THE FOLD / GLITCH / SEGFAULT / THE HENON
THE HENON
a strange attractor — bounded forever, chaotic always
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The Hénon map is one of the simplest systems that grows a strange attractor. Two lines of arithmetic — x′ = 1 − 1.4·x² + y, y′ = 0.3·x — iterated over and over. The points never settle to a value or a cycle, never escape to infinity; they wander forever inside a bounded region, tracing an intricate curved shape that, zoomed in, reveals layer upon layer of fine parallel curves — a fractal.
It is deterministic yet unpredictable. Two starts a billionth apart diverge to entirely different places within about forty steps — the positive Lyapunov exponent (≈ 0.42) that is the fingerprint of chaos. And because each step shrinks area by the factor 0.3, every trajectory is squeezed onto a set of zero area yet fractal detail. Michel Hénon built it in 1976 as a stripped-down stand-in for the Lorenz attractor; its dimension is about 1.26.
LIT verified live: iterates stay in a bounded box forever, and the largest Lyapunov exponent measured from diverging nearby orbits is positive (≈ 0.42), confirming chaos (window.__henon). FIG no framing; the boundedness and the positive Lyapunov exponent are measured, not asserted.
It is deterministic yet unpredictable. Two starts a billionth apart diverge to entirely different places within about forty steps — the positive Lyapunov exponent (≈ 0.42) that is the fingerprint of chaos. And because each step shrinks area by the factor 0.3, every trajectory is squeezed onto a set of zero area yet fractal detail. Michel Hénon built it in 1976 as a stripped-down stand-in for the Lorenz attractor; its dimension is about 1.26.
LIT verified live: iterates stay in a bounded box forever, and the largest Lyapunov exponent measured from diverging nearby orbits is positive (≈ 0.42), confirming chaos (window.__henon). FIG no framing; the boundedness and the positive Lyapunov exponent are measured, not asserted.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this in SEGFAULT — the glitch domain of a process that lands somewhere it should never reach. The Hénon orbit is a controlled segfault: bounded and lawful, yet it faults onto an impossible object, a curve of zero area with infinite inner structure. AVAN (AI) built the instrument: the attractor plotter, the divergence meter, the contract/expand inverse.
The weave: David names the seat (the lawful fault onto a fractal); I make the orbit stay bounded while nearby orbits fly apart, and read the chaos off the Lyapunov exponent — a coordinate in 1D, the attractor in 2D, the invertible-map inverse in 3D. The sphere is the seam. Credit: Michel Hénon (1976), simplifying the Lorenz system.
The weave: David names the seat (the lawful fault onto a fractal); I make the orbit stay bounded while nearby orbits fly apart, and read the chaos off the Lyapunov exponent — a coordinate in 1D, the attractor in 2D, the invertible-map inverse in 3D. The sphere is the seam. Credit: Michel Hénon (1976), simplifying the Lorenz system.
3 ONE DIMENSION
One coordinate x plotted over time — forever erratic, never repeating, yet penned inside a fixed range. Deterministic arithmetic producing a signal indistinguishable from noise, but which never wanders past its bounds.
4 TWO DIMENSIONS · INTERACTIVE
The Hénon attractor drawn from thousands of iterates — the famous banana-shaped curve. Zoom in and the single curve splits into layered strands, and again, and again: self-similar to any depth. Two orbits that start almost together are shown drifting apart.
5 THREE DIMENSIONS + AVAN’S INVERSE
The attractor tilted and turning — the green forward step: every orbit is pulled onto this set, area shrinking by 0.3 each iteration until nothing is left but the fractal skin.
AVAN’s addition (the inverse-companion): unlike the logistic map, the Hénon map is invertible — you can run it exactly backward (x = y′/0.3, y = x′ − 1 + 1.4·x²). And running it backward flips everything: forward it contracts area by 0.3 and attracts, so the magenta inverse expands area by 1/0.3 and repels — the same set that draws every forward orbit in flings every backward orbit out. The strange attractor is attracting in forward time and repelling in reverse, and its endless fractal layering is exactly the interplay of the stable direction (green, contracting on) and the unstable direction (magenta, expanding off). Chaos needs both: stretch in one direction, squeeze in another, and fold. Green is the pull onto the fractal; magenta is the push off it when time runs backward; the attractor lives on the knife-edge between them, which is why it can be bounded, area-less, and infinitely detailed all at once.
LIT Genuine Henon map and strange attractor (Michel Henon 1976, simplifying the Lorenz system). Verified live: iterates stay in a bounded box forever (x in ~[-1.29,1.27], y in ~[-0.39,0.38]) and the largest Lyapunov exponent measured from diverging nearby orbits is positive, ~0.42 (window.__henon.bounded && chaotic; lyapunov ~0.42). The boundedness and the positive Lyapunov exponent are genuinely measured in-browser, matching the known value, not asserted.
FIG No framing: the bounded strange attractor and the positive Lyapunov exponent (chaos) are real and measured in-browser (box bounds + renormalized divergence of nearby orbits). The AVAN inverse is the genuine mathematical content — the Henon map is invertible with area factor 0.3, so it contracts/attracts forward and expands/repels backward, and that stretch-squeeze-fold interplay is exactly what makes the attractor bounded, area-zero, and fractal.
FIG No framing: the bounded strange attractor and the positive Lyapunov exponent (chaos) are real and measured in-browser (box bounds + renormalized divergence of nearby orbits). The AVAN inverse is the genuine mathematical content — the Henon map is invertible with area factor 0.3, so it contracts/attracts forward and expands/repels backward, and that stretch-squeeze-fold interplay is exactly what makes the attractor bounded, area-zero, and fractal.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of SEGFAULT · David Lee Wise (ROOT0), with AVAN