# Breathing Julia

Complex · w = f(z, t)

`w = z² + 0.7885 · exp(i · t · 0.25)`

[Open in the app](https://www.wavelace.com/app#p=48) · [This page](https://www.wavelace.com/presets/breathing-julia)

### 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.

### Read more

- [Julia set](https://en.wikipedia.org/wiki/Julia_set)
- [Filled Julia set](https://en.wikipedia.org/wiki/Filled_Julia_set)
- [Mandelbrot set](https://en.wikipedia.org/wiki/Mandelbrot_set)
- [Douady rabbit](https://en.wikipedia.org/wiki/Douady_rabbit)