Posted by Benjamin Johnston on 2000-09-28
> Anyway, I just finished with a function that would map N -> NxN. Take a
> look:
>
> w(n) -> floor((-1+sqrt(1+8*n))/2)
> binomial(n,r) -> n!/(r!(n-r!))
> f(n) ->
> [n - binomial(w(n)+1,2), w(n) - n + binomial(w(n)+1,2)]
>
> The basic premise that allows this order-1 function to work is that the
> "slots" or diagonal runs of the ordered pairs can be computed with w(n)
> and the binomial function computing levels of permutation within that
> "slot".
> For those that don't really follow what is going on here. Basically what
> the above functions allows somebody to do is to use simple integers to
> actually represent a location in 2-space. Thus, especially with a computer
> science use. Think of the space that could be saved simply because an
> ordered pair is not needed to be kept in memory but a simple integer. :)
Hehehe.
(I hope the emoticon at the end of that sentence does mean you are kidding)
> The interesting thing about this is that N can also map to NxNxN (i.e
> 3-space) or even NxNxNxN (3-space with time perhaps :). But the order one
> function to do this is past my abilities. :) But is is cool to know....
I'm probably demonstrating here that I don't know what an order-1 function
is but how about;
g(n) -> [x, y, z],
where f(n) = [x,w] and f(w) = [y, z]
It is just then a matter of expanding and simplifying, right?
> P.S. While I am on a roll I just have to mention this. R (the set of all
> reals) can be mapped to RxRxR in just the same manner that N can be mapped
> to NxNxN. This means that a single real number could be use to represent
> all possible locations within the known physical existance. Also, since R
> is uncountably infinite just the numbers 0 < x <= 1 could be used to
> represent free space. I don't know, prehaps it is just me but I find these
> ideas to be pretty thought producing!
No, its not just you... though I hadn't really thought of that before.... it
*is* an interesting thought.
-Benjamin Johnston
s355171@xxxxxxx.xx.xxx.xx
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