z squared
w = z²
Open in the app The dials and keys named below are the app's.
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.