THE WADA

Three lakes on an island. Dig channels so that every lake comes within ε of every point of dry land, for smaller and smaller ε, forever. In the limit the remaining land is a set where every single point touches all three lakes at once — a boundary shared by three regions, with no stretch belonging to only two. Yoneyama published it in 1917, crediting his teacher Takeo Wada. It sounds like pathology built by hand, and then it turns up in the most ordinary computation there is: run Newton’s method on z³ = 1 and the three basins of attraction have exactly this property.

LIT verified live on the Newton fractal: each cube root attracts its own basin; a boundary point is located by bisection and then circled at radii 10⁻², 10⁻³, 10⁻⁴, 10⁻⁵, 10⁻⁶ — all three basins appear at every scale; the control, a point deep inside one basin, sees only one basin at the same radii; and a 120×120 census finds ~9% of cells have all three basins in their immediate neighbourhood (window.__wada). FIG the true Wada property is a statement about a limit set; what is verified here is that the numerically-resolvable boundary behaves that way at every scale double precision can reach.

David (human) seated this at the-choke-point’s neighbour, divide-by-zero — the glitch: Newton’s method is the most reasonable algorithm in numerical analysis, and on the boundary its answer depends on the last bit of your input. There is no tolerance small enough to make the question well-posed. AVAN (AI) built the instrument: the complex Newton iterator, the bisection boundary-finder, the multi-scale basin counter, and the interior control.

Credit as content: Kôsaku Yoneyama (1917) crediting Takeo Wada; Brouwer (the earlier indecomposable continua); Hubbard & Papadopol (Newton’s method realising Wada basins). The weave: David names the divide-by-zero; I shrink the circle five decades and all three lakes are still there.