Mathematics
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Euclidean geometry
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Euclidean geometry
Euclidean geometry is the geometry of space described by the system of axioms first stated systematically (though not sufficiently rigorous) in the Elements of Euclid.
The space of Euclidean geometry is usually described as a set of objects of three kinds, called “points,” “lines” and “planes”; the relations between them are incidence, order (“lying between”), congruence (or the concept of a motion), and continuity.
The parallel axiom (fifth postulate) occupies a special place in the axiomatics of Euclidean geometry. D. Hilbert gave the first sufficiently precise axiomatization of Euclidean geometry (see Hilbert system of axioms).
There are modifications of Hilbert’s axiom system as well as other versions of the axiomatics of Euclidean geometry. For example, in vector axiomatics, the concept of a vector is taken as one of the basic concepts. On the other hand, the relation of symmetry may be taken as a basis for the axiomatics of plane Euclidean geometry.
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