THE JULIA SET — a dart

THE JULIA SET the boundary between falling in and flying out — iterate z → z²+c

Fix a constant c and iterate z → z²+c from each starting point. Some starts spiral into a fixed point or cycle (bounded forever); others fly off to infinity. The Julia set is the boundary between the two fates — the infinitely intricate fractal coastline separating capture from escape, and every point on it is a place where the tiniest nudge decides which way you fall. For c=0 the map has fixed points at 0 (attracting) and 1 (repelling); for c=−1 the attractor is a 2-cycle 0 ↔ −1. The whole zoo of shapes comes from one quadratic and one constant.

THE TECHNIQUE z → z²+c; the boundary between bounded (fixed point/cycle) and escape

The demo iterates z→z²+c on the real line: c=0 pulls 0.5 into the fixed point 0; c=−1 settles into a 2-cycle: live demo

HISTORY & CREDIT Gaston Julia & Pierre Fatou, 1918

“A quadratic is simple.” — iterate z²+c and the boundary between capture and escape is a fractal of infinite detail. One line, an endless coastline. cited

1918 · Gaston Julia & Pierre Fatou — the iteration theory of rational maps (decades before it could be drawn).
the set · the boundary of the basin of the attracting fixed point / cycle.
connected? · iff c is in the Mandelbrot set (dart 372) — the two are locked together.

Fall into the fixed point or fly to infinity — and the border between the fates is a coastline without end. Julia & Fatou 1918

RECOMMEND FOR I-13 capture vs cycle, on the compiler

On the canonical compiler, c=0 pulls 0.5 into the attracting fixed point 0; c=−1 settles into the 2-cycle 0 ↔ −1:

$ i13 run f_juliaset.i13 # z -> z^2 + c RUN OK · 3118 step(s) · peak stack 5 · call depth 62 attracted_to_0 = 0 -- c=0: 0.5 spirals into the fixed point 0 cyc_even = 0 cyc_odd = -1 -- c=-1: the attractor is the 2-cycle 0 <-> -1

Recommend: the Julia set is the fixed point's basin boundary — the fractal edge between falling in and flying out — and i13 shows both fates (capture at c=0, a 2-cycle at c=−1). Not a keeper (which fate a point meets is witnessed by iteration, and the boundary is a theorem of the map). The dart that shows a fixed point has a shape: the coastline of everything drawn to it.