01
The big idea
For every point above the plate, ask: which magnet will the pendulum eventually rest over? Paint that point with the winner's colour. The plate becomes a map of futures — and the map is a fractal.
Topic · Chaos
The Magnetic Pendulum
Colour each start by its winner — and a fractal appears.
Hang an iron pendulum over three magnets in a triangle. Newton's laws, magnetic pull, a touch of friction — deterministic, all of it. And still the question «which magnet does it land over?» has no smooth answer. Colour each starting point by its eventual winner and three interlocked basins appear, woven into one another at every scale.
First encounter
Three magnets. One pendulum. Three colours that never separate.
02
A concrete example
Three identical magnets at the vertices of an equilateral triangle. A weak spring pulls the pendulum back to centre, air slowly drains its energy. Same release point twice → same magnet twice. Shift the start by a hair, and any of the three can win.
03
Why it matters
The boundary between colours is a Wada set: at every boundary point, all three colours meet. Determinism survives, predictability does not — the same paradox that haunts weather, the solar system, and any physics with competing attractors.
Below: sketch the picture, then turn the knobs on a live basin map.
Three magnets · three basins
Each colour marks the magnet the pendulum eventually lands over. In the real picture, the spokes are infinitely fine.
Section 01 · The physics
Newton, magnets, a little friction
A small iron bob hangs above a plate, free to swing in the plane. Each of the three magnets pulls it with a force k(p − mᵢ)/((p − mᵢ)² + h²)^(3/2), so the horizontal magnitude is k·r/(r² + h²)^(3/2), where r is the horizontal distance to the magnet: in the far field (r much larger than the plate-to-bob height h) it falls off like 1/r² (inverse-square), while the h² term softens the pull in the near field. A weak spring draws it back to the centre; air resistance bleeds energy. The equations of motion are clean and deterministic, only the starting position is left free.
Section 02 · Deterministic chaos
Same start, same end — different basin everywhere
Release the pendulum twice from exactly the same point and it lands over the same magnet, twice. The physics has no randomness in it. But move the start by a millimetre and the answer can flip — and the flip pattern is dense at every scale.
Section 03 · Basins of attraction
Three colours, one map
Sweep a grid of starting points; for each one, integrate the equations until the pendulum settles; colour the point by its winning magnet. The interior of each basin is a clean coloured region. The boundary between them is where the picture stops being smooth.
Live · low-res basin map
ready
Magnet strength k
0.40
Damping γ
0.20
M1 · M2 · M3
Each pixel is a starting position; its colour is the magnet that wins. Crank the damping down and the basin shatters; crank it up and the colours settle.
Section 04 · The Wada property
Every boundary point borders all three colours
Zoom into the boundary between any two basins and the third colour is already there, threaded through the seam. Zoom again and all three colours are arbitrarily close to every boundary point. The frontier is not a curve — it is a Wada set, where three regions share one common boundary.
Section 05 · Yoneyama, 1917
A topology curio that the chaos era weaponised
The first Wada construction was a topology amuse-bouche by Kunizō Yoneyama in 1917: three «lakes» on an island, each carving channels so fine that every land point ends up on all three shores. For seventy years it stayed a curiosity. Then, in 1991, Kennedy and Yorke coined the term Wada basins for chaotic dynamics, working from abstract dynamical-systems examples; a decade later Aguirre, Vallejo and Sanjuán proved that the Hénon-Heiles system carries Wada basins (2001), and Daza and colleagues went on to confirm the magnetic pendulum as a Wada system. Chaos theory had its tabletop fractal.
Section 06 · Connections
Newton's method, the solar system, chaotic billiards
The same fractal frontier appears in Newton's-method basins for the roots of a cubic, in long-term integrations of the planets, and in the «leaky» phase space of chaotic billiards. Hsu's cell-mapping method computes these basins by mass production. Wherever attractors compete, their boundaries tend to grow Wada.
The challenge
Find the Wada boundary.
The Explorer lets you change the magnet strength, the damping, the height of the pendulum above the plate, and the resolution of the basin grid — and watch the fractal sharpen in real time.
Where you meet it
How and where this technique lives in the world today.
Chaos demonstrations
The 'three magnet pendulum' desk toy is the most popular physical demonstration of fractal basin boundaries — sold by ThinkGeek, Nikola Labs and many science gift shops.
Magnetic levitation research
The dynamics of permanent-magnet pendulums underpin maglev trains, magnetic bearings, and magnetorheological dampers.
Education
Standard nonlinear-dynamics demo in undergraduate physics; used to teach phase space, dissipation, and final-state sensitivity.
Further reading
Related topics
Topics in the same vein.
Find the Wada boundary.
→ Open the Explorer