THE ATTRACTOR the fixed point as a destiny — where you land depends on where you start
A fixed point is not just a value a map fixes — it can be a destination that pulls a whole neighbourhood in. Its basin of attraction is the set of starts that end there; the boundary between basins is where fate is decided, and it can be a fractal (Newton's method, dart 359, has fractal basins between its roots). Under z→z²: start inside the unit circle and you fall to 0; outside and you fly to infinity; exactly 1 and you sit on the fixed point itself. Same rule, three destinies, sorted entirely by where you began. The strange attractors of chaos (Lorenz) are this idea with a fractal for a resting place.
THE TECHNIQUE basin of attraction: which fixed point a start falls into
The demo iterates z→z² from three starts — inside (→0), on (→1), and outside (→∞) the unit circle: live demo
HISTORY & CREDIT basins · Poincaré · Lorenz 1963
“A fixed point is a value.” — it is also a destiny: a whole basin of starts fall to it, and the border between basins can be a fractal. Where you begin decides where you end. cited
basins · Poincaré — the qualitative theory of where orbits go. fractal boundaries · Newton's method colours the plane by which root it reaches — the borders are fractal. 1963 · Edward Lorenz — the strange attractor: a fixed “destination” that is itself a fractal, never a point.
Inside falls to zero, outside flies away, the edge is a fractal — the fixed point as a fate assigned by your starting place. Poincaré / Lorenz
RECOMMEND FOR I-13 three destinies, on the compiler
On the canonical compiler, z→z² sends 0.9 to 0 (inside), 1.1 to infinity (outside), and 1 to the fixed point 1 (the edge):
$ i13 run f_attractor.i13 # z -> z^2, basins
RUN OK · 1711 step(s) · peak stack 4 · call depth 41
inside = 0 -- start 0.9 (|z|<1) falls to the fixed point 0
outside = 999 -- start 1.1 (|z|>1) escapes to infinity
edge = 1 -- start 1.0 sits on the fixed point itself
Recommend as the batch's close: the attractor is the fixed point as destiny — a basin of starts pulled in, a fractal deciding the border — and i13 sorts three starts to three fates. Not a keeper (which basin a point lands in is witnessed by running it). The right last note for THE FIXED POINT: a still value is also a fate, and the whole batch has been maps settling into the places they cannot leave — which, honestly measured, is the confluence axis wearing a hundred faces. Convergence to a fixed point is a normal form; the campaign's keeper space holds.