Mathematica 9 is now available
Services & Resources / Wolfram Forums / MathGroup Archive
-----

MathGroup Archive 2009

[Date Index] [Thread Index] [Author Index]

Search the Archive

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


  • Prev by Date: Show[list] does not work - details of the script
  • Next by Date: Re: Computing resources
  • Previous by thread: AMERICAN MATHEMATICAL MONTHLY -April 2009:Transformations Between
  • Next by thread: Re: AMERICAN MATHEMATICAL MONTHLY -April 2009:Transformations Between