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