Breathing Julia
w = z² + 0.7885 · exp(i · t · 0.25)
Open in the app The dials and keys named below are the app's.
What it draws
w = z² + 0.7885 · exp(i · t · 0.25) is pushed through from every point of the square, with the added constant the same everywhere on the sheet. That makes this a filled Julia set and not a Mandelbrot set. Iterations sets how many steps a surviving orbit has to take. Span of z is 1.8, and nothing of a filled Julia set for this constant lies outside |z| = 2.
The constant runs round the circle of radius 0.7885 at a quarter radian per second. One lap takes 8π ≈ 25.1 seconds at Speed 1, and that lap is the breathing.
Connected or dust
The shape of a filled Julia set is decided by one thing: whether its constant lies in the Mandelbrot set. Inside, the set is connected, a single piece. Outside, it shatters into a Cantor dust with no interior at all. The circle of radius 0.7885 is chosen to cross that boundary again and again.
It spends roughly a seventh of each lap inside. The longest stretch is the two-cycle disc round −0.7885, about 1.2 seconds wide and centred on t = 4π ≈ 12.6, when the set closes into one connected body. Near t ≈ 6.4 to 7.5 the constant crosses the three-cycle bulb and the set grows the three-lobed look of Douady's rabbit. The rest of the lap the set is dust, and at sixty steps the finest dust reads as a haze rather than as separate specks.
Try
- Top for the flat view, straight down on the plane.
- Freeze the constant: type z² − 0.123 + 0.745i, Douady's rabbit, a connected set with three lobes at every joint.
- Shrink the circle, z² + 0.2 · exp(i · t · 0.25): the constant now stays inside the main cardioid for the whole lap and the set never breaks up.
- Change the 0.25 to 1 and the picture breathes four times as fast.
- Raise Iterations toward 300: the dust separates into specks and the plateau's edge sharpens.