16 Facts About Euclidean geometry

Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the Elements.

Elements begins with plane Euclidean geometry, still taught in secondary school as the first axiomatic system and the first examples of mathematical proofs.

Today many other self-consistent non-Euclidean geometry geometries are known, the first ones having been discovered in the early 19th century.

An implication of Albert Einstein's theory of general relativity is that physical space itself is not Euclidean geometry, and Euclidean geometry space is a good approximation for it only over short distances .

Euclidean geometry is an example of synthetic geometry, in that it proceeds logically from axioms describing basic properties of geometric objects such as points and lines, to propositions about those objects.

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Euclidean geometry is an axiomatic system, in which all theorems are derived from a small number of simple axioms.

Euclidean geometry allows the method of superposition, in which a figure is transferred to another point in space.

Euclidean geometry has two fundamental types of measurements: angle and distance.

The fundamental types of measurements in Euclidean geometry are distances and angles, both of which can be measured directly by a surveyor.

Geometric optics uses Euclidean geometry to analyze the focusing of light by lenses and mirrors.

Some classical construction problems of Euclidean geometry are impossible using compass and straightedge, but can be solved using origami.

Euclidean geometry proved equations for the volumes and areas of various figures in two and three dimensions, and enunciated the Archimedean property of finite numbers.

Euler discussed a generalization of Euclidean geometry called affine geometry, which retains the fifth postulate unmodified while weakening postulates three and four in a way that eliminates the notions of angle and of equality of length of line segments in general while retaining the notions of parallelism as an equivalence relation between lines, and equality of length of parallel line segments .

Since non-Euclidean geometry is provably relatively consistent with Euclidean geometry, the parallel postulate cannot be proved from the other postulates.

Euclidean geometry's axioms do not guarantee that the circles actually intersect, because they do not assert the geometrical property of continuity, which in Cartesian terms is equivalent to the completeness property of the real numbers.

Euclid frequently used the method of proof by contradiction, and therefore the traditional presentation of Euclidean geometry assumes classical logic, in which every proposition is either true or false, i e, for any proposition P, the proposition "P or not P" is automatically true.