15 Facts About Plane geometry

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

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

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

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

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

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

An application of Euclidean solid Plane geometry is the determination of packing arrangements, such as the problem of finding the most efficient packing of spheres in n dimensions.

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

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

Plane 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 Plane geometry called affine Plane 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 .

Century's most significant development in Plane geometry occurred when, around 1830, Janos Bolyai and Nikolai Ivanovich Lobachevsky separately published work on non-Euclidean Plane geometry, in which the parallel postulate is not valid.

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

Plane 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.