STUDY #15 · 2026 · IN OBSERVATION

Magnetic Pendulum Basins

A model-driven visual study of fractal basins of attraction.

MOVING IMAGE — THE ATLAS RESOLVES γ 0.15 · d 0.3 · one pendulum per pixel · lodestone

WHAT IS THIS

A steel pendulum swings above three magnets. Friction slowly drains its energy until it hangs over one of them — but which one is practically unpredictable: releases a hair's-width apart end on different magnets. This study colours every possible starting point by its final magnet, turning the plane into an atlas of fate whose three territories interlock along a fractal boundary (a "Wada" boundary: every point of it touches all three).

The model is the standard planar magnetic pendulum — friction, a restoring pull toward the centre, and three regularised magnetic wells. This study integrates one pendulum per pixel on the GPU (about a million at once) and runs in real time, so the map visibly resolves out of the undecided shimmer.

the atlas — the classic three-magnet fate map γ 0.15 · d 0.3 · ω₀² 0.5 · window 2.4 · lodestone

Motif magnetic pendulum / basins of attraction / final-state sensitivity / Wada boundaries

Method A small simulator was generated and modified with AI assistance, then ported to a real-time GPU (GLSL) renderer. The visual output was selected through parameter exploration.

Observation Friction is the dial of fate: heavy damping cuts the plane into three smooth territories, light damping shatters the boundary into fractal filaments (uncertainty exponent α ≈ 0.62, D_b ≈ 1.38). Raising the magnets births a fourth, magnet-less fate — rest at a silent equilibrium. The longest-hesitating points glow brightest: the shining part of the map is exactly where fate cannot be told.

Reference S. W. McDonald, C. Grebogi, E. Ott & J. A. Yorke, "Fractal basin boundaries," Physica D, vol.17, 125-153 (1985).

Tools Python / NumPy / three.js / React / GLSL / ffmpeg / AI coding assistant

Year 2026

This is not a scientific simulation result, but a visual interpretation of the phenomenon.

THE SAME FAMILY, A DIFFERENT QUESTION

Lorenz follows one orbit. This study asks all of them at once.

Study #05 — Lorenz Study #15 — Pendulum basins

The question where does one orbit go? — a single trajectory, followed forever where does every start end? — a million trajectories, asked once each

What is drawn the path itself, accumulated into orbit density only the verdict — each start coloured by its final magnet

The chaos never settles — sensitivity along one endless path settles every time — the sensitivity lives in which end you get

PARAMETERS EXPLORED

param meaning effect on the image

γ friction — the dissipation rate the fractality dial: heavy damping gives three smooth lobes, light damping shatters the boundary into filaments that invade the whole map

d the bob's height above the magnet plane the sharpness of the wells — raised high, a third kind of rest appears away from every magnet, woven in as pewter ribbons (the null fate)

ω₀² the restoring pull toward centre tightens the map inward and raises rings of equal settling time around the rim

N the number of magnets the map's symmetry — 3 the classic Wada atlas · 4 a compass chart · 5 a rosette

window the frame of the map zoom — the boundary holds structure at every magnification

settle time how long each start hesitated (measured) the brightness — undecided points glow, so the fractal boundary is the shining part

Each image below records its exact parameter set.

SELECTED STILLS — 6

the atlas — three fates interlocked γ 0.15 · d 0.3 · window 2.4 · lodestone

the null fate — pewter ribbons where the bob stops over no magnet γ 0.18 · d 0.5 · lodestone

filigree — low friction γ 0.08 · window 1.7 · weaver

trinity — heavy friction γ 0.45 · window 2.4 · lodestone

compass — four magnets N 4 · γ 0.2 · window 2.0 · chart

shore — every coastline hides another atlas γ 0.15 · window 0.75 · centre (0.9, 0.9)

PROCESS — PARAMETER SWEEPS

Friction γ against magnet height d — the phase map of fate. Down the friction axis the boundary shatters from smooth pie to all-invading filigree; along the height axis the fourth, magnet-less fate is born.

the γ × d morphology map rows = γ (friction) · columns = d (magnet height)

SIGNATURE — WHERE FATE CANNOT BE TOLD, IT SHINES

The boundary is measured, not just admired.

Brightness in these maps is hesitation: each point glows by how long its pendulum wavered before settling. The fractal boundary — where releases a hair's-width apart end on different magnets — is therefore the part that shines. The map's darkness is certainty; its light is doubt.

And the doubt is quantified: the uncertainty exponent of McDonald-Grebogi-Ott-Yorke, measured inside this model as α ≈ 0.62, gives the boundary dimension D_b ≈ 1.38 at the hero's friction — final-state sensitivity made visible, then measured.

how to read the atlas — colour is fate, brightness is hesitation α ≈ 0.62 · D_b ≈ 1.38 · isochrones = swings needed

COLOUR = IRON OXIDES

The three basins are painted with the three iron-oxide pigments — hematite red (Fe₂O₃), goethite ochre (FeOOH), magnetite blue-black (Fe₃O₄). This is the magnet's own mineral chemistry: magnetite is lodestone, the first magnet humanity ever knew — and these same oxides were the first pigments humanity ever painted with. A map of magnetism, painted in magnetite's family colours.

Brightness maps settle time — hesitation glows — and the pewter regions are a real fourth fate: equilibria where the bob stops away from every magnet.

the fourth fate in pewter — rest, over no magnet at all d 0.5 · null-fate ribbons + centre pupil · lodestone

Pigment hues are artistic approximations of the minerals, not measurements.

REFERENCES

S. W. McDonald, C. Grebogi, E. Ott & J. A. Yorke, "Fractal basin boundaries," Physica D, vol.17, 125-153 (1985).
A. E. Motter, M. Gruiz, G. Károlyi & T. Tél, "Doubly Transient Chaos: Generic Form of Chaos in Autonomous Dissipative Systems," Physical Review Letters, vol.111, 194101 (2013).