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MathGroup Archive 2007

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Sierpinski's thing

  • To: mathgroup at smc.vnet.net
  • Subject: [mg76685] Sierpinski's thing
  • From: "Anolethron" <abacus78685 at tin.it>
  • Date: Sat, 26 May 2007 04:18:22 -0400 (EDT)

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




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