Re: Why is euclidean geometry so important?
From: Shmuel (Seymour J.) Metz (spamtrap_at_library.lspace.org.invalid)
Date: 09/24/04
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Date: Fri, 24 Sep 2004 13:27:37 -0300
In <200409240053.i8O0r4132276@proapp.mathforum.org>, on 09/24/2004
at 01:01 AM, swamijs@tamu.edu (Swami) said:
>According to Klein's erlanger program, Euclidean geometry(in two
>dimensions) is a study of invariants of the groups of rigid motions
>of the plane. Hence , isnt it true that if we begin with just the
>vector space R^2 and the group of rigid motions(taken as an abstract
>group), we should be able to recover all the interesting invariants
>including the distance between two points (given by
>sqrt((x1-x2)^2+(y1-y2)^2)) and the inner product?
No, because there is an arbitrary scaling factor and the group won't
single out the origin. You recover the properties of the Euclidean
Plane, but not the algebraic properties of R^2.
>I have seen that almost all descriptions of the Euclidean geometry
>start of mysteriously by specifying the usual inner product
Then you've been looking at analytic rather than synthetic approaches.
Lots of things are easier that way, but it's not the Erlanger program.
>I wonder if there is some treatment, where we
>dont start of with the standard inner product, but rather with the
>group of rigid motions or the notion of zero curvature at all points
>or the parallel postulate or any such intuitive idea
That's the synthetic approach.
>finally obtain the usual inner product as an invariant of that
>theory along with all other invariants.
The Euclidean plane is not a vector space. If you want inner products
then you'll have to define them in terms of the Euclidean plane, and
that will be more work than using an analytic approach.
>I find that the usual proofs of the Pythagoras theorem appear as
>miraculous coincidences and not as byproducts of a transparent theory.
EXPN?
>I mean is it possible to start of with say Hyperbolic geometry with no
>knowledge of Euclidean geometry whatsoever and then relate
>NonHyperbolic geometry(say Euclidean geometry) to it , in the same
>manner as we relate non euclidean geometry to euclidean geo.
I'm not sure what you asking. When we construct a model of the
Hyperbolic plane, we start knowing what properties the model should
have. If you wanted to model the Euclidean plane in the Hyperbolic
plane, you'd also start knowing what properties the model would have.
Otherwise how would you prove that it is a model?
IMHO, if you want a synthetic approach then it makes more sense to
start with Projective Geometry and derive all three (Elliptic,
Hyperbolic and Euclidean) from that.
-- Shmuel (Seymour J.) Metz, SysProg and JOAT <http://patriot.net/~shmuel> Unsolicited bulk E-mail subject to legal action. I reserve the right to publicly post or ridicule any abusive E-mail. Reply to domain Patriot dot net user shmuel+news to contact me. Do not reply to spamtrap@library.lspace.org
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