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Title: surreal numbers?
Post by: kram1032 on December 18, 2009, 01:05:59 AM

Do you think, Surreal Numbers or their extensions, "surcomplex" numbers could be interesting for some kinds of fractals?
http://en.wikipedia.org/wiki/Surreal_number
http://en.wikipedia.org/wiki/Surcomplex_number

if not they then maybe the unordered set of "Games" (http://en.wikipedia.org/wiki/Surreal_number#Games ) which, as the name implies somewhat, help to analyze certain games like chess or Go...

Of course, as they include standard real numbers, any nice fractal based on real numbers is included in the possible fractals of surreal numbers. Though, maybe there are some fractals exclusive to surreal numbers... (or their what-ever-degree complex extensions)

Maybe also related: Could there be a representation of surcomplex numbers in the hyperbolic plane? - as infinity kind of sits on the unit circle, but if you go beyond, you would come to those surcomplex numbers which define bigger kinds of infinity...

Title: Re: surreal numbers?
Post by: Timeroot on January 09, 2010, 07:26:06 AM

I doubt they would have much application... you can't perform too many interesting operations with them. Even in the hyperbolic plane, all the transfinite numbers would be clustered at the outer edge. Maybe something could be done with the slog base infinity (slog here refers to the inverse operation of tetration), but I think that would require a lot of arguing among googologists first before there is good definition of that... :-\ Until then, maybe you can come up with some definition of your own...? Cool idea, though!

Title: Re: surreal numbers?
Post by: kram1032 on January 11, 2010, 10:56:45 PM

There might be some nice functions which return non-infinite values if you use them on different kinds of infinity... Can't really think of any yet but formulae using inverses could lead to interesting results under circumstances.

I actually wonder what kinds of infinity turn out on, say, $\log(+-\epsilon)$ or how to test if two kinds of infinity are equal to each other...

like: is $\log(+-\epsilon)$ equal to $1\over-\epsilon$ ?

Title: Re: surreal numbers?
Post by: Timeroot on January 12, 2010, 03:42:58 AM

Quote from: kram1032 on January 11, 2010, 10:56:45 PM

There might be some nice functions which return non-infinite values if you use them on different kinds of infinity... Can't really think of any yet but formulae using inverses could lead to interesting results under circumstances.

I actually wonder what kinds of infinity turn out on, say, <Quoted Image Removed> or how to test if two kinds of infinity are equal to each other...

like: is <Quoted Image Removed> equal to <Quoted Image Removed>?

I would think that $\log(-\epsilon) = i \pi + \frac{\epsilon^{-n}}{n}$ , as $n \rightarrow 0$ . But since $\epsilon$ is defined as being infitesimal, we could say $\log(-\epsilon) = i \pi + \frac{\epsilon^{-\epsilon}}{\epsilon} = i \pi + \epsilon^{-(\epsilon + 1)}$ . Interesting result... the logarithim is equivalent to the reciprocal of epsilon raised itself plus one.

Title: Re: surreal numbers?
Post by: kram1032 on January 12, 2010, 03:30:26 PM

that would be a slightly higher order infinitesimal I guess (as |n|<1 raised by |k|>1 would make n even smaller) and the inverse of that which would give a "more infinite" infinity than the standard infinity lim 1/n , n -> 0+

one idea to map infinity and infiniesimal would be with new axes...

first order infinity would be at 1 on the infinity axis, first order infinitesimal at 1 of the infinitesimal axis.

It would definitely be hard to do some kinds of fractals from that...

an other idea is to use games which on one hand are even harder to handle but on the other hand "directly" are related to fractals:
Games with always equal chances and bound rules which still allow a lot of behaviours and are iterated over some counts of steps...
Games like chess, go and co :)

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Fractal Math, Chaos Theory & Research => General Discussion => Topic started by: kram1032 on December 18, 2009, 01:05:59 AM