The Mystic Ranger

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..

Update: This ques­tion brings up a rather old doubt of mine — can I call an equal pitched spi­ral, a frac­tal? Or does it have too “sim­ple a definition”?