Colour every point of the plane by which root Newton's method sends it to — and the boundaries are impossible.
Which root does it find?
Newton’s method for finding a root of f(z) = zⁿ − 1:
z ← z − f(z)/f′(z) = ((n−1)·zⁿ + 1) / (n·zⁿ⁻¹)
The equation has n roots, evenly spaced on the unit circle. Start from any point of
the plane, iterate, and you will end up at one of them. Colour the starting point by
which one, and the picture is this.
The question the picture answers is which basin does each point belong to — and the
answer is that the basins are not simple regions.
Three steps, and almost nothing has committed. The tolerance is 1e−6, and reaching
six correct digits from a cold start takes a few iterations however fast the method is.
Five steps, and the basins arrive all at once. Newton’s method converges
quadratically — the number of correct digits roughly doubles each step — so a point
goes from nowhere near a root to within 1e−6 of it in the space of one or two
iterations. There is no gradual approach to watch.
The boundary is where all three meet
Eight steps. Look at the boundary between two basins and you find bulbs of the third
one embedded in it; look at the boundary of those and you find bulbs of the first.
That is not decoration. The Julia set of a Newton map has the property that every
boundary point is on the boundary of all n basins simultaneously. There is no point
anywhere in the picture where exactly two colours meet along a curve — wherever two
meet, the third is present too, at every scale.
That is a strange topological object to have in a picture, and it is why the boundary
cannot be drawn as a line: at any resolution, the pixels along it contain all three
colours.
Sixteen steps.
The full budget of sixty-four. The undecided set — points that have not reached any
root — has measure zero, so it occupies no area and is nonetheless everywhere along
the boundaries.
What raising the budget actually does
Because convergence is quadratic, the budget makes almost no difference to the interior
of a basin: a point a reasonable distance from a boundary is decided within five or six
steps and would be decided identically with a budget of a thousand.
What the budget buys is entirely the boundary. Points there sit between attractors and
take arbitrarily long to commit, so each extra iteration decides a thinner shell of
them — which is why the progression above changes almost nothing except the fineness of
the filigree.
Five degrees
| Rendering | Polynomial | Roots | Symmetry |
|---|
| Cubic | z³ − 1 | 3 | 3-fold |
| Quartic | z⁴ − 1 | 4 | 4-fold |
| Quintic | z⁵ − 1 | 5 | 5-fold |
| Sextic | z⁶ − 1 | 6 | 6-fold |
| Septic | z⁷ − 1 | 7 | 7-fold |
The degree sets the rotational symmetry of the whole picture, because the roots are the
nth roots of unity and the map commutes with rotating by 2π/n. Raising the degree
does not make the fractal more complicated in kind — the every-boundary-touches-every-
basin property holds for all of them — only more crowded.
Determinism
Nothing is seeded and nothing is random. Each pixel runs an independent fixed-point
iteration in double, tested against a tolerance of 1e−6, and drawn as a solid-filled
rectangle in scanline order.
