Euclidean space
Their work was collected by the ancient Greek mathematician Euclid in his Elements,[2] with the great innovation of proving all properties of the space as theorems, by starting from a few fundamental properties, called postulates, which either were considered as evident (for example, there is exactly one straight line passing through two points), or seemed impossible to prove (parallel postulate).
After the introduction at the end of the 19th century of non-Euclidean geometries, the old postulates were re-formalized to define Euclidean spaces through axiomatic theory.
In 1637, René Descartes introduced Cartesian coordinates, and showed that these allow reducing geometric problems to algebraic computations with numbers.
This reduction of geometry to algebra was a major change in point of view, as, until then, the real numbers were defined in terms of lengths and distances.
Ludwig Schläfli generalized Euclidean geometry to spaces of dimension n, using both synthetic and algebraic methods, and discovered all of the regular polytopes (higher-dimensional analogues of the Platonic solids) that exist in Euclidean spaces of any dimension.
[4] Despite the wide use of Descartes' approach, which was called analytic geometry, the definition of Euclidean space remained unchanged until the end of 19th century.
One way to think of the Euclidean plane is as a set of points satisfying certain relationships, expressible in terms of distance and angles.
In order to make all of this mathematically precise, the theory must clearly define what is a Euclidean space, and the related notions of distance, angle, translation, and rotation.
This notation is not ambiguous, as, to distinguish between the two meanings of +, it suffices to look at the nature of its left argument.
The fact that the action is free and transitive means that, for every pair of points (P, Q), there is exactly one displacement vector v such that P + v = Q.
More precisely, given a Euclidean space E of dimension n, the choice of a point, called an origin and an orthonormal basis of
They are called affine properties and include the concepts of lines, subspaces, and parallelism, which are detailed in next subsections.
The inner product and the norm allows expressing and proving metric and topological properties of Euclidean geometry.
Two segments AB and AC that share a common endpoint A are perpendicular or form a right angle if the vectors
[7] Given a Euclidean space E, a Cartesian frame is a set of data consisting of an orthonormal basis of
An affine basis of a Euclidean space of dimension n is a set of n + 1 points that are not contained in a hyperplane.
That is, a closed subset of a Euclidean space is compact if it is bounded (that is, contained in a ball).
In reality, Euclid did not define formally the space, because it was thought as a description of the physical world that exists independently of human mind.
The presentation of Euclidean spaces given in this article, is essentially issued from his Erlangen program, with the emphasis given on the groups of translations and isometries.
On the other hand, David Hilbert proposed a set of axioms, inspired by Euclid's postulates.
In Geometric Algebra, Emil Artin has proved that all these definitions of a Euclidean space are equivalent.
One must thus prove that this length satisfies properties that characterize nonnegative real numbers.
Since the ancient Greeks, Euclidean space has been used for modeling shapes in the physical world.
Most non-Euclidean geometries can be modeled by a manifold, and embedded in a Euclidean space of higher dimension.
It is common to represent in a Euclidean space mathematical objects that are a priori not of a geometrical nature.
For example, a circle and a line have always two intersection points (possibly not distinct) in the complex affine space.
Non-Euclidean geometry refers usually to geometrical spaces where the parallel postulate is false.
Their introduction in the second half of 19th century, and the proof that their theory is consistent (if Euclidean geometry is not contradictory) is one of the paradoxes that are at the origin of the foundational crisis in mathematics of the beginning of 20th century, and motivated the systematization of axiomatic theories in mathematics.
Generally, straight lines do not exist in a Riemannian manifold, but their role is played by geodesics, which are the "shortest paths" between two points.
In this case, geodesics are arcs of great circle, which are called orthodromes in the context of navigation.
