Geometry

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.

Euclid’s postulates

  1. A straight line segment can be drawn joining any two points.
  2. Any straight line segment can be extended indefinitely in a straight line.
  3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center.
  4. All right angles are congruent.
  5. If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough. This postulate is equivalent to what is known as the parallel postulate.

Euclid’s fifth postulate cannot be proven as a theorem, although this was attempted by many people. Euclid himself used only the first four postulates (“absolute geometry”) for the first 28 propositions of the Elements, but was forced to invoke the parallel postulate on the 29th.

In 1823, Janos Bolyai and Nicolai Lobachevsky independently realized that entirely self-consistent “non-Euclidean geometries” could be created in which the parallel postulate did not hold. (Gauss had also discovered but suppressed the existence of non-Euclidean geometries.)

Non-Euclidean geometry

Non-Euclidean geometry means, in the literal sense — all geometric systems distinct from Euclidean geometry; usually, however, the term “non-Euclidean geometries” is reserved for geometric systems (distinct from Euclidean geometry) in which the motion of figures is defined, and this with the same degree of freedom as in Euclidean geometry.

The degree of freedom of motion of figures in the Euclidean plane is characterized by the condition that every figure can be moved, without changing the distances between its points, in such a way that any selected point of the figure can be made to occupy a previously-designated position; moreover, every figure can be rotated about any of its points.

In the Euclidean three-dimensional space, every figure can be moved in such a way that any selected point of the figure will occupy any prescribed position; besides, every figure can be rotated about any axis through any of its points.

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