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> >Lefty wrote:
> >
> >> Einstein's Relativity is interesting because it uses independent frames
of
> >> reference w/respect to motion. How can you do this with numbers ? How
to
> >> think of a space which has no geometric origin, but still has all the
nice
> >> properties of R(n) ?
> >
> >Euclidean space?
> >
> >Unless I am mistaken, Euclidean space has no preferred origin.
>
> You are not mistaken, but many people are, and persistently make
> the error of taste and judgment of identifying the vectorspace
> R^n (or R(n) as Lefty writes it) with the affine space E_n, which
> they then call Euclidean space. Of course Euclid's lines, planes,
> and space only become affine spaces (of dimensions 1, 2, and 3,
> respectively) upon interpretation--but they can only become
> vectorspaces upon *mis*interpretation.
So Euclidean space has no origin ? Just a well ordering and a metric ? Sure
about that ?
According to
http://en.wikipedia.org/wiki/Euclidean_space
Real coordinate space together with dot product and the associated norm and
metric is called Euclidean space often denoted by En. (Many authors refer to
Rn itself as Euclidean space, with the Euclidean structure being
understood). The Euclidean structure on En gives it the structure of an
inner product space (in fact a Hilbert space), a normed vector space, and a
metric space.
The contrast I'm trying to draw is the difference between En, (or Rn), and
the physical universe. The distinction between En and Rn is perhaps somewhat
less signifigant in this effort - but certainly worth noting.
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