This is a test.

test

This is a test.

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

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

{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

{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

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

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

{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