Using Degree (Degree Table[x, {x, 0, 172.5, 7.5}]) we get a good alignment on several of the points of a "concentric circle" for a special set of k. Enter

Remove[burnsStrip]
burnsStrip[k_?NumericQ, w_?NumericQ] := 
 Block[{m = RotationTransform[k, {-1, 0}]}, 
  Polygon[{m[{-5/2, -w}], m[{1/2, -w}], m[{1/2, w}], m[{-5/2, w}]}]]

Graphics[{{Green, burnsStrip[0, 1/40]}, {Yellow, 
   Sequence @@ (burnsStrip[#, 1/40] & /@
      (Degree Table[x, {x, 0, 172.5, 7.5}]))},
  Table[k = 2 (3 n^3); 
   Point /@ 
    Evaluate[
     ReIm[(-1)^
        k ((Flatten[Evaluate[Block[{x}, Solve[x^k == k, x]]]])[[All, 
           2]] - 1)]], {n, 1, 5}]}, Frame -> True, AspectRatio -> 3/3,
  PlotRange -> All, PlotLabel -> "Let the sunshine in!"]

and get,

For more detail and more accuracy in the top half, try the code,

Remove[burnsStrip]
burnsStrip[k_?NumericQ, w_?NumericQ] := 
 Block[{m = RotationTransform[k, {-1, 0}]}, 
  Polygon[{m[{-5/2, -w/4}], m[{1/2, -w/4}], m[{1/2, w/4}], 
    m[{-5/2, w}/4]}]]

Graphics[{{Green, burnsStrip[0, 1/40]}, {Yellow, 
   Sequence @@ (burnsStrip[#, 1/40] & /@
      (Degree Table[x, {x, 0, 180, 7.5/8}]))},
  Table[k = 2 (3 n^3); 
   Point /@ 
    Evaluate[
     ReIm[(-1)^
        k ((Flatten[Evaluate[Block[{x}, Solve[x^k == k, x]]]])[[All, 
           2]] - 1)]], {n, 1, 4}]}, Frame -> True, AspectRatio -> 3/3,
  PlotRange -> All, PlotLabel -> "Let the sunshine in!", 
 ImageSize -> 2000]

for what a portion of which looks like the following:

The bottom half of which looks like the points are shifted over by one unit:

I think an important thought we can take home about the points, or "terms of the partial sums of the many satellites (on straight lines through -1 + 0 I or on straight lines through 1 + 0 I) of the MRB constant," or roots of x^k=k is that for every k there are a finite number of unique ones! or at least so it seems !!!!!!!!