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Topic: This Week's Finds in Mathematical Physics (Week 236)
Replies: 29   Last Post: Aug 24, 2006 9:00 AM

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 Alec Edgington Posts: 9 Registered: 3/21/06
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 1-1 order-preserving 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_(n-1)*omega^(n-1) + ... + 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_(n-1) 1s, followed by a 0, followed by
a_(n-2) 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_(n-1) + k^3*c^a_(n-2) + ...

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?