       Re: Sierpinski's thing

• To: mathgroup at smc.vnet.net
• Subject: [mg76758] Re: Sierpinski's thing
• From: Szabolcs <szhorvat at gmail.com>
• Date: Sun, 27 May 2007 04:45:20 -0400 (EDT)
• Organization: University of Bergen
• 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
>
>
> 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

I don't understand completely what you were doing above, but here's a
function that constructs a Sierpinski carpet:

In:= pieces = Complement[
Join@@Table[{i, j}, {i, 0, 2}, {j, 0, 2}],
{{1, 1}}]

Out= {{0,0},{0,1},{0,2},{1,0},{1,2},{2,0},{2,1},{2,2}}

In:= sierp[cornerPt_, sideLen_, n_] :=
sierp[cornerPt + #1*(sideLen/3), sideLen/3, n-1] & /@ pieces

In:= sierp[cornerPt_, sideLen_, 0] :=
Rectangle[cornerPt, cornerPt + sideLen*{1, 1}]

In:= Graphics[sierp[{0, 0}, 1, 5], AspectRatio -> Automatic]//Show

Could someone please explain why is this SO MUCH slower in Mathematica 6
than in Mathematica 5.2?

Szabolcs

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