Re: why is euclidean geometry so important?
From: Arnold Neumaier (Arnold.Neumaier_at_univie.ac.at)
Date: 09/27/04
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Date: Mon, 27 Sep 2004 08:32:10 +0000 (UTC)
Swami wrote:
> 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?
If you assume the group and look for algebraic invariants of degree d
in k variables for small d and k you find none for d=1 and k=1, but
a unique one for d=k=2 (up to a constant factor).
Does this derivation satisfy you?
Arnold Neumaier
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