Editing Euclidean space (section)

==Definition==
===History of the definition===
Euclidean space was introduced by [[Greek mathematics|ancient Greeks]] as an abstraction of our physical space. Their great innovation, appearing in [[Euclid's Elements|Euclid's ''Elements'']] was to build and ''[[proof (mathematics)|prove]]'' all geometry by starting from a few very basic properties, which are abstracted from the physical world, and cannot be mathematically proved because of the lack of more basic tools. These properties are called [[postulate]]s, or [[axiom]]s in modern language. This way of defining Euclidean space is still in use under the name of [[synthetic geometry]].

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 number]]s were defined in terms of lengths and distances.

Euclidean geometry was not applied in spaces of dimension more than three until the 19th century. [[Ludwig Schläfli]] generalized Euclidean geometry to spaces of dimension {{mvar|n}}, using both synthetic and algebraic methods, and discovered all of the regular [[polytope]]s (higher-dimensional analogues of the [[Platonic solid]]s) that exist in Euclidean spaces of any dimension.{{sfn|Coxeter|1973|}}

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. The introduction of abstract [[vector space]]s allowed their use in defining Euclidean spaces with a purely algebraic definition. This new definition has been shown to be equivalent to the classical definition in terms of geometric axioms. It is this algebraic definition that is now most often used for introducing Euclidean spaces.

===Motivation of the modern definition===
One way to think of the Euclidean plane is as a [[Point set|set of points]] satisfying certain relationships, expressible in terms of distance and angles. For example, there are two fundamental operations (referred to as [[motion (geometry)|motions]]) on the plane. One is [[translation (geometry)|translation]], which means a shifting of the plane so that every point is shifted in the same direction and by the same distance. The other is [[rotation (mathematics)|rotation]] around a fixed point in the plane, in which all points in the plane turn around that fixed point through the same angle. One of the basic tenets of Euclidean geometry is that two [[Figure (geometry)|figures]] (usually<!-- not always --> considered as [[subset]]s) of the plane should be considered equivalent ([[congruence (geometry)|congruent]]) if one can be transformed into the other by some sequence of translations, rotations and [[reflection (mathematics)|reflection]]s (see [[#Euclidean group|below]]).

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. Even when used in [[physics|physical]] theories, Euclidean space is an [[abstraction]] detached from actual physical locations, specific [[frame of reference|reference frames]], measurement instruments, and so on. A purely mathematical definition of Euclidean space also ignores questions of [[unit of length|units of length]] and other [[dimensional analysis|physical dimensions]]: the distance in a "mathematical" space is a [[number]], not something expressed in inches or metres.

The standard way to mathematically define a Euclidean space, as carried out in the remainder of this article, is as a set of points on which a [[real vector space]] [[group action (mathematics)|acts]] – the ''space of translations'' which is equipped with an [[inner product space|inner product]].{{sfn|Solomentsev|2001}} The action of translations makes the space an [[affine space]], and this allows defining lines, planes, subspaces, dimension, and [[parallel (geometry)|parallelism]]. The inner product allows defining distance and angles.

The set <math>\R^n</math> of {{mvar|n}}-tuples of real numbers equipped with the [[dot product]] is a Euclidean space of dimension {{mvar|n}}. Conversely, the choice of a point called the ''origin'' and an [[orthonormal basis]] of the space of translations is equivalent with defining an [[isomorphism]] between a Euclidean space of dimension {{mvar|n}} and <math>\R^n</math> viewed as a Euclidean space.

{{anchor|Standard}}It follows that everything that can be said about a Euclidean space can also be said about <math>\R^n.</math> Therefore, many authors, especially at elementary level, call <math>\R^n</math> the '''''standard Euclidean space''''' of dimension {{mvar|n}},{{sfn|Berger|1987|loc=Section 9.1}} or simply ''the'' Euclidean space of dimension {{mvar|n}}.

[[File:Blender3D BW Grid 256.png|thumb|Origin-free illustration of the Euclidean plane]]
A reason for introducing such an abstract definition of Euclidean spaces, and for working with <math>\mathbb{E}^n</math> instead of <math>\R^n</math> is that it is often preferable to work in a ''coordinate-free'' and ''origin-free'' manner (that is, without choosing a preferred basis and a preferred origin). Another reason is that there is no standard origin nor any standard basis in the physical world.

===Technical definition===

A '''{{vanchor|Euclidean vector space}}''' is a finite-dimensional [[inner product space]] over the [[real number]]s.{{sfn|Berger|1987|loc=Chapter 9}}

A '''Euclidean space''' is an [[affine space]] over the [[real number|reals]] such that the associated vector space is a Euclidean vector space. Euclidean spaces are sometimes called ''Euclidean affine spaces'' to distinguish them from Euclidean vector spaces.{{sfn|Berger|1987|loc=Chapter 9}}

If {{mvar|E}} is a Euclidean space, its associated vector space (Euclidean vector space) is often denoted <math>\overrightarrow E.</math> The ''dimension'' of a Euclidean space is the [[dimension (vector space)|dimension]] of its associated vector space.

The elements of {{mvar|E}} are called ''points'', and are commonly denoted by capital letters. The elements of <math>\overrightarrow E</math> are called ''[[Euclidean vector]]s'' or ''[[free vector]]s''. They are also called ''translations'', although, properly speaking, a [[translation (geometry)|translation]] is the [[geometric transformation]] resulting from the [[group action|action]] of a Euclidean vector on the Euclidean space.

The action of a translation {{mvar|v}} on a point {{mvar|P}} provides a point that is denoted {{math|''P'' + ''v''}}. This action satisfies

<math display="block">P+(v+w)= (P+v)+w.</math>

'''Note:''' The second {{math|+}} in the left-hand side is a vector addition; each other {{math|+}} denotes an action of a vector on a point. This notation is not ambiguous, as, to distinguish between the two meanings of {{math|+}}, 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 {{math|(''P'', ''Q'')}}, there is exactly one [[displacement (geometry)|displacement vector]] {{mvar|v}} such that {{math|1=''P'' + ''v'' = ''Q''}}. This vector {{mvar|v}} is denoted {{math|''Q'' − ''P''}} or <math>\overrightarrow {PQ}\vphantom{\frac){}}.</math>

As previously explained, some of the basic properties of Euclidean spaces result from the structure of affine space. They are described in {{slink||Affine structure}} and its subsections. The properties resulting from the inner product are explained in {{slink||Metric structure}} and its subsections.