Orbiting Poles
w = z³/(z² + 0.5 · exp(i · t))
Open in the app The dials and keys named below are the app's.
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.