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A complex implicit conformal map of the tent mapping function to the complex plane:
(* Mathematica*)
Clear[f, g, h, k, v, ff, gg, hh, w]
f[x_] := 2*x /; 0 <= x <= 1/2
f[x_] := 2 - 2*x /; 1/2 < x <= 1
ff[x_] = f[Mod[Abs[x], 1]]
g[x_] := 1 - 2*x /; 0 <= x <= 1/2
g[x_] := -1 + 2*x /; 1/2 < x <= 1

gg[x_] = g[Mod[Abs[x], 1]]
Plot[{ff[x], gg[x]}, {x, 0, 2}]
s0 = Log[2]/Log[3]
hh[x_] = Sum[ff[2^k*x]/2^(s0*k), {k, 0, 20}];
kk[x_] = Sum[gg[2^k*(x + 1/2)]/2^(s0*k), {k, 0, 20}];
ContourPlot[Abs[hh[t + I*w] + I*kk[t + I*w]], {t, -2, 2}, {w, -2, 2},
Axes -> False, ImageSize -> {1000, 1000}, PlotPoints -> 30,
Contours -> 30, ColorFunction -> "BrightBands"]
(* end*)
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Uda - Summer
MyFractalArt by Myriam Elorza
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A bit off topic, but nice to see :-)

https://www.youtube.com/watch?v=bZ_RHI1TFek

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Trying to simulate the Julia with the Tent map:
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I got this 3d version of the fractal-topologist curve working:
(* Mathematica*)
Clear[f, g, h, k, s0, ff, ll, kk, mm, a, g3, ga]
f[x_] := (x/2)Sin[2/x] /; 0 <= x <= 1/2

f[x_] := ((1 - x)/2)*Sin[2/(1 - x)] /; 1/2 < x <= 1
ff[x_] = f[Mod[Abs[x], 1]]

Plot[ff[x], {x, 0, 4}]
s0 = N[0.51]

kk[x_] = N[Sum[ff[3^k*x]/3^(s0*k), {k, 0, 20}]];
Plot[kk[x], {x, 0, 4}]
hh[x_] = N[Sum[ff[3^k(x) + 1/3]/3^(s0*k), {k, 0, 20}]];
ll[x_] = N[Sum[ff[3^k*(x) + 1/6]/3^(s0*k), {k, 0, 20}]];
Plot[hh[x], {x, 0, 4}]
Plot[ll[x], {x, 0, 4}]
rotate[theta_] := {{Cos[theta], -Sin[theta]},
{Sin[theta], Cos[theta]}};
w = {hh[t], kk[t], ll[t]};
g1 = ParametricPlot3D[w, {t, 0, 1}, Boxed -> False, Axes -> False,
ColorFunction -> "BrightBands", ImageSize -> 1000, PlotPoints -> 500]
a = Table[{Hue[Norm[{hh[n/10000], kk[n/10000], ll[n/10000]}]],
PointSize[0.001],
Sphere[{hh[n/10000], kk[n/10000], ll[n/10000]}, 0.005]}, {n, 1.01,
10000}];
ga = Show[Graphics3D[a], Axes -> False, ImageSize -> 1000,
ViewPoint -> Top, Boxed -> False]
Show[{g1, ga}]
(* end*)
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This picture is the cusp end Julia of the Mandelbrot set cardioid:
f[z_]=z^2+1/4
that is said in Indras Pearls (page 291) to have the cusp-like Makit slice border. It is easy to see that this Julia set is related to the tent map period doubling set as well.
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Simply adding a single iteration of:

z = pow(z, -4) - .25;

before:

z = pow(z, 3) - (pow(-z, 2.00001) - 1.0008875);

Give this most interesting fractal.

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https://plus.google.com/101799841244447089430/posts/Cibp2h6nDjG

#Fractal #Julia #Hybrid #Space #Math
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My cubic Julia in Xaos. So far, I have it working fine in several of my C and C++ programs, along with Xaos. I need to get it to work in Ultra Fractal damn it! I am a newbie in that program. Fwiw, the formula from that works in Xaos is:

Z^3 - ((-Z)^2.00001 - 1.0008875)

https://plus.google.com/101799841244447089430/posts/ASbWXo8Da6g

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#Fractal #Cubic #Julia #Math #Art
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