Julia Dials
w = z² +((a + b · i)/5)
Open in the app The dials and keys named below are the app's.
What it draws
w = z² + (a + b · i)/5 is applied over and over from every point of the square, with the added constant the same everywhere. That makes this a filled Julia set rather than a Mandelbrot set. The two sliders are the real and imaginary parts of that constant, and the division by five is what makes them usable. One step of a or b moves the constant by 0.02, fine enough to walk across the small bulbs one at a time.
It opens at a 1.5 and b 2.5, so the constant is 0.3 + 0.5i. There the orbit of zero settles into a four-step cycle, and the set grows four lobes at every joint. It is one connected body, and it fills about a tenth of the square on screen.
Connected or dust
One fact decides the whole shape. The filled Julia set is connected, a single body, exactly when its constant lies in the Mandelbrot set, and otherwise it is a scattered Cantor dust with no interior. So dragging the two sliders is dragging a point around the Mandelbrot set, with the Julia set for wherever it lands drawn in front of you.
Inside, the orbit of zero settles onto a cycle, and the length of that cycle is a whole number the sliders choose. At the opening values it is four, and four is also the number of lobes meeting at each joint. Elsewhere on the two sliders the cycle runs to one step, two, three, five and further, each with its own joint count.
History
Pierre Fatou and Gaston Julia built the theory of iterating these maps in 1917 and 1918, with no way to draw one. The connected-or-dust dichotomy is theirs. The parameter picture it points at, the Mandelbrot set, was not seen until the late 1970s. For sixty years the theorem was known and the map of where it applies was not.
Try
- Set a to −0.5 and b to 3.5: the constant is −0.1 + 0.7i, the cycle is three steps long, and three lobes meet at each joint. That is Douady's rabbit.
- Drag a to −5 with b at 0: the constant is −1, the cycle is two steps long, and the set is two big lobes joined at a point.
- Drag b to 5 with a at 0: the constant is i. The set is still connected but has no interior at all, just a dendrite of threads.
- Drag a to −2 with b at 1: the constant is −0.4 + 0.2i, the cycle is a single step, and the set is one plump blob with no joints.
- Hold a at 1.5 and push b from 2.5 to 3.1: the constant leaves the Mandelbrot set at 2.9, and what was a solid body is scattered dust by the end.