Yea, that's right. The central issue is in the expression .

So you're breaking up the circle into sectors of equal size starting at the point with a root at each terminal point of each sector so of course they're going to be equally spaced around a circle of radius and therefore at the points of a regular polygon.

Here's an idea. Write a Mathematica application which illustrates the roots of as red points in the complex plane for a variable sized z for n running from 2 to 10. Sort of like this:

Code:

Manipulate[
pts = N[w /. Solve[r*Exp[I*tval] == w^n,
w]]; points =
(Point[{Re[#1], Im[#1]}] & ) /@ pts;
Show[Graphics[{Red, PointSize[0.02],
points}], PlotRange ->
{{-3, 3}, {-3, 3}}, Axes -> True],
{r, 1, 5}, {tval, 0, Pi/2, 0.1},
{n, 2, 10, 1}]