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Shmuel (Seymour J.) Metz wrote:
> In <1158075306.847084.259440@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>, on
> 09/12/2006
> at 08:35 AM, dgoldsmith_89@xxxxxxxxxxxxxxxx said:
>
> >Hi! I can't remember (and can't seem to easily prove, disprove, or
> >find in the lit): in Euclidean geometry, are the Parallel Postulate
> >and the similarity of all circles (and thus the constancy of pi)
> >equivalent?
>
> All but the last part. Pi is constant, period. What is lost if the 5th
> Postulate does not hold is the relevance of 2 Pi R and of Pi R^2.
Right, the limit of the series 4 - 4/3 + 4/5 - 4/7 + 4/9 - ... is a
constant, but you've missed what I'm really interested in, which _is_
the (Euclidean) geometric property of this number and whether there are
_any_ non-Euclidean metric geometries (esp. non-Pythogorean
finite-dimensional metric spaces) with _analogous_ global constantants
(i.e., not just any global constants - I know, e.g., that there are
non-Euclidean geometries of globally constant curvature different from
0). In more formal terms, are there any non-Pythagorean metric, D(x,y)
!= <x-y,x-y>^1/2, spaces such that, considering _all_ the point-subsets
C of the space, {C: there exists an element P of the space and a
positive real number r such that D(C,P) = r}, i.e., all the "circles"
in the space, the ratio of the (minimum total) arclength of C (using
the metric to define the arclength differential, of course) to r is the
same for all such "circles"? In less formal terms, are there any
non-Euclidean geometries in which the ratio of the "circumference" of
any "circle" to its "radius" is (globally) constant? If the answer is
"No" for constant (i.e., non-spatially varying) metrics, what about for
(spatially) variable metrics? Also, if the answer is "Yes," are the
any such spaces for which the ratio of the "area" of the "interior" of
the "circle" (assuming the interior to be measurable) to the square of
r is half of the "circumfrential" ratio? Etc., etc. Have these
question even been investigated in the literature? Thanks!
DG
>
> --
> Shmuel (Seymour J.) Metz, SysProg and JOAT <http://patriot.net/~shmuel>
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