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The factoring of the Logistic powers to give tent map
powers reminded me of the von Koch
fractals I have made with Besicovich Ursell functions in the past.
Logistic generalization:
f[x_,n_]:=1-2^(2n)*(x-1/2)^(2*n)
Tent Map generalization ( positive curvature):
f[x_, n_] := 1 + 2^n*(x - 1/2)^n /; 0 <= x <= 1/2
f[x_, n_] := 1 - 2^n*(x - 1/2)^n /; 1/2 < x <= 1
Tent Map generalization( negative curvature):
f[x_, n_] := (2*x)^((2 + n)/3) /; 0 <= x <= 1/2
f[x_, n_] := (2 - 2*x)^((2 + n)/3) /; 1/2 < x <= 1
I multiplied the two parts of the von Koch unit square function and
got a quadratic/ Logistic von Koch.
f[x_] := 0 /; 0 <= x <= 1/3
f[x_] := (6*x - 2)*(-6*x + 4) /; 1/3 < x ? 2/3
f[x_] := 0 /; 2/3 < x <= 1
I have no idea how this affects the fractal dimension,
but it does give a new type of fractal.
It is clear that there are two different types of von Koch:
those with double points/ multiple points in their functional definition
as in an IFS
and those with "single" point definitions as in the Biscuit functions
without parametric phases.
In both cases the results are "fractals", but of two different types.
What made me think of this function was the resemblance of a von Koch to
a Normal/ Gaussian
type curve and Mandelbrot's extensive work on unit square type cartoon
multifractal constructions to get fractional Brownian sets.
Von Koch unit square function ( called a cartoon):
f[x_] := 0 /; 0 <= x <= 1/3
f[x_] := 6*x - 2 /; 1/3 < x <= 1/2
f[x_] := -6*x + 4 /; 1/2 < x <= 2/3
f[x_] := 0 /; 2/3 < x <= 1
Mathematica:
Clear[f, g, h, k, x]
f[x_] := 0 /; 0 <= x <= 1/3
f[x_] := (6*x - 2)*(-6*x + 4) /; 1/3 < x ? 2/3
f[x_] := 0 /; 2/3 < x <= 1
ff[x_] := f[Mod[Abs[x]
, 1]]
Plot[ff[x], {x, 0, 4}]
s0 = Log[3]/Log[4];
kk[x_] = Sum[ff[3^k*x]/3^(s0*k), {k, 0, 20}];
Plot[kk[x], {x, 0, 4}]
ll[x_] = Sum[ff[3^k*(x + 1/2)]/3^(s0*k), {k, 0, 20}];
Plot[ll[x], {x, 0, 4}]
ParametricPlot[{ll[t], kk[t]}, {t, 0, 1/2}]
g1 = ListPlot[Table[{ll[n/10000], kk[n/10000]}, {n, 1, 10000}],
PlotStyle -> {AbsolutePointSize[0.3],
Blue}, PlotRange -> {{-2, 2}, {0, 2}}, Axes -> False]
g2 = ListPlot[Table[{-ll[n/10000], kk[n/10000]}, {n, 1, 10000}],
PlotStyle -> {AbsolutePointSize[0.3], Blue},
PlotRange -> {{-2, 2}, {0, 2}}, Axes -> False]
Respectfully, Roger L. Bagula
11759Waterhill Road, Lakeside,Ca 92040-2905,tel: 619-5610814
:http://www.geocities.com/rlbagulatftn/Index.html
<http://www.geocities.com/rlbagulatftn/Index.html>
alternative email: rlbagula@[EMAIL PROTECTED]
>
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The factoring of the Logistic powers to give tent map<br>
powers reminded me of the von Koch<br>
fractals I have made with Besicovich Ursell functions in the past.<br>
Logistic generalization:<br>
f[x_,n_]:=1-2^(2n)*(x-1/2)^(2*n)<br>
Tent Map generalization ( positive curvature):<br>
f[x_, n_] := 1 + 2^n*(x - 1/2)^n /; 0 <= x <= 1/2<br>
f[x_, n_] := 1 - 2^n*(x - 1/2)^n /; 1/2 < x <= 1<br>
Tent Map generalization( negative curvature):<br>
f[x_, n_] := (2*x)^((2 + n)/3) /; 0 <= x <= 1/2<br>
f[x_, n_] := (2 - 2*x)^((2 + n)/3) /; 1/2 < x <= 1<br>
I multiplied the two parts of the von Koch unit square function and<br>
got a quadratic/ Logistic von Koch.<br>
f[x_] := 0 /; 0 <= x <= 1/3<br>
f[x_] := (6*x - 2)*(-6*x + 4) /; 1/3 < x ≤ 2/3<br>
f[x_] := 0 /; 2/3 < x <= 1<br>
I have no idea how this affects the fractal dimension,<br>
but it does give a new type of fractal.<br>
It is clear that there are two different types of von Koch:<br>
those with double points/ multiple points in their functional
definition <br>
as in an IFS<br>
and those with "single" point definitions as in the Biscuit functions <br>
without parametric phases.<br>
In both cases the results are "fractals", but of two different types.<br>
What made me think of this function was the resemblance of a von Koch
to <br>
a Normal/ Gaussian<br>
type curve and Mandelbrot's extensive work on unit square type cartoon<br>
multifractal constructions to get fractional Brownian sets.<br>
Von Koch unit square function ( called a cartoon):<br>
f[x_] := 0 /; 0 <= x <= 1/3<br>
f[x_] := 6*x - 2 /; 1/3 < x <= 1/2<br>
f[x_] := -6*x + 4 /; 1/2 < x <= 2/3<br>
f[x_] := 0 /; 2/3 < x <= 1<br>
<br>
Mathematica:<br>
Clear[f, g, h, k, x]<br>
f[x_] := 0 /; 0 <= x <= 1/3<br>
f[x_] := (6*x - 2)*(-6*x + 4) /; 1/3 < x ≤ 2/3<br>
f[x_] := 0 /; 2/3 < x <= 1<br>
ff[x_] := f[Mod[Abs[x]
<p><wbr>, 1]]<br>
Plot[ff[x], {x, 0, 4}]<br>
s0 = Log[3]/Log[4]<wbr>;<br>
kk[x_] = Sum[ff[3^k*x]<wbr>/3^(s0*k)<wbr>, {k, 0, 20}];<br>
Plot[kk[x], {x, 0, 4}]<br>
ll[x_] = Sum[ff[3^k*(<wbr>x + 1/2)]/3^(s0*<wbr>k), {k, 0, 20}];<br>
Plot[ll[x], {x, 0, 4}]<br>
ParametricPlot[<wbr>{ll[t], kk[t]}, {t, 0, 1/2}]<br>
g1 = ListPlot[Table[<wbr>{ll[n/10000]<wbr>, kk[n/10000]}<wbr>, {n, 1,
10000}],<br>
PlotStyle -> {AbsolutePointSize[<wbr>0.3],<br>
Blue}, PlotRange -> {{-2, 2}, {0, 2}}, Axes -> False]<br>
g2 = ListPlot[Table[<wbr>{-ll[n/10000]<wbr>, kk[n/10000]}<wbr>, {n, 1,
10000}],<br>
PlotStyle -> {AbsolutePointSize[<wbr>0.3], Blue},<br>
PlotRange -> {{-2, 2}, {0, 2}}, Axes -> False]<br>
<br>
Respectfully, Roger L. Bagula<br>
11759Waterhill Road, Lakeside,Ca 92040-2905,tel: 619-5610814 :<a
href="http://www.geocities.com/rlbagulatftn/Index.html">http://www.geocitie<wbr>s.com/rlbagulatf<wbr>tn/Index.<wbr>html</a><br>
alternative email: <a
href="mailto:rlbagula%40sbcglobal.net">rlbagula@[EMAIL PROTECTED]
>net</a></p>
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