
Re: Pictures of infinity
Posted:
Aug 2, 2006 3:54 PM


John Baez wrote: > If any hacker out there creates nice pictures of omega^omega > and/or epsilon_0, I'll put them on my website. If you do > both and I think they're really nice, I'll also give you a > signed copy of the new (corrected) version of "Gauge Fields, > Knots and Gravity", as soon I can buy it from World Scientific > (I got my copy a while ago, so it should be coming out soon.) > Or, if you prefer, some other book of comparable price.
Just an idea for how it might be done with omega^omega:
First, here's a nice 11 orderpreserving map f from omega^omega onto a subset S of the dyadic rationals: map 0 to 0, and given the ordinal
x = a_n*omega^n + a_(n1)*omega^(n1) + ... + a_0
where a_i < omega and a_n>0,
write down a (binary) point, followed by n 1s, followed by a 0, followed by (a_n)1 1s, followed by a 0, followed by a_(n1) 1s, followed by a 0, followed by a_(n2) 1s, followed by a 0, followed by ... a_0 1s.
Call this number f(x).
Then, find some way to represent the 'visibility' of x (the heights of the lines): for example,
v(x) = c^n + k*c^a_n + k^2*c^a_(n1) + k^3*c^a_(n2) + ...
where 0<k,c<1 (there may be prettier ways to do it).
Then, plot v(x) against f(x), where x ranges over some subset of omega^omega constructed so that it includes all x with v(x) > delta > 0 (this is where the hacking comes in and I give up!)
Could binary expansions be used in a similar (but more complex) way to represent epsilon_0?

