I see that the original aim of my map: to be able to pinpoint any location on the map with just one piece of information instead of two, was rather randomly far-fetched. What I was imagining, was creating a mapping of all the points in two dimensions to a system in one dimension. Which per se sounds pretty bizarre.
In using something like a simple map, what becomes simple is that I only have a limited surface area to cover. So, if the spiral I use to traverse over the map has a very very tiny pitch, I can approximately pinpoint each point on the map. But that’s still not perfect.
So, to do the same, we can change two things: the nature of the surface on which the map is plotted; and the description of the line. I did the latter in the previous, figurative attempt, by making my “single dimension” along a (well defined spiral). I currently, cannot imagine how to do the former.
I thought of numerous possibilities for changing the nature of the single line I’m using. The best so far, seems to make it some sort of a fractal. That is applicably, second to creating a much more brilliant definition of a spiral, but also much far-sounding. But theoretically, it wouldn’t be as simple as a spiral.
The inspiration to choose them to be a fractal, comes from the fact that most space-filling curves, are fractals (apart from the fact that I love them, of course). And space filling curves do exactly what I want this construct of mine to do – cover all the points in 2 dimensions, using a single continuous 1-D curve.
All I want, is an equation which for a certain iteration, given one parameter, can give me back a unique value, and this equation for different parameters can cover all the points belonging to the above. That’s it. Easy, innit?
More for later..