Re: Sierpinski carpet

• To: mathgroup at smc.vnet.net
• Subject: [mg76789] Re: Sierpinski carpet
• From: Jean-Marc Gulliet <jeanmarc.gulliet at gmail.com>
• Date: Sun, 27 May 2007 05:01:24 -0400 (EDT)
• Organization: The Open University, Milton Keynes, UK
• References: <f38tqu\$j16\$1@smc.vnet.net>

```Anolethron wrote:
> What I'm trying to do is basically constructing a Sierpinski's carpet with
> an algorithm that can be generalized to the construction of a Menger Sponge.
>
> e1 = {1, 0}; e2 = {0, 1}; p1 = {0, 0}; p2 = {1, 0}; p3 = {1, 1}; p4 = {0,
> 1};
>
> Sierpinski[{p1_, p2_, p3_, p4_}] :=
> Delete[Flatten[
> Table[{p1 + m e1 + n e2, p2 + m e1 + n e2, p3 + n e2 + m e1,
> p4 + m e1 + n e2}, {n, 0, 2}, {m, 0, 2}], 1], 5];
>
>
> Sierpinski1 = Sierpinski[{p1, p2, p3, p4}]
>
>
> Sierpinski2[ls_] := Flatten[Map[Sierpinski, ls], 1]
>
>
> S2 = Sierpinski2[Sierpinski1]
>
>
> Sierpinski3[n_] := Nest[Sierpinski2, {{p1, p2, p3, p4}}, n]
>
>
> Sierpinski3[3]
>
>
>
> Now, I'm not good enough to think of a much more complicated construction
> and the problem is that with this algorithm the lengths of the squares I
> construct at each step does not scale down with the level of the carpet I'm
> constructing: e.g. He builds 9 squares from the big one at the beginning and
> deletes the central one, it's ok. But as I Iterate the process at each
> smaller square It builds squares of the same size, so what I get is just a
> big black figure. It obviously does this way because in the algorithm
> there's no instruction to decrease the size of the base vectors (e1,e2).
> Thing is I can't think of a way to give mathematica that instruction inside
> the Nest or in the definition of  the basic "Sierpinski" function. I need
> some help. Thanks in advance. This is the expected result:
> http://mathworld.wolfram.com/SierpinskiCarpet.html

Under version 6.0, the following code will draw some nice Sierpinski
Carpets that can be controlled thanks to the Manipulate function.

rules = {0 -> {{0, 0, 0}, {0, 0, 0}, {0, 0, 0}},
1 -> {{1, 1, 1}, {1, 0, 1}, {1, 1, 1}}};
f[m_: 1] := ArrayFlatten[m /. rules]
drawSerp[n_] := MatrixPlot[Nest[f, 1, n], FrameTicks -> None]
Manipulate[drawSerp[n], {n, 1, 6, 1}]

You can check directly the files at
http://homepages.nyu.edu/~jmg336/mathematica/SierpinskiCarpet.nb
http://homepages.nyu.edu/~jmg336/mathematica/SierpinskiCarpet.pdf
http://homepages.nyu.edu/~jmg336/mathematica/SierpinskiCarpet.png

Regards,
Jean-Marc

```

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