It is my opinion that the concept of coordinate relations stem from how we perceive the universe and this stems from the nature of the universe itself and how it is perceived or measured. A perception is a measure of things. To make sense of what we perceive and how that relates to what is actually taking place, we must transform coordinate systems.
One of the first things that any creature must do is translate the fact that objects are smaller by 1/x and 1/y at a distance z, and that rotation about an axis leads to a perception of the length that is a transform based upon its rotation with respect to the observer. A baby must make this transform in their mind before they can deal with space. By the time we learn math we have completely forgotten this transform and cannot recall it.
Strangely I can actually recall that transform mentally and apply it or not depending on my mood or whim. It is not difficult for me to look at a room and say that the wall across from me is smaller than the one closer. Mentally, I can cancel that transform and reapply it. I rarely do that as it makes it difficult to do many common tasks and is very much like when I was 14 and I grew nearly a foot in height in one summer. I had to recalculate all of the world coordinates to my enlarged dimensions.
The point of this is that energy travels along straight lines in all directions and we live in a universe with many straight lines. The relationship of the perception of the length and width of an object at a distance in rotation is determined by the lines that intersect with our space. The positional rotational transform is one of the most important mental transforms and also the most important mathematically to devise rules to apply to systems of measure to deduce the real structure relationship of its parts.
This then allows us to apply very simple transforms to the state and then convert it back to the relative perception ( or measure ).
The concept of the triangular relationships is key to that conversion and is probably the only system of measure to which I would attach a great significance. Much like the identity (A=A), it describes the identity of relationships in space. By further decomposing its nature to the physicality of the 'material' that forms the universe it is easier for me to follow the progress of relationships which form as a consequence.
I was doing some puzzles earlier and there is a topology problem about how to connect gas, electric, and water to three houses in 2D space without crossing. Most people know this riddle and some know the topological foundations, however I am always looking for a way to trick the universe into doing something which others think is impossible. I have not found a solution that would solve that riddle in the context it is presented, however it did make me think of something very odd in the genotype to phenotype development of complex organisms. They must get gas and electricity and water too and there is an interesting topological trick that nature has devised.
I can't go into detail of every thing I discover and document it twice. I spend a great deal of time documenting the code for 'antfarmgl' and it serves as a reference, a proof, and a way to observe the algorithms with the explanations so that they can be understood. When the source is integrated enough that I am able to release it as something that needs polish, but has no inherent flaws that would require a complete rewrite, then it should be obvious to anybody who can do a "svn co" of the source. It has Doxygen so it is just following the HTML to see the graphic relationships and dependencies. It also does its own self documentation and graphic demonstration as the videos show.
I added Doxygen to my Linux kernel source and that took a long time to get done and created a 1.5 Gigabyte data base of the inter-relationships. Strangely as it progressed I saw all the things that I know must happen. Some things are not a symbolic match and cannot be coordinated with some tools. The relationship of SMM, Virtual 8086, PM, and Paged virtual memory and cache is a topological nightmare and is clear in my mind, however I would rather not think about a Doxygen that could unscramble those relationships, as it might just tell me 42.
Xulrunner, the coyote is after you. Xulrunner, if it catches you, you're through!
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