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programming - Generating a Sierpinski carpet


I am trying to draw a Sierpinski_carpet. I have code that works, but I think there is a more elegant way to do than my way. Maybe I couls use Tuples or Permutations or some similar function to simplify my code.


enter image description here



f[{{x1_, y1_}, {x2_, y2_}}] := Map[Mean, {
    {{{x1, x1, x1}, {y1, y1, y1}}, {{x1, x1, x2}, {y1, y1, y2}}},
    {{{x1, x1, x1}, {y1, y1, y2}}, {{x1, x1, x2}, {y1, y2, y2}}},
    {{{x1, x1, x1}, {y1, y2, y2}}, {{x1, x1, x2}, {y2, y2, y2}}},
    {{{x1, x1, x2}, {y1, y1, y1}}, {{x1, x2, x2}, {y1, y1, y2}}},
    {{{x1, x1, x2}, {y1, y2, y2}}, {{x1, x2, x2}, {y2, y2, y2}}},
    {{{x1, x2, x2}, {y1, y1, y1}}, {{x2, x2, x2}, {y1, y1, y2}}},
    {{{x1, x2, x2}, {y1, y1, y2}}, {{x2, x2, x2}, {y1, y2, y2}}},
    {{{x1, x2, x2}, {y1, y2, y2}}, {{x2, x2, x2}, {y2, y2, y2}}}
    }, {3}];

d = Nest[Join @@ f /@ # &, {{{0., 0.}, {1, 1}}}, 3];
Graphics[Rectangle @@@ d]
Clear["`*"]

Answer



Version 11.1 introduces MengerMesh:


MengerMesh[3]

enter image description here


enter image description here





This seems the most natural to me:


Mathematica graphics


carpet[n_] := Nest[ArrayFlatten[{{#, #, #}, {#, 0, #}, {#, #, #}}] &, 1, n]

ArrayPlot[carpet @ 5, PixelConstrained -> 1]

Mathematica graphics


Shorter (in InputForm), but perhaps harder to read and slightly slower, though speed hardly matters given the geometric memory usage:


carpet[n_] := Nest[ArrayFlatten @ ArrayPad[{{0}}, 1, {{#}}] &, 1, n]



Style by level


With a minor change we can increment the values with each fractal level allowing identification such as styling, or other processing.


Wild colors are but a few commands away:


carpet2[n_] := Nest[ArrayFlatten[{{#, #, #}, {#, 0, #}, {#, #, #}}] &[1 + #] &, 1, n]

Table[
  ArrayPlot[carpet2 @ 4, PixelConstrained -> 1, ColorFunction -> color],
  {color, ColorData["Gradients"]}

]

Mathematica graphics



Extension to three dimensions


A Menger sponge courtesy of chyanog, with refinements:


carpet3D[n_] :=
   With[{m = # (1 - CrossMatrix[{1,1,1}])}, Nest[ArrayFlatten[m, 3] &, 1, n]]

Image3D[ carpet3D[4] ]


enter image description here



Element coordinates


If you wish to get coordinates for display with graphics primitives or analysis this can be done efficiently using SparseArray Properties:


coords = SparseArray[#]["NonzeroPositions"] &;

Example usages:


Graphics @ Point @ coords @ carpet @ 4


Mathematica graphics


Graphics3D[Cuboid /@ coords @ carpet3D @ 3]

Mathematica graphics


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