Julias from Pc type polynomial solutions for n=2 the Mandelbulb
- To: mathgroup at smc.vnet.net
- Subject: [mg105805] Julias from Pc type polynomial solutions for n=2 the Mandelbulb
- From: Roger Bagula <rlbagula at sbcglobal.net>
- Date: Sun, 20 Dec 2009 06:52:25 -0500 (EST)
- Reply-to: rlbagulatftn at yahoo.com
In investigating the claims that the Mandelbulb n forms
are "true" 3d Mandelbrots I did some 2d projections of
Pc polynomial solutions of cycle 2 and cycle 3 using programs
programmed in Mathematica by Paul Nylander. ( I'm posting this
with his specific permission).
The cycle two Julia points are:(pc in Nylander's notation)
{-1,0,0}
http://www.facebook.com/photo.php?pid=4182775&id=787071498
{1/2,0,Sqrt[3]/2}
{1/2,0,-Sqrt[3]/2}
The Douady's rabbit cycle 3 is:
pc = {-0.122561, 0.744862, 0}
http://www.flickr.com/photos/fractalmusic/4196769055/
This last one was found by trying the Douady values in the Julia
program. not by calculation.
My conclusion is that they ( Mandlbulbs) are Mandelbrot like
but scaled to near a unit sphere instead of the complex plane bigger values.
I've also verified that the Siegel disk from 2d works in the 3d projection
Julia:( figure 7 The Beauty of Fractals)
pc = {-0.39054, -0.58679, 0};
A rescaled n=2 Mandelbulb would be:
{x,y,z}'={(x^2 - y^2 - z^2) /(1 - z^2), 2 x y, -2 *(x^2 + y^2)^(3/2)*
z/(1 - z^2)
Daniel White and Paul Nylander seem to have been
the principle investigators
but there also seems to be a cast of thousands.
They are all very good at 3d rendering but
somewhat sketchy about Mandelbrot set theory
and how to actually "prove" their claims?
I call this the great taffy pulling contest of modern fractals...
("Taffy" is the nick name the 3d renders that Terry Gintz has been doing for the last 10 years have gotten: no disrespect meant:a discription of what the surfaces look like.)
I also did a toral inverse transform on the Mandelbulb algorithm
and it appeared as you would expect.
They certainly have the endorsement of most of the fractal community
encluding Terry Gintz.
Respectfully, Roger L. Bagula
11759 Waterhill Road, Lakeside,Ca 92040-2905,tel: 619-5610814 :
http://www.google.com/profiles/Roger.Bagula
alternative email: roger.bagula at gmail.com
Mathematica: Pc polynomials for mandelbulb n=2:
Clear[g, gg, a, p, q, r, x, y, z, x0, y0, z0]
(*definition of a 3d Mandelbrot Mandelbulb algorithm*)
p[x_, y_, z_] = (x^2 - y^2)(1 - z^2/(x^2 + y^2)) + x0;
q[x_, y_, z_] = 2*x*y(1 - z^2/(x^2 + y^2)) + y0;
r[x_, y_, z_] = -2*z*Sqrt[x^2 + y^2] + z0;
g[x_, y_, z_] = {p[x, y, z], q[x, y, z], r[x, y, z]};
(*definition of recursive 3d polynomials*)
gg[0] := {x0, y0, z0}
gg[1] := g[x0, y0, z0]
gg[n_] := gg[n] = g[gg[n - 1][[1]], gg[n - 1][[2]], gg[n - 1][[3]]]
a = Table[Table[FullSimplify[gg[n][[i]]] == 0, {i, 1, 3}], {n, 0, 3}];
(*First level solution :*)
NSolve[a[[1]], {x0, y0, z0}]
{{x0 -> 0, y0 -> 0, z0 -> 0}}
(*Second level solution :*)
NSolve[a[[2]], {x0, y0, z0}]
NSolve[a[[3]], {x0, y0, z0}]
Mathematica: Paul Nylander's Julia program:( try n=50 for a fast render)
http://www.facebook.com/photo.php?pid=4182775&id=787071498
n = 300; norm[x_] := x.x; pc = {-1, 0, 0};
BagulaSquare[{x_,
y_, z_}] :=
Module[{rxy2 = x^2 + y^2, rxz2 = x^2 +
z^2, a}, a = 1 - z^2/rxy2; If[rxz2 == 0, {0, 0,
0}, {a(x^2 - y^2), 2 a x y, -2 Sqrt[rxy2] z}]];
gradient = {{0., {0, 0, 0.5}}, {0.1, {0, 0, 1}}, {0.4, {0, 1, 1}}, {
0.6, {1, 1, 0}}, {0.9, {1, 0, 0}}, {1, {0.5, 0, 0}}};
Gradient2[x_, grad_] := Module[{i = 1, n =
Length[grad]}, While[i â?¤ n && gradã??i, 1ã?? <
x, i++]; RGBColor @@
If[1 < i â?¤ n, Module[{x1 = gradã??i - 1,
1ã??, x2 = gradã??i, 1ã??}, ((x2 - x) gradã??i - 1, 2ã?? + (
x - x1)gradã??i, 2ã??)/(x2 - x1)], gradã??Min[i, n], 2ã??]];
Julia3D[p0_] := Module[{p = p0, i = 0},
While[i < 24 && norm[p] < 4, p = BagulaSquare[p] + pc; i++]; i];
image = Table[z = 1.25;
While[z â?¥ -0.1 && Julia3D[{x,
y, z}] < 24, z -= 3.5/n]; z, {y, -1.75, 1.75, 3.5/
n}, {x, -1.75, 1.75, 3.5/n}];
ListDensityPlot[image, Mesh ->
False, Frame -> False,
PlotRange -> {-0.1, 1.25},
ColorFunction -> (Gradient2[#, gradient] &), ImageSize -> 1000];
gr = ListPlot3D[image, Mesh ->
False, AspectRatio -> Automatic, Boxed -> False,
Axes -> False, ViewPoint -> {-0.884, -1.543, 2.879}];