Rational Map
w = (z² − 1) · (z − 2 − i)/(z² + 2 + 2i)
Open in the app The dials and keys named below are the app's.
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.