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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*)

(* 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

MyFractalArt by Myriam Elorza

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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)

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*)

(* 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.

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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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

#Fractal #Cubic #Julia #Math #Art

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

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

*___*#Fractal #Cubic #Julia #Math #Art

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