Re: Transcendental Dimensions

gsax wrote:
> Thanks for this example,...I shall try to see if I can come up with
> any other objects with transcendental dimensions..

In some sense, "most" objects with non-integer dimension can be
expected to have transcendental dimension.

Dimensions of fractals often have the form (log p / log q) for some
integers p and q. If the ratio isn't "obviously" a rational then it's
irrational and hence transcendental. If log p / log q is a rational
then p^n = q^m with m > 1, which tends to be pretty obvious. That's a
fairly uncommon relationship, requiring that p and q be different
powers of a common base.

If that relationship doesn't hold, then log p / log q is
transcendental.


> Also I read somewhere that most of the numbers are
> transcendental,... it is funny then that it takes so much trouble
> to produce their examples..

It's easy to produce unlimited examples of transcendental numbers.
It's also easy to come up with unlimited examples of numbers of
unknown transcendentality.


- Tim
.

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