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Re: AMERICAN MATHEMATICAL MONTHLY -April 2009:Transformations Between
*To*: mathgroup at smc.vnet.net
*Subject*: [mg98369] Re: AMERICAN MATHEMATICAL MONTHLY -April 2009:Transformations Between
*From*: David Bailey <dave at removedbailey.co.uk>
*Date*: Wed, 8 Apr 2009 05:01:51 -0400 (EDT)
*References*: <grcgfr$pil$1@smc.vnet.net>
Roger Bagula wrote:
> I haven't had fun like this since I typed in Barnley's original Byte
> article IFS's.
> The Fern just blew me away
> back then! Today this affine rearranged Sierpinski gasket is great work.
> http://www.geocities.com/rlbagulatftn/barnsley_affine123.gif
> http://www.geocities.com/rlbagulatftn/barnsley_affine_234.gif
> http://www.geocities.com/rlbagulatftn/barnsley_affine_134.gif
>
> http://www.maa.org/pubs/monthly_apr09_toc.html
>
>
> April 2009
>
> *Transformations Between Self-Referential Sets*
> By: Michael F. Barnsley
> mbarnsley at aol.com <mailto:mbarnsley at aol.com>
> Did you know that there are continuous transformations from a fractal
> fern onto a filled square? Also, there are functions of a similar wild
> character that map from a filled triangle onto itself. We prove that
> these /fractal transformations/ may be homeomorphisms, under simple
> conditions, and that they may be calculated readily by means of a
> coupled Chaos Game. We illustrate several examples of these beautiful
> functions and show how they exemplify basic notions in topology,
> probability, analysis, and geometry. Thus they are worthy of the
> attention of the mathematics community, both for aesthetic and
> pedagogical reasons.
>
> Mathematica:
> Clear[f, dlst, pt, cr, ptlst, M, p, a, b, c]
> n0 = 3;
> dlst = Table[ Random[Integer, {1, n0}], {n, 100000}];
> a = 0.65; b = 0.3; c = 0.4;
> M = {{{-1 + b, -1/2 + b/2 + a/2}, {0, a}}, {{b + c/
> 2 - 1/2, b/2 - c/4 + 1/4}, {1 - c, c/2 - 1/2}}, {{c/2, -1/2 + a/2 - \
> c/4}, {-c, -1 + a + c/2}}, {{b + c/2 - 1/2, -3/4 + b/4 + a/2 - 1/4}, {
> 1 - c, a - 1/2 - c/4}}}
> in = {{1 - b, 0}, {1 - b, 0}, {1/2, 1}, {1 - b, 0}};
> Length[in]
> f[j_, {x_, y_}] := M[[j]]. {x, y} + in[[j]]
> pt = {0.5, 0.5};
> cr[n_] := Flatten[Table[If[i == j == k == 1, {}, RGBColor[i, j, k]], {i, 0,
> 1}, {j, 0, 1}, {k, 0, 1}]][[1 + Mod[n, 7]]];
> ptlst[n_] := Table[{cr[dlst[[j]]], Point[pt = f[dlst[[j]], Sequence[pt]]]},
> {j, Length[dlst]}];
> Show[Graphics[Join[{PointSize[.001]},
> ptlst[n]]], AspectRatio -> Automatic, PlotRange -> All]
>
>
Roger,
Which parameter do you vary to go between the two types of fractal -
your calculation takes a bit too long to figure this out by guesswork!
David Bailey
http://www.dbaileyconsultancy.co.uk
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