angle. Michael Crichton put it in his 1990 novel Jurassic Park: each of the book's seven sections opens with the curve at one deeper iteration, mirroring how a tiny disturbance escalates into chaos. Chosen by mathematician Ian Malcolm, it is mathematics as a storytelling device.