## Author Archive for

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.

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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.

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 “Squarry” 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. Link : Mandy Squarry, Minibrot Squarry – by David Makin 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

29
Jul
09

### 4D Bicomplex Julia Set

{x,y,z,w}2 = {x2-y2-z2+w2, 2(xy-zw), 2(xz-yw), 2(xw+yz)}
c={-0.2,0.8,0,0}, imax=12

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