Newton Petals
w = z −((z^n − 1)/(n · z^(n−1)))
Open in the app The dials and keys named below are the app's.
What it draws
w = z − (z^n − 1)/(n · z^(n−1)) is one step of Newton's method for the equation z^n = 1: subtract the value of z^n − 1 divided by its derivative. The step is taken from every point of the square, and the colour of a point says which answer its run reached. The slider opens at 5, so the answers are the five fifth roots of one, spaced 72° apart on the unit circle, and the view breaks into five basins pinwheeling about the origin.
About a seventh of the square reaches no root at all. The step divides by n · z^(n−1), so a start close enough to the origin is flung out of the picture in a single move.
How many petals
The fixed points of the step are the numbers whose n-th power is 1, since those are where the correction is zero. At a whole n there are exactly n of them, evenly spaced round the unit circle, and the picture has n basins and n-fold symmetry. Nine different pictures sit on the slider between 2 and 10.
Off a whole number the count stops following the slider. A fractional power is read on the principal branch, which only reaches angles within half a turn of the positive real axis. The fixed points are the unit-circle points at 360° · k / n that fall inside it, and that count is always odd. There are three basins at 3.5 just as at 3, and five at 4.5. Only at the whole values do the count and the slider agree.
History
Arthur Cayley posed the question in 1879. Newton's method for a quadratic he could settle: the plane splits along a straight line, and each half runs to the root on its side. For the cube roots of one he could not say which start reaches which root, and wrote that the problem appeared to present considerable difficulty. It did. The answer needed the complex dynamics Fatou and Julia built forty years later. The picture here is why: every point of the boundary has points of every basin arbitrarily close to it, so no curve can separate them.
Try
- Drag n to 3 for Cayley's own case, the cube roots of one in three colours.
- Drag n to 2: the one tame member of the family. The plane splits along a straight line, and the only points that reach no root are the ones sitting on it.
- Drag n to 10: ten slender petals, and the share of the square that reaches no root grows to about a fifth.
- Drag n to 4.5 and count the basins. There are five, not four, because a half step keeps an odd number of fixed points.
- Drop Iterations to 4: most points have not converged yet, so the basins read as blurred bands instead of flat colour.