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Re: AMERICAN MATHEMATICAL MONTHLY -April 2009:Transformations Between

  • To: mathgroup at smc.vnet.net
  • Subject: [mg98558] Re: AMERICAN MATHEMATICAL MONTHLY -April 2009:Transformations Between
  • From: Roger Bagula <rlbagula at sbcglobal.net>
  • Date: Sun, 12 Apr 2009 03:50:08 -0400 (EDT)
  • References: <grcgfr$pil$1@smc.vnet.net> <grhp5r$mg6$1@smc.vnet.net> <grn1jc$ms7$1@smc.vnet.net>

I found what appears to be a new tile associated with Barnsley's
 triangular affine.
The fourth transform algorithm is my own.
Riddle rotation of Barnsley affine: fractal animation

http://www.mathematica-users.org/mathematica/images/d/db/affine_riddle.avi

Clear[f, dlst, pt, cr, ptlst, M, p, a, b, c, n]
rotate[theta_] := {{Cos[theta], -Sin[theta]}, {Sin[theta], Cos[theta]}};
n0 = 4;
dlst = Table[ Random[Integer, {1, n0}], {n, 10000}];
a = b = c = 0.5;
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}}, rotate[-Pi/((7/4 -
a/2 - b/2 - c/2)) + n*Pi/20].{{1/2 - a/2 -
  b/2 - c/2, 3/4 - a/4 - b/4 - c/4}, {1 - a/3 - b/3 - c/3, 1/2 - a/6 - b/6 -
   c/6}}}
a0 = Table[Det[M[[i]]], {i, 1, 4}]
FullSimplify[Apply[Plus, a0]]
rotate[-Pi/((7/4 - a/2 - b/2 - c/2))]
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 +
         1, 7]]];
ptlst[n_] := Table[{cr[dlst[[j]]], Point[pt = f[dlst[[j]], Sequence[pt]]]},
  {j, Length[dlst]}];
Table[Show[Graphics[Join[{PointSize[.001]}, ptlst[n]]], AspectRatio ->
       Automatic, PlotRange -> All], {n, 0, 20}]

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