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MathGroup Archive 2007

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Re: Sierpinski's thing

  • To: mathgroup at smc.vnet.net
  • Subject: [mg76776] Re: Sierpinski's thing
  • From: Jean-Marc Gulliet <jeanmarc.gulliet at gmail.com>
  • Date: Sun, 27 May 2007 04:54:40 -0400 (EDT)
  • Organization: The Open University, Milton Keynes, UK
  • References: <f38qg9$hq4$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

Hi,

If you have version 6.0, the following code will draw a nice Sierpinski 
Carpet.

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]
drawSerp[5]

You can see the resulting picture at 
http://homepages.nyu.edu/~jmg336/Sierpinski%20Carpet.png

If you do not have access to version 6.0 and are interested by this 
approach, let me know so I can tell you how to tweak the code for 
version 5.2.

Regards,
Jean-Marc


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