# Rational Map

Complex · w = f(z, t)

`w = (z² − 1) · (z − 2 − i)/(z² + 2 + 2i)`

[Open in the app](https://www.wavelace.com/app#p=53) · [This page](https://www.wavelace.com/presets/rational-map)

### Zeros and poles

`(z² − 1)(z − 2 − i)/(z² + 2 + 2i)` is a ratio of two polynomials, and everything on screen is set by where each of them vanishes. The numerator is already factored: it has simple zeros at `z = 1`, `z = −1` and `z = 2 + i`. At each of the three the sheet drops to the ground and the colour wheel turns once anticlockwise.

The denominator vanishes where `z² = −2 − 2i`. That number has modulus `2√2 ≈ 2.83` and argument `−3π/4`, so its two square roots have modulus `√(2√2) ≈ 1.68` and arguments `−3π/8` and `5π/8`. The simple poles stand at `0.644 − 1.554i` and `−0.644 + 1.554i`, two spires with the wheel running backwards around each. `Span of z` opens at 3, wide enough to hold all five points. Between them the sheet is gentle: at the origin `w = 0.75 − 0.25i`, a height of `0.8 · log(1.79) ≈ 0.47` at the opening `Height`.

### Counting on the sphere

Counting the zeros and poles is how a rational function is read. The numerator has degree 3 and the denominator degree 2, so far from the origin `w ≈ z`: at `z = 100` the value is `98.0`. That excess of one degree is a third pole, out at infinity, and it balances the three zeros. The rule is general: a rational map of degree `n` takes every value exactly `n` times, counted on the sphere and with multiplicity. That is why every colour of the wheel is found in three places.

### Try

- `Top`: the flat view, where the three anticlockwise wheels and the two backward ones are easiest to tell apart.
- Delete a factor: `(z² − 1)/(z² + 2 + 2i)` leaves two zeros against two poles, and the growth at the edge disappears with the pole at infinity.
- Turn the numerator over: `(z² + 2 + 2i)/((z² − 1)(z − 2 − i))` swaps every spire for a dimple and every dimple for a spire.
- Move a zero onto a pole with `(z² − 1)(z − 2 − i)/((z² + 2 + 2i)(z − 1))`: the dimple at `z = 1` cancels and only two zeros are left.
- Raise `Iterations` to 20 to feed the formula its own output over and over. Points whose orbit escapes drop to the outside tone, shaded by how soon they left, and the rest stand on a flat plateau, coloured by where the orbit ended up.

### Read more

- [Rational function](https://en.wikipedia.org/wiki/Rational_function)
- [Zeros and poles](https://en.wikipedia.org/wiki/Zeros_and_poles)
- [Domain coloring](https://en.wikipedia.org/wiki/Domain_coloring)
- [Riemann sphere](https://en.wikipedia.org/wiki/Riemann_sphere)