# z squared

Complex · w = f(z, t)

`w = z²`

[Open in the app](https://www.wavelace.com/app#p=51) · [This page](https://www.wavelace.com/presets/z-squared)

### What it draws

`w = z²` squares the modulus and doubles the argument, and both halves of that are on show. Walk once round the origin in the plane below and the colour wheel turns twice. That double sweep of hue is exactly what a zero of order two looks like, and counting hue cycles is how the eye reads the order of any zero or pole. Since `|w|` goes as `|z|²`, the brightness rings come at `|z|` multiplied by `√2 ≈ 1.414` each time, half as often on the page as the hue.

### Two to one, and one branch point

The doubling also means the map is two-to-one: `z` and `−z` land on the same `w`. Everywhere but the origin the derivative `2z` is non-zero and the map is conformal, so small squares stay square and only turn and stretch. At the origin the derivative vanishes and right angles are opened into straight angles. The inverse, the square root, cannot be defined continuously all the way round, and that single point is the branch point a Riemann surface exists to tidy up.

### The height

Height follows `log(1 + |w|)`, which for this formula is a gentle bowl. `Span of z` is 2, so the plot reaches `log 5 ≈ 1.61` at the middle of an edge, where `|z| = 2`. In the corners, where `|z| = 2√2 ≈ 2.83`, it reaches `log 9 ≈ 2.20`. At the origin it is zero, and the picture is mostly in the colour.

### Try

- `Top` for the bare colour wheel, turning twice as it goes once round the origin.
- Type `z³`: three hue cycles, a zero of order three, and a steeper bowl.
- Type `1/z`: a pole of order one. The hue now runs backwards round the origin, and the height climbs into a spike.
- Turn `Iterations` up to 20 so the squaring is repeated. Everything outside the unit circle runs off to infinity and everything inside falls to zero, so the closed unit disc stands up as a round plateau.
- Lower `Span of z` to 0.5 to watch the brightness rings pile up toward the origin.

### Read more

- [Domain coloring](https://en.wikipedia.org/wiki/Domain_coloring)
- [Argument (complex analysis)](https://en.wikipedia.org/wiki/Argument_(complex_analysis))
- [Conformal map](https://en.wikipedia.org/wiki/Conformal_map)
- [Zeros and poles](https://en.wikipedia.org/wiki/Zeros_and_poles)
- [Riemann surface](https://en.wikipedia.org/wiki/Riemann_surface)