What's the minimum amount of extra "structure" do we need to add to the general concept of an affine space to get Euclidean space? That includes the concepts of angle and distance, in which we can describe things like polygons and circles, and in which we can derive all of the familiar theorems of Euclidean geometry.
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From Wikipedia: "Euclidean Space is an affine space over the reals, equipped with a metric." https://en.wikipedia.org/wiki/Affine_space – TomGrubb Dec 23 '15 at 23:41
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How does that reproduce the notion of angle, which is absent in general affine spaces? – user300646 Dec 23 '15 at 23:43
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Euclidean space and the space studied in Euclid's Elements are not actually the same: there is no need to define the whole of the real numbers for the Elements, for example. – Chappers Dec 23 '15 at 23:44
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The axioms of Euclid's elements are incomplete, so technically "the space of Euclid's elements" doesn't make sense. There are more modern axiomatic systems that better nail down the space that Euclid had in mind, though, most notably due to Hilbert. – Arthur Dec 23 '15 at 23:49
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@user300646 If you have a metric, you can define a norm and vice-versa. If this norm satisfies the parallelogram law, you can use it to define an inner product, which also gives you angles. See http://math.stackexchange.com/a/38591/265466 for a discussion of the connection between metric, norm, and inner product. – amd Dec 24 '15 at 03:22
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In VI Arnold's classical mechanics page 5 he defines a Euclidean space as an affine space with a norm derived from the inner product on a real vector space. http://users.uoa.gr/~pjioannou/mech1/READING/Arnold_Clas_Mech_ch_1_2.pdf
Bernard W
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He defines a Euclidean structure on $R^n$ and then uses that to define a Euclidean space $E^n$ below. – Bernard W Dec 23 '15 at 23:53