Why does this pattern occur on this plot?

Question

I have attached below a real imaginary plot. These are all of the nth roots of 5+5i, from n = 2 to n=50. This general shape occured no matter the numbers used. Why is it a circle, and why does it have these parts flying off the shape. The black dots are the 2nd root, blue dots are the even roots, and red dots are odd roots.

Answer 1

All the roots of a complex number lie on a circle. You can write your number in polar form $5+5i=5\sqrt 2 e^{i(\frac { \pi}4+2k\pi)}$, where $k$ is any integer. The $n^{th}$ roots of this are $(5\sqrt 2)^{\frac 1n}e^{i(\frac { \pi}{4n}+2\pi\frac kn)}$. They lie on a circle of radius $(5\sqrt 2)^{\frac 1n}$ and are equally spaced around it. The same behavior will occur for any complex number you take the root of.

The radius of the circle decreases with $n$.

Answer 2

Your number has one first root (the point with largest modulus¹), two square roots (two points of next largest modulus), three cube roots, four fourth roots, and so on. The modulus decreases with $n$, going to $1$. By the time you reach the 50th roots, they clearly form a circle.

¹ The first root is not in your picture, since you started with $n=2$.