# Orbiting Poles

Complex · w = f(z, t)

`w = z³/(z² + 0.5 · exp(i · t))`

[Open in the app](https://www.wavelace.com/app#p=177) · [This page](https://www.wavelace.com/presets/orbiting-poles)

### What it draws

`w = z³ / (z² + 0.5 · exp(i · t))` is applied to every point of the plane 20 times, the preset's `Iterations`. The map has a triple zero at the origin and two poles where `z² = −0.5 · exp(i · t)`. The poles sit opposite each other at a distance `√0.5 ≈ 0.71` from the centre.

Near the origin the map is close to `z³ / 0.5`, so a point's distance from 0 is roughly cubed and doubled at every step. A point that starts close enough falls into 0 at a furious rate: the flat band through the middle is that basin.

Far out the map is nearly `z − 0.5 · exp(i · t) / z`, almost the identity. A distant point creeps by about `0.5 / |z|` per step, outward in some directions and inward in others, and many steps pass before it gets anywhere. The coloured wings are points still creeping when the steps run out.

### The fractal edge

Between the points that fall into 0 and the points that drift away lies the Julia set of the map, the boundary where the two fates meet. Every neighbourhood of a point on it holds points of both kinds, which is why the edge of the band is scalloped at every scale rather than smooth.

### Why it turns and never warps

Turning the plane by `t/2` turns the pole term by `t`, so the map at time `t` is the map at time 0 seen in a rotated frame. The whole picture turns rigidly, once every `4π ≈ 12.6` seconds at `Speed` 1, and its shape never changes. Only the colours move on, since the hue is the argument of `w` and that turns with the frame. The same argument with a scale shows that every nonzero constant in place of `0.5 · exp(i · t)` gives this one fractal, turned and resized.

### History

Pierre Fatou and Gaston Julia founded the study of iterated rational maps in memoirs of 1918 and 1919. The slow creep at a fixed point where the map is nearly the identity is described by the Leau–Fatou flower theorem, after Léopold Leau in 1897 and Fatou.

### Try

- Set `Iterations` to 0: the map itself, the triple zero at the centre and the two poles circling it at half the clock's pace.
- Replace `0.5 · exp(i · t)` with `0.5`: the picture stops turning and nothing else about it changes.
- Replace it with `2 · exp(i · t)`: the same fractal edge at twice the size, since `√(2 / 0.5) = 2`. The wings change, because each point now takes a different number of steps to leave.
- Raise `Iterations` to 120: more of the creeping points have time to leave, and the wings shrink.

### Read more

- [Julia set](https://en.wikipedia.org/wiki/Julia_set)
- [Rational function](https://en.wikipedia.org/wiki/Rational_function)
- [Classification of Fatou components](https://en.wikipedia.org/wiki/Classification_of_Fatou_components)
- [Pierre Fatou](https://en.wikipedia.org/wiki/Pierre_Fatou)