3D Mandelbrot Set Pickover Stalks

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

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

• 3D Mandelbrot set orbit trap – animation by David Makin
• 3D Orbit Trap – by Ramiro Perez

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 Which is Associated? – volumetric rendering, by Krzysztof Marczak Mandy Cousin – by David Makin
Mandelbrot Set, imax=12	Minibrot, imax=20

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

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

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

• TetraBrot Explorer – deep zooming program by Etienne Martineau and Dominic Rochon, see the animations
• similar renderings – by Jason McGuiness

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

3D Julia Set

based on Daniel White’s formula for squaring a 3D hypercomplex number

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

4D “Roundy” 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

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

• Genuine Mandelbrot 3D, Minibrot 3D, New Mandelbrot 3D, True 3D Julia, Patch with Julia Fractals – volumetric renderings, by Krzysztof Marczak, be sure to see his animation
• Space Station – volumetric mini-minibrot, by Krzysztof Marczak
• 3D Mandlebrot – by Thomas Ludwig
• 4D Mandy, Minibrot – by David Makin