Cantor type iterations in Mathematica

Question

I am trying to draw the iterations of the IFS in which from an interval middle third is removed and in the resulting figure left line is shifted 1/6th to the right and right line is shifted 1/6th to the left.

The following code I tried, doesn't seems right from the output image:

    cantorRule = 
 Line[{{a_, n_}, {b_, n_}}] :> 
  With[{d = b - a, 
    np = n - 0.1`}, {Line[{{a + d/6, np}, {a + d/3 + d/6, np}}], 
    Line[{{b - d/3 - d/6, np}, {b - d/6, np}}]}]

Graphics[{CapForm["Butt"], Thickness[.05], 
  Flatten@NestList[# /. cantorRule &, Line[{{0., 0}, {1., 0}}], 6]}]

In the above code how to change the color of the left shifting and right shifting lines different in each step.

Update:

cantorRule = 
 Line[{{a_, n_}, {b_, n_}}] :> 
  With[{d = b - a, 
    np = n - 0.1`}, {Line[{{a + d/6, np}, {a + d/3 + d/6, np}}], 
    Line[{{b - d/3 - d/6, np}, {b - d/6, np}}]}]

Graphics[{CapForm["Butt"], Thickness[.05], 
  Flatten@NestList[# /. cantorRule &, Line[{{0., 0}, {1., 0}}], 6]}]

I changed the code as above and got this image: How to get the end points of each intervals labelled?

Answer 1

For labeling you can post-process the output to inject Texts:

label = # /. l : Line[p : {{_, _}, {_, _}}] :> {l, Black,
      Text[Rationalize[#[[1]]], Offset[{0, -20}, #], {Center, Top}] & /@ p} &;
lines = Flatten @ NestList[# /. cantorRule &, Line[{{0., 0}, {1., 0}}], 3];
label @ Graphics[{CapForm["Butt"], Thickness[.05], lines}]

Needless to say, if you have many layers there won't be enough space to avoid overlaps.

For coloring, you can Partition the list of lines (starting from the second) and Riffle desired colors into each pair of lines:

coloredlines = Flatten[Riffle[{Red, Yellow}, #] & /@ Partition[lines, 2, 2, {-1, 1}, {}]]
label @ Graphics[{CapForm["Butt"], Thickness[.05], coloredlines}]

Alternatively, use RandomColor[2] instead of {Red, Yellow} to color each piece separately: