----- Original Message ----- From: "Russell Walsmith" <russw@lycos.com> To: "Fractint and General Fractals Discussion" <fractint@mailman.xmission.com> Subject: Re: [Fractint] Triterions revisited Date: Sat, 02 Jul 2005 15:14:38 -0500 I wrote
The basic idea is that a cyclic group of odd order j has an identity element (I) and j-1 elements (e) of degree j (i.e., e^j=I). If each element is also given a negative sign, then we have in toto a group C2j. Thus, opposing those elements of degree j across the origin from elements of degree 2j is the same as ruling each axis with positive and negative numbers. But you don't necessarily have to do that; e.g., here is C6 with degree 6 elements on the same axis:
but I should have said that a cyclic group of PRIME order j... For odd j in general though, and cyclic groups of order 2j, it does hold that elements of degree k can be juxtaposed across the origin from elements of degree 2k. This seems to be a sufficient condition for reducing a group table to ordered j-tuples that are then easily adapted to Fractitnt frms. Of course many groups other than the cyclics exist, and the larger the group, often the more ways that its elements may be juxtaposed on a given axis. Different arrangements typically have a profound effect on the shape of the fractal object that a formula generates. Clearly, there is depth to this subject. It will be interesting to see to what extent it can be comprehensively and systematically investigated, and what sort of general understanding may be attained.