Archive for the 'Mandelbrot' Category

18
Aug
09

3D Mandelbrot Set Pickover Stalks

Pickover Stalks are cross-shaped orbit traps invented by Clifford Pickover. See also my 2D Pickover stalks fractal.

18
Aug
09

3D Mandelbrot Set Orbit Trap

Orbit trapping is a popular technique for mapping arbitrary shapes to a fractal. This 3D orbit trap is shaped like a sphere. See also my 2D orbit trap fractals.

Links

30
Jul
09

4D “Mandy Cousin” Julia and Mandelbrot Sets

Julia : c={-0.2,0.8,0,0}, imax=12

{x,y,z,w}2 = {x2-y2-z2+w2, 2(xy+zw), 2(xz+yw), 2(xw+yz)}
David Makin suggested this formula to me for squaring a 4D hypercomplex number.

Links

Mandelbrot Set, imax=12

Minibrot, imax=20

30
Jul
09

4D Julia and Mandelbrot Sets Variations

Julia : c={-0.2,0.8,0,0}, imax=12

{x,y,z,w}2 = {x2-y2-z2-w2, 2(xy+zw), 2(xz-yw), 2(xw+yz)}
Here is another formula for squaring a 4D hypercomplex number that David Makin suggested to me.

Mandelbrot Set, imax=12

Minibrot, imax=20

30
Jul
09

4D Bicomplex Mandelbrot Set

{x,y,z,w}2 = {x2-y2-z2+w2, 2(xy-zw), 2(xz-yw), 2(xw+yz)}
This “Tetrabrot” reminds me of a Bismuth crystal.

Here is some Mathematica code for the minibrot:
(* runtime: 2.6 minutes, increase n for higher resolution *)
n = 100; Norm[x_] := x.x; Square[{x_, y_, z_, w_}] := {x^2 - y^2 - z^2 + w^2, 2 (x y - z w), 2 (x z - y w), 2 (x w + y z)};
Mandelbrot4D[qc_] := Module[{q = {0, 0, 0, 0}, i = 0}, While[i < 20 && Norm[q] < 4, q = Square[q] + qc; i++]; i];
image = Table[z = 0.06; While[z >= 0 && Mandelbrot4D[{x, y, z, 0}] < 20, z -=0.12/n]; If[z < 0, -0.06, z], {y, -0.06, 0.06, 0.12/n}, {x, -1.82, -1.7, 0.12/n}];
ListDensityPlot[image, Mesh -> False, Frame -> False, PlotRange -> {-0.02, 0.06}]

Tetrabrot, imax=12

Minibrot, imax=20

power=3, imax=12

Bismuth Crystal

Links

24
Jul
09

4D “Roundy” Mandelbrot Set

Mandelbrot Set, imax=24

Minibrot, imax=21

{x,y,z,w}2 = {x2-y2-z2-w2, 2(xy+zw), 2(xz+yw), 2(xw+yz)}
This formula for squaring a 4D hypercomplex number is probably the most popular method for creating 3D Mandelbrot sets. Here is some Mathematica code:
(* runtime: 1.2 minutes, increase n for higher resolution *)
n = 100; Norm[x_] := x.x; Square[{x_, y_, z_, w_}] := {x^2 - y^2 - z^2 - w^2, 2 (x y + z w), 2 (x z + y w), 2 (x w + y z)};
Mandelbrot4D[qc_] := Module[{q = {0, 0, 0, 0}, i = 0}, While[i < 24 && Norm[q] < 4, q = Square[q] + qc; i++]; i];
image = Table[z = 1.5; While[z >= 0 && Mandelbrot4D[{x, y, z, 0}] < 24, z -= 3/n]; If[z < 0, -1.5, z], {y, -1.5, 1.5, 3/n}, {x, -2.0, 1.0, 3/n}];
ListDensityPlot[image, Mesh -> False, Frame -> False, PlotRange -> {-0.1,1.5}]

Links

07
Jul
09

4D Quaternion Mandelbrot Set

Mandelbrot Set, imax=12

Minibrot, imax=18

{x,y,z,w}2 = {x2-y2-z2-w2, 2xy, 2xz, 2xw}
Quaternions are 4D hypercomplex numbers, discovered in 1843 by Sir William Rowan Hamilton. They are mathematically elegant, but unfortunately, they produce axisymmetric results when used to calculate the 3D Mandelbrot set.

Link: Quat Minibrot – by David Makin




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