Order from Order: 3D Fractals from Quaternions and Hypercomplex Numbers

 
Fractals from Quaternions as wholes
 
3D Fractals more generally (even with no cylindrical symmetry) can be obtained thanks to generalizations of the usual complex numbers like, e.g., Quaternions or Hypercomplex numbers. In both cases the number of degrees of freedom are raised from 2 (1 real and 1 imaginary part of usual complex numbers) up to 4 (1 real and 3 imaginary parts of quaternions or hypercomplex numbers). Since only 3 co-ordinates can be represented in a plot, the image resulting will be a projection onto the usual 3D space of a 4D fractal structure.


 

Julia 3d (C =-0.7454294)
POV Ray 3.7 code

Julia 3d (C =-0.7454294 + j 0.25)
POV Ray 3.7 code

 

 

Fractals from Hypercomplex Numbers as wholes

Julia 3d (C =-0.7454294)
 
POV Ray 3.7 code

Julia 3d (C =-0.7454294 + i o.113089 +
+ j 0.113089)
POV Ray 3.7 code

 

 

Sequentially ordered "point by point" generation allows to combine Mandelbrot or Julia like recurring law according to several different combinations so obtaining e.g. either Mandelbrot-Mandelbrot-Mandelbrot, or Julia-Julia-Julia, or Julia-Mandelbrot-Mandelbrot, etc. structures.

 

Fractals from Quaternions plotted sequentially

 

Mandelbrot-Mandelbrot-Mandelbrot

section 1

section 2

section 3

 

POV Ray 3.7 code
view animation

open section
 

 

 

Julia-Julia-Julia (C = -0.7454294)

section 1

section 2

section 3

 

POV Ray 3.7 code
view animation

open section
 

 

 

Julia-Mandelbrot-Mandelbrot (C = -0.7454294)

section 1

section 2

section 3

 

POV Ray 3.7 code
view animation

open section