Wada Basins
Three regions sharing the exact same boundary
The Paradox
Imagine three lakes. In ordinary geometry, if Lake A borders Lake B along some shoreline, then Lake C must be somewhere else—it can't also share that same stretch of shore.
But in 1917, mathematician Takeo Wada discovered something impossible: you can construct three disjoint regions where every single boundary point touches all three regions. The three lakes share the exact same boundary!
This isn't just a mathematical curiosity—it appears naturally in chaos theory. The Newton fractal for z³ - 1 demonstrates this beautifully: three basins of attraction with a shared fractal boundary.
Newton Fractal: z³ - 1
Each color shows which of three roots that starting point converges to. Click to zoom in!
The Wada Property in Action
Zoom into any boundary region. No matter how close you look, you'll find ALL THREE colors interleaved. There is no boundary between just two colors—the third always squeezes in between. This is the essence of the Wada property: three regions, one shared fractal boundary.
How It Works
The Newton-Raphson method finds roots of equations by iteration:
For f(z) = z³ - 1, this becomes:
The polynomial z³ - 1 has three roots equally spaced around the unit circle:
- z₁ = 1 (red)
- z₂ = e^(2πi/3) ≈ -0.5 + 0.866i (green)
- z₃ = e^(4πi/3) ≈ -0.5 - 0.866i (blue)
The Surprising Result
Arthur Cayley posed this problem in 1879: which starting points converge to which root? He expected simple 120° "pie slices" around each root. The actual answer—revealed by computers a century later—is infinitely complex.
The Julia set forms the boundary between all three basins. On this boundary, arbitrarily small perturbations can send you to any of the three roots. That's why the boundary is shared by all three regions!
Why the Third Color Always Intrudes
Think about it geometrically:
- If red and green regions meet along a border...
- Then somewhere nearby, there must be a point that goes to blue
- But blue must also border red and green!
- So blue must squeeze in between red and green
- This creates new red-blue and green-blue borders...
- Which themselves need the third color between them...
- This continues infinitely!
The result is a fractal boundary of infinite complexity. The boundary has zero area (it's a 1-dimensional curve in a sense) but infinite length—just like the coastline paradox!
The Original Lakes of Wada
Takeo Wada's 1917 construction was different but achieved the same property:
- Day 1: Dig Lake 1, extending tendrils within distance a₁ of all land
- Day 2: Dig Lake 2, extending within distance a₂ of all remaining land
- Day 3: Dig Lake 3, extending within distance a₃ of all remaining land
- Day 4: Extend Lake 1 within distance a₄...
- Continue cycling through lakes, with distances a₁, a₂, a₃, ... → 0
After infinitely many days, the remaining "dry land" is the common boundary of all three lakes! This boundary is an indecomposable continuum—a bizarre topological object that cannot be split into two proper subcontinua.
Wada Basins in Nature
The Wada property isn't just a mathematical curiosity—it appears in real physical systems:
- Forced pendulums: The basins of attraction for different periodic states can have Wada boundaries
- Light reflection: Four reflective spheres create Wada-like patterns
- Magnetic pendulums: Multiple magnets create chaotic Wada basins
- Chaotic scattering: Particle trajectories in certain potentials
Anywhere you have three or more attractors in a chaotic system, Wada basins can emerge!