E1: Initialize! n = 2. Choose play: the edge point of the quadratic array will shift along the unit circle. How often does its image in u revolve for one revolution in z?
(The u-plane has 2 Riemann sheets).
E2: Contract the array to a point and measure how points inside and outside of the unit circle are mapped.
r(u) = [r(z)] n , φ(u) = n φ(z)
E3: Pull the red point of the array to the unit circle and observe where it is mapped (to the unit circle with twice the original angle).
E3: Observe how straight lines and intersecting angles of the quadratic array are mapped: conformal mapping ⇔ intersecting angles are preserved
E4: Choose n = 0, observe where points are mapped while you shift the array: (all points of the z-plane are mapped to (1, 0))
E5: Choose the radius of the circular array as 1. Choose an arbitrary power. Where is the circle mapped? (on itself under rotation). In how many Riemann sheets?
E6: Choose a positive power! Where is the interior of the unit circle mapped, where the exterior? (On itself, unter rotation and expansion or contraction).
E7: Choose a negative power, as n = -1, u = 1/z. What happens now to the exterior and interior of the unit circle?
The origin is mapped to infinity, infinity to the origin, with infinity understood as a circular limit .
E8: Shift both arrays and study what causes the observed distortions.
E9: Choose a rational number for the power (as 1 < n < 2); Characterize the Riemann sheets as to their number and coverage.
E10: choose n < 1 (e.g.. u = z 2/3 as inverse to u = z 3/2).
E11: Initialize! Contract the quadratic array to a point and choose the radius of the circular array as 1. Choose a high power, as 20 or 50. You will see only a few points at the mapped unit circle Why, and how many?
Shift the center of the circle a bit. Now you will see all 100 points in n Riemann sheets.
Regular patterns may be generated by the superposition of points in different sheets. Experiment with the circle diameter, the circle center and the power value to get interesting patterns.