Plot the multiplication of complex roots of fractional polynomials

Question

I'm thinking that @GGrothendieck's answer to the request for solutions to fractional roots of negative numbers deserves a graphical addendum:

Can someone plot the roots in a unit complex circle. as well as add the "graphical sum" of some of the roots, i.e. the sequential products of the same 5 roots of -8, vectors multiplied in sequence?

x <- as.complex(-8)  # or x <- -8 + 0i
      # find all three cube roots
xroot5 <- (x^(1/5) * exp(2*c(0:4)*1i*pi/5))
plot(xroot5, xlim=c(-8, 2),  ylim=c(-5,5))
abline(h=0,v=0,lty=3)

Originally I was thinking this would be some sort of head to tail illustration but complex multiplication is a series of expansions and rotations around the origin.

Answer 1

The circle is centered at 0, 0. The roots all have the same radius and picking any one of them, the radius is

r <- Mod(xroot[1])

The following gives a plot which looks similar to the plot in the question except we have imposed an aspect ratio of 1 in order to properly draw it and there is a circle drawn through the 5 points:

plot(Re(xroot5), Im(xroot5), asp = 1)

library(plotrix)
draw.circle(0, 0, r) 

Multiplying any root by

e <- exp(2*pi*1i/5) 

will rotate it into the next root. For example, this plots xroot5[1] in red:

i <- 0
points(Re(xroot5[1] * e^i), Im(xroot5[1] * e^i), pch = 20, col = "red") 

and then repeat the last line for i = 1, 2, 3, 4 to see the others successively turn red.