Simple Pole
w = 1/z
Open in the app The dials and keys named below are the app's.
What it draws
1/z is the smallest example of a pole: one point where the function has no value and near which it grows without bound. Because |w| = 1/|z|, the modulus depends on the distance from the origin alone, so every brightness ring is a circle. |w| = 1 on the unit circle, |w| = 2 at |z| = 0.5, and |w| = 0.5 at |z| = 2, the edge of the square set by Span of z. The height follows the same circles, a bowl rising to a spire at the centre.
Why it is simple
The hue depends on the argument alone: arg w = −arg z, so walking once anticlockwise around the origin runs the colour wheel once backwards. A zero does the opposite, and that reversal is what tells the two apart on sight. The single turn is what makes this pole simple, of order one. A pole of order two winds the wheel twice, and one of order n winds it n times.
The spire never ends. The sheet climbs without limit as the origin is approached, and the point itself has no value to draw. On the Riemann sphere that missing point is filled in by the north pole, and 1/z becomes a well-behaved map of the sphere to itself, swapping 0 and infinity.
Try
- Top: the plane flat on, with the whole colour wheel wrapped once around the origin.
- z: a simple zero in place of the pole. The wheel now turns the same way you walk, and the sheet rises outward instead of inward.
- 1/z²: the wheel goes round twice for one turn of z, and the rings close in as |w| = 1/|z|².
- 1/z + 1/(z − 1): two simple poles, at 0 and 1, with the sheet sagging to a saddle between them.
- Raise Iterations to 20 so the formula is applied over and over. A small disc round the origin escapes on the first step and takes the outside tone, while every other point alternates between z and 1/z for ever and stands on the plateau.