Editing Euclidean geometry (section)

==As a description of the structure of space==
Euclid believed that his [[axioms]] were self-evident statements about physical reality. Euclid's proofs depend upon assumptions perhaps not obvious in Euclid's fundamental axioms,<ref name=Trudeau>{{cite book |title=The Non-Euclidean Revolution|author=Richard J. Trudeau |pages=39 ''ff'' |chapter-url=https://books.google.com/books?id=YRB4VBCLB3IC&pg=PA39 |chapter=Euclid's axioms |publisher= Birkhäuser |year=2008 |isbn=978-0-8176-4782-7}}</ref> in particular that certain movements of figures do not change their geometrical properties such as the lengths of sides and interior angles, the so-called ''Euclidean motions'', which include translations, reflections and rotations of figures.<ref name=Euclidean_Motion>

See, for example: {{cite book |title=Shape analysis and classification: theory and practice |author1=Luciano da Fontoura Costa |author2=Roberto Marcondes Cesar |page=314 |url=https://books.google.com/books?id=x_wiWedtc0cC&pg=PA314 |isbn=0-8493-3493-4 |year=2001 |publisher=CRC Press}} and {{cite book |title=Computational Line Geometry |author1=Helmut Pottmann |author2=Johannes Wallner |url=https://books.google.com/books?id=3Mk2JIJKsGwC&pg=PA60 |page=60 |isbn=978-3-642-04017-7 |year=2010 |publisher=Springer}} The ''group of motions'' underlie the metric notions of geometry. See {{cite book |title=Elementary Mathematics from an Advanced Standpoint: Geometry |author=Felix Klein |url=https://books.google.com/books?id=fj-ryrSBuxAC&pg=PA167 |page=167 |isbn=0-486-43481-8 |publisher=Courier Dover |year=2004 |edition=Reprint of 1939 Macmillan Company}}</ref> Taken as a physical description of space, postulate 2 (extending a line) asserts that space does not have holes or boundaries; postulate 4 (equality of right angles) says that space is [[isotropic]] and figures may be moved to any location while maintaining [[congruence (geometry)|congruence]]; and postulate 5 (the [[parallel postulate]]) that space is flat (has no [[intrinsic curvature]]).<ref name=Penrose>{{cite book |author=Roger Penrose |title= The Road to Reality: A Complete Guide to the Laws of the Universe |year=2007 |page= 29 |url=https://books.google.com/books?id=coahAAAACAAJ&q=editions:cYahAAAACAAJ |isbn=978-0-679-77631-4 |publisher=Vintage Books}}</ref>

As discussed above, [[Albert Einstein]]'s [[theory of relativity]] significantly modifies this view.

The ambiguous character of the axioms as originally formulated by Euclid makes it possible for different commentators to disagree about some of their other implications for the structure of space, such as whether or not it is infinite<ref name="Heath, p.&nbsp;200">Heath, p.&nbsp;200.</ref> (see below) and what its [[topology]] is. Modern, more rigorous reformulations of the system<ref>e.g., Tarski (1951).</ref> typically aim for a cleaner separation of these issues. Interpreting Euclid's axioms in the spirit of this more modern approach, axioms 1–4 are consistent with either infinite or finite space (as in [[elliptic geometry]]), and all five axioms are consistent with a variety of topologies (e.g., a plane, a cylinder, or a [[torus]] for two-dimensional Euclidean geometry).