roots of unity Visualization

Questions
1

Hey there,

I’m a german student and very new here. Wikipedia brought me here because of a skilled work in mathematics. It’s about visualizing roots of unity and I found a graphic made with wolfram mathematica: Einheitswurzel – Wikipedia

It would be very pretty if I could visualize my roots of unity in a similar way. I have already visualized the unity roots in a coordinate system but don’t know how to colorize it like wikipedia’s one. Maybe with the colorschemes but I don’t know how to put them in my already written input.
I added a file of my already written one and hope it works.

Maybe someone can help me. It would be very great.

Thank you.

2

I get something vaguely similar with

DensityPlot[Sin[3 Arg[(x + I y)^3 - 1]], {x, -2, 2}, {y, -2, 2}, 
 ColorFunction -> "Rainbow"]

I don’t quite understand it, though.

3

Yeah this looks really good. But couldn’t I plot those colors in my already written input?

Thanks a lot.

4

Yes, you can overlay a DensityPlot and your original plot with Show[plot1,plot2]. Be careful of the order: plot1 is at the bottom, plot2 is on top.

5

thx

6

Here it is:

Show[DensityPlot[Sin[3 Arg[(x + I y)^3 - 1]], {x, -2, 2}, {y, -2, 2}, 
  ColorFunction -> "Rainbow", Exclusions -> None, PlotPoints -> 61], 
 ListPlot[Table[{Re[E^(I*2 \[Pi]*n/12)], Im[E^(I*2 \[Pi]*n/12)]}, {n, 
    0, 20}], AspectRatio -> Automatic, 
  PlotStyle -> {{ Black, PointSize[.02]}}],
 Table[ 
  Graphics [{Thick, Dashed, Black, 
    Line[{{Re[E^(I*2 \[Pi]*n/12)], Im[E^(I*2 \[Pi]*n/12)]},
      {Re[E^(I*2 \[Pi]*(n + 1)/12)], 
       Im[E^(I*2 \[Pi]*(n + 1)/12)]}}]}], {n, 0, 20}]]