Hausdorff of hypecomplex J-M sets

[snayperx]

Hello,

I tried to find any information on the calculation of Hausdorff dimension of quaternionic J-M sets and there is no any academic papers or even trials of determination of it. By the analogy to the classical J-M sets (i.e. on the complex plane) it could be expected that the "hypersurface" of their quaternionic versions have HD=4. Could anyone here did such studies?

[David Makin]

I think it unlikely for quaternions since they most definitely do not exhibit fractal structure in all directions across their "surface" - it seems more possible for what I think of as "standard" hypercomplex otherwise known as bi-complex fractals and even more so in the case of the Mandelbulbs - though of course Mandelbulbs are 3D rather than 4D.

[Endemyon]

You can try to estimate the dimension by the "Box-Counting Method" with Quaternonic  boxes .

http://en.wikipedia.org/wiki/Box_counting_dimension

If you input log(N(eta))as a function of  log(1/eta) you have two options :
- You don't find a straight line, the shape is not a fractal (it doesn't follow a power law where the exponent is the  minkowski-bouligand dimension).
- If it does follow a straight line, the slope of this line is its dimension