END OF AN ERA, FRACTALFORUMS.COM IS CONTINUED ON FRACTALFORUMS.ORG

I believe that the key phrase from the PDF is "This novel approach to 3-space may
eventually prove to be as fertile as are complex numbers in the plane." but since the "T set" is basically just an extruded M set, I don't see that it would be worth the effort to build a fractal renderer based on it at this time (when you can just pick 3 of the 4 axes of a quaternion and get much more interesting results, or even rendering 3D height maps of 2D fractals).

It's one possible way to provide a group on R3, but by no means the only one (in their example, it's all about transposition from the generating symmetry group C6, as opposed to being based on axes - a quick perusal of http://en.wikipedia.org/wiki/Hypercomplex_number shows all sorts of concepts).

At the very least, it seems a bit of a misnomer to call them triternions, since they've got very little in common with quaternions (complex numbers are a subset of quaternions which in turn are a subset of octernions, but complex numbers are not a subset of triternions, nor are triternions a subset of quaternions - and at this point I really wish everything was unicode compliant so I could just type real "blackboard" characters and other math operators).

Perhaps we ought to take this to a different thread...

Thomas Ludwig (lycium) wrote:
>
>    i added a little blue light inside the julia, just for fun;

The added color was a nice touch to the image!!   
 

O can be viewed as H^2 (just like H can be viewed as C^2), and if you have quaternion (A,B,C,D), it is equivalent to octonion (A,B,C,D, 0,0,0,0)

And while octonions are non-associative, octonions of the form (A,B,C,D, 0,0,0,0) are associative (just like quaternions).

You can take it one step further with Sedenions http://www.geocities.com/zerodivisor/, but at that point, all that you've got left is power associativity...  (and it isn't a normed algebra, because there are zero divisors, i.e., there are sedenions S and T such that S != 0 and T != 0, but S x T = 0 (so |S| x |T| != |S x T|)

Similarly, complex ({a + bi}, where i*i = -1) and dual numbers ({a + be}, where e*e = 0, e != 0) are both subsets of complex duals ({{a + bi} + {c + di}e}) (complex (A,B) is complex dual (A,B,0,0) and dual number (A,B) is complex dual (A,0,B,0) - see "Dual Polynomials and Complex Dual Numbers for Analysis of Spatial Mechanisms" in "Proceedings of 1996 ASME Design Engineering Technical Conference and Computers in Engineering Conference")

fantasitc image!