TheInfoList

Euclidean space is the fundamental space of classical geometry. Originally, it was the
three-dimensional space Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called parameters) are required to determine the position of an element (i.e., point). This is the informal meaning of the ...
of
Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandria Alexandria ( or ; ar, الإسكندرية ; arz, اسكندرية ; Coptic Coptic may refer to: Afro-Asia * Copts, an ethnoreligious group mainly in the area of modern ...
, but in modern
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
there are Euclidean spaces of any nonnegative integer
dimension In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular s ...
, including the three-dimensional space and the ''Euclidean plane'' (dimension two). It was introduced by the
Ancient Greek Ancient Greek includes the forms of the Greek language Greek ( el, label=Modern Greek Modern Greek (, , or , ''Kiní Neoellinikí Glóssa''), generally referred to by speakers simply as Greek (, ), refers collectively to the diale ...
mathematician
Euclid of Alexandria Euclid (; grc, Wikt:Εὐκλείδης, Εὐκλείδης – ''Eukleídēs'', ; floruit, fl. 300 BC), sometimes called Euclid of Alexandria to distinguish him from Euclid of Megara, was a Greek mathematics, Greek mathematician, often refe ...

, and the qualifier ''Euclidean'' is used to distinguish it from other spaces that were later discovered in
physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical scie ...

and modern mathematics. Ancient Greek geometers introduced Euclidean space for modeling the
physical universe In religion Religion is a social system, social-cultural system of designated religious behaviour, behaviors and practices, morality, morals, beliefs, worldviews, religious text, texts, shrine, sanctified places, prophecy, prophecies, ethics ...
. Their great innovation was to ''
prove Proof may refer to: * Proof (truth), argument or sufficient evidence for the truth of a proposition * Alcohol proof, a measure of an alcoholic drink's strength Formal sciences * Formal proof, a construct in proof theory * Mathematical proof, a co ...
'' all properties of the space as
theorem In mathematics, a theorem is a statement (logic), statement that has been Mathematical proof, proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the the ...
s by starting from a few fundamental properties, called ''
postulate An axiom, postulate or assumption is a statement that is taken to be truth, true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Greek ''axíōma'' () 'that which is thought worthy or fit' or ...
s'', which either were considered as evident (for example, there is exactly one
straight line 290px, A representation of one line segment. In geometry, the notion of line or straight line was introduced by ancient mathematicians to represent straight objects (i.e., having no curvature In mathematics, curvature is any of several str ...

passing through two points), or seemed impossible to prove (
parallel postulate In geometry, the parallel postulate, also called Euclid's fifth postulate because it is the fifth postulate in Euclid's Elements, Euclid's ''Elements'', is a distinctive axiom in Euclidean geometry. It states that, in two-dimensional geometry: ' ...

). After the introduction at the end of 19th century of
non-Euclidean geometries In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
, the old postulates were re-formalized to define Euclidean spaces through
axiomatic theory In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
. Another definition of Euclidean spaces by means of
vector space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
s and
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrix (mat ...
has been shown to be equivalent to the axiomatic definition. It is this definition that is more commonly used in modern mathematics, and detailed in this article. In all definitions, Euclidean spaces consist of points, which are defined only by the properties that they must have for forming a Euclidean space. There is essentially only one Euclidean space of each dimension; that is, all Euclidean spaces of a given dimension are
isomorphic In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

. Therefore, in many cases, it is possible to work with a specific Euclidean space, which is generally the real -space $\R^n,$ equipped with the
dot product In mathematics, the dot product or scalar productThe term ''scalar product'' is often also used more generally to mean a symmetric bilinear form, for example for a pseudo-Euclidean space. is an algebraic operation that takes two equal-length seque ...
. An isomorphism from a Euclidean space to $\R^n$ associates with each point an -tuple of
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s which locate that point in the Euclidean space and are called the ''
Cartesian coordinates A Cartesian coordinate system (, ) in a plane Plane or planes may refer to: * Airplane or aeroplane or informally plane, a powered, fixed-wing aircraft Arts, entertainment and media *Plane (Dungeons & Dragons), Plane (''Dungeons & Dragons'') ...

'' of that point.

# Definition

## History of the definition

Euclidean space was introduced by
ancient Greeks Ancient Greece ( el, Ἑλλάς, Hellás) was a civilization belonging to a period of History of Greece, Greek history from the Greek Dark Ages of the 12th–9th centuries BC to the end of Classical Antiquity, antiquity ( AD 600). This era was ...
as an abstraction of our physical space. Their great innovation, appearing in Euclid's ''Elements'' was to build and ''
prove Proof may refer to: * Proof (truth), argument or sufficient evidence for the truth of a proposition * Alcohol proof, a measure of an alcoholic drink's strength Formal sciences * Formal proof, a construct in proof theory * Mathematical proof, a co ...
'' 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 An axiom, postulate or assumption is a statement that is taken to be truth, true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Greek ''axíōma'' () 'that which is thought worthy or fit' or ...
s, or
axiom An axiom, postulate or assumption is a statement that is taken to be truth, true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Greek ''axíōma'' () 'that which is thought worthy or fit' o ...

s in modern language. This way of defining Euclidean space is still in use under the name of
synthetic geometry Synthetic geometry (sometimes referred to as axiomatic geometry or even pure geometry) is the study of geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is conce ...
. In 1637,
René Descartes René Descartes ( or ; ; Latinized Latinisation or Latinization can refer to: * Latinisation of names, the practice of rendering a non-Latin name in a Latin style * Latinisation in the Soviet Union, the campaign in the USSR during the 1920s ...

introduced
Cartesian coordinates A Cartesian coordinate system (, ) in a plane Plane or planes may refer to: * Airplane or aeroplane or informally plane, a powered, fixed-wing aircraft Arts, entertainment and media *Plane (Dungeons & Dragons), Plane (''Dungeons & Dragons'') ...

and showed that this allows reducing geometric problems to algebraic computations with numbers. This reduction of geometry to
algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In its most ge ...

was a major change of point of view, as, until then, the
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s were defined in terms of lengths and distances. Euclidean geometry was not applied in spaces of more than three dimensions until the 19th century. generalized Euclidean geometry to spaces of ''n'' dimensions using both synthetic and algebraic methods, and discovered all of the regular
polytope In elementary geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relativ ...
s (higher-dimensional analogues of the
Platonic solid In three-dimensional space, a Platonic solid is a Regular polyhedron, regular, Convex set, convex polyhedron. It is constructed by Congruence (geometry), congruent (identical in shape and size), regular polygon, regular (all angles equal and all sid ...
s) that exist in Euclidean spaces of any number of dimensions. Despite the wide use of Descartes' approach, which was called
analytic geometry In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measur ...
, the definition of Euclidean space remained unchanged until the end of 19th century. The introduction of abstract
vector space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
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 set of
point Point or points may refer to: Places * Point, LewisImage:Point Western Isles NASA World Wind.png, Satellite image of Point Point ( gd, An Rubha), also known as the Eye Peninsula, is a peninsula some 11 km long in the Outer Hebrides (or Western I ...
s satisfying certain relationships, expressible in terms of distance and angles. For example, there are two fundamental operations (referred to as
motions 300px, Motion involves a change in position In physics, motion is the phenomenon in which an object changes its position over time. Motion is mathematically described in terms of displacement, distance Distance is a numerical measurement of ...
) on the plane. One is
translation Translation is the communication of the meaning Meaning most commonly refers to: * Meaning (linguistics), meaning which is communicated through the use of language * Meaning (philosophy), definition, elements, and types of meaning discusse ...
, 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 A rotation is a circular movement of an object around a center (or point) of rotation. The plane (geometry), geometric plane along which the rotation occurs is called the ''rotation plane'', and the imaginary line extending from the center an ...
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 figures (usually considered as
subset In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

s) of the plane should be considered equivalent (
congruent Congruence may refer to: Mathematics * Congruence (geometry), being the same size and shape * Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure * In modu ...
) if one can be transformed into the other by some sequence of translations, rotations and
reflectionReflection or reflexion may refer to: Philosophy * Self-reflection Science * Reflection (physics), a common wave phenomenon ** Specular reflection, reflection from a smooth surface *** Mirror image, a reflection in a mirror or in water ** Signal r ...
s (see
below Below may refer to: *Earth *Ground (disambiguation) *Soil *Floor *Bottom (disambiguation) *Less than *Temperatures below freezing *Hell or underworld People with the surname *Fred Below (1926–1988), American blues drummer *Fritz von Below (1853 ...
). 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
physical Physical may refer to: *Physical examination, a regular overall check-up with a doctor *Physical (album), ''Physical'' (album), a 1981 album by Olivia Newton-John **Physical (Olivia Newton-John song), "Physical" (Olivia Newton-John song) *Physical ( ...

theories, Euclidean space is an
abstraction Abstraction in its main sense is a conceptual process where general rules Rule or ruling may refer to: Human activity * The exercise of political Politics (from , ) is the set of activities that are associated with Decision-making, mak ...

detached from actual physical locations, specific
reference frames In physics, a frame of reference (or reference frame) consists of an abstract coordinate system and the set of physical reference points that uniquely fix (locate and orient) the coordinate system and standardize measurements within that frame . ...

, measurement instruments, and so on. A purely mathematical definition of Euclidean space also ignores questions of
units of length A unit of length refers to any arbitrarily chosen and accepted reference standard for measurement of length. The most common units in modern use are the metric system, metric units, used in every country globally. In the United States the U.S. ...
and other physical dimensions: the distance in a "mathematical" space is a
number A number is a mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and with which one may do deduct ...

, 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 to define a Euclidean space as a set of points on which
acts The Acts of the Apostles ( grc-koi, Πράξεις Ἀποστόλων, ''Práxeis Apostólōn''; la, Actūs Apostolōrum), often referred to simply as Acts, or formally the Book of Acts, is the fifth book of the New Testament The New Te ...
a
real vector space Real may refer to: * Reality, the state of things as they exist, rather than as they may appear or may be thought to be Currencies * Brazilian real (R\$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish col ...
, the ''space of translations'' which is equipped with an
inner product In mathematics, an inner product space or a Hausdorff space, Hausdorff pre-Hilbert space is a vector space with a binary operation called an inner product. This operation associates each pair of vectors in the space with a Scalar (mathematics), ...
. The action of translations makes the space an
affine space In mathematics, an affine space is a geometric Structure (mathematics), structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping on ...
, and this allows defining lines, planes, subspaces, dimension, and parallelism. The inner product allows defining distance and angles. The set $\R^n$ of -tuples of real numbers equipped with the
dot product In mathematics, the dot product or scalar productThe term ''scalar product'' is often also used more generally to mean a symmetric bilinear form, for example for a pseudo-Euclidean space. is an algebraic operation that takes two equal-length seque ...
is a Euclidean space of dimension . Conversely, the choice of a point called the ''origin'' and an
orthonormal basisIn linear algebra, two vectors in an inner product space are orthonormal if they are orthogonal (or perpendicular along a line) unit vectors. A set of vectors form an orthonormal set if all vectors in the set are mutually orthogonal and all of u ...
of the space of translations is equivalent with defining an
isomorphism In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

between a Euclidean space of dimension and $\R^n$ viewed as a Euclidean space. It follows that everything that can be said about a Euclidean space can also be said about $\R^n.$ Therefore, many authors, specially at elementary level, call $\R^n$ the ''standard Euclidean space'' of dimension , or simply ''the'' Euclidean space of dimension . A reason for introducing such an abstract definition of Euclidean spaces, and for working with it instead of $\R^n$ 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 origin nor any basis in the physical world.

## Technical definition

A is a finite-dimensional
inner product space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
over the
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s. A Euclidean space is an
affine space In mathematics, an affine space is a geometric Structure (mathematics), structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping on ...
over the reals such that the associated vector space is a Euclidean vector space. Euclidean spaces are sometimes called ''Euclidean affine spaces'' for distinguishing them from Euclidean vector spaces. If is a Euclidean space, its associated vector space is often denoted $\overrightarrow E.$ The ''dimension'' of a Euclidean space is the
dimension In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular s ...
of its associated vector space. The elements of are called ''points'' and are commonly denoted by capital letters. The elements of $\overrightarrow E$ are called ''
Euclidean vector In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s'' or ''
free vector In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
s''. They are also called ''translations'', although, properly speaking, a
translation Translation is the communication of the meaning Meaning most commonly refers to: * Meaning (linguistics), meaning which is communicated through the use of language * Meaning (philosophy), definition, elements, and types of meaning discusse ...
is the
geometric transformation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
resulting of the
action ACTION is a bus operator in , Australia owned by the . History On 19 July 1926, the commenced operating public bus services between Eastlake (now ) in the south and in the north. The service was first known as Canberra City Omnibus Se ...
of a Euclidean vector on the Euclidean space. The action of a translation on a point provides a point that is denoted . This action satisfies $P+(v+w)= (P+v)+w.$ (The second in the left-hand side is a vector addition; all other denote an action of a vector on a point. This notation is not ambiguous, as, for distinguishing between the two meanings of , it suffices to look on the nature of its left argument.) The fact that the action is free and transitive means that for every pair of points there is exactly one vector such that . This vector is denoted or $\overrightarrow .$ As previously explained, some of the basic properties of Euclidean spaces result of the structure of affine space. They are described in and its subsections. The properties resulting from the inner product are explained in and its subsections.

# Prototypical examples

For any vector space, the addition acts freely and transitively on the vector space itself. Thus a Euclidean vector space can be viewed as a Euclidean space that has itself as associated vector space. A typical case of Euclidean vector space is $\R^n$ viewed as a vector space equipped with the
dot product In mathematics, the dot product or scalar productThe term ''scalar product'' is often also used more generally to mean a symmetric bilinear form, for example for a pseudo-Euclidean space. is an algebraic operation that takes two equal-length seque ...
as an
inner product In mathematics, an inner product space or a Hausdorff space, Hausdorff pre-Hilbert space is a vector space with a binary operation called an inner product. This operation associates each pair of vectors in the space with a Scalar (mathematics), ...
. The importance of this particular example of Euclidean space lies in the fact that every Euclidean space is
isomorphic In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

to it. More precisely, given a Euclidean space of dimension , the choice of a point, called an ''origin'' and an
orthonormal basisIn linear algebra, two vectors in an inner product space are orthonormal if they are orthogonal (or perpendicular along a line) unit vectors. A set of vectors form an orthonormal set if all vectors in the set are mutually orthogonal and all of u ...
of $\overrightarrow E$ defines an isomorphism of Euclidean spaces from to $\R^n.$ As every Euclidean space of dimension is isomorphic to it, the Euclidean space $\R^n$ is sometimes called the ''standard Euclidean space'' of dimension .

# Affine structure

Some basic properties of Euclidean spaces depend only of the fact that a Euclidean space is an
affine space In mathematics, an affine space is a geometric Structure (mathematics), structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping on ...
. They are called affine properties and include the concepts of lines, subspaces, and parallelism, which are detailed in next subsections.

## Subspaces

Let be a Euclidean space and $\overrightarrow E$ its associated vector space. A ''flat'', ''Euclidean subspace'' or ''affine subspace'' of is a subset of such that $\overrightarrow F = \left\$ is a
linear subspace In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...
of $\overrightarrow E.$ A Euclidean subspace is a Euclidean space with $\overrightarrow F$ as associated vector space. This linear subspace $\overrightarrow F$ is called the ''direction'' of . If is a point of then $F = \left\.$ Conversely, if is a point of and is a
linear subspace In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...
of $\overrightarrow E,$ then $P + V = \left\$ is a Euclidean subspace of direction . A Euclidean vector space (that is, a Euclidean space such that $E = \overrightarrow E$) has two sorts of subspaces: its Euclidean subspaces and its linear subspaces. Linear subspaces are Euclidean subspaces and a Euclidean subspace is a linear subspace if and only if it contains the zero vector.

## Lines and segments

In a Euclidean space, a ''line'' is a Euclidean subspace of dimension one. Since a vector space of dimension one is spanned by any nonzero vector a line is a set of the form $\left\,$ where and are two distinct points. It follows that ''there is exactly one line that passes through (contains) two distinct points.'' This implies that two distinct lines intersect in at most one point. A more symmetric representation of the line passing through and is $\left\,$ where is an arbitrary point (not necessary on the line). In a Euclidean vector space, the zero vector is usually chosen for ; this allows simplifying the preceding formula into $\left\.$ A standard convention allows using this formula in every Euclidean space, see . The ''
line segment In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position ...

'', or simply ''segment'', joining the points and is the subset of the points such that in the preceding formulas. It is denoted or ; that is $PQ = QP = \left\.$

## Parallelism

Two subspaces and of the same dimension in a Euclidean space are ''parallel'' if they have the same direction. Equivalently, they are parallel, if there is a translation vector that maps one to the other: $T= S+v.$ Given a point and a subspace , there exists exactly one subspace that contains and is parallel to , which is $P + \overrightarrow S.$ In the case where is a line (subspace of dimension one), this property is
Playfair's axiom In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space th ...
. It follows that in a Euclidean plane, two lines either meet in one point or are parallel. The concept of parallel subspaces has been extended to subspaces of different dimensions: two subspaces are parallel if the direction of one of them is contained in the direction to the other.

# Metric structure

The vector space $\overrightarrow E$ associated to a Euclidean space is an
inner product space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
. This implies a
symmetric bilinear formA symmetric bilinear form on a vector space is a bilinear map from two copies of the vector space to the field of scalars such that the order of the two vectors does not affect the value of the map. In other words, it is a bilinear function B tha ...
$\begin \overrightarrow E \times \overrightarrow E &\to \R\\ (x,y)&\mapsto \langle x,y \rangle \end$ that is
positive definiteIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
(that is $\langle x,x \rangle$ is always positive for ). The inner product of a Euclidean space is often called ''dot product'' and denoted . This is specially the case when a
Cartesian coordinate system A Cartesian coordinate system (, ) in a plane Plane or planes may refer to: * Airplane An airplane or aeroplane (informally plane) is a fixed-wing aircraft A fixed-wing aircraft is a heavier-than-air flying machine Early fly ...
has been chosen, as, in this case, the inner product of two vectors is the
dot product In mathematics, the dot product or scalar productThe term ''scalar product'' is often also used more generally to mean a symmetric bilinear form, for example for a pseudo-Euclidean space. is an algebraic operation that takes two equal-length seque ...
of their
coordinate vector In linear algebra, a coordinate vector is a representation of a vector as an ordered list of numbers that describes the vector in terms of a particular ordered basis. Coordinates are always specified relative to an ordered basis. Bases and their a ...
s. For this reason, and for historical reasons, the dot notation is more commonly used than the bracket notation for the inner product of Euclidean spaces. This article will follow this usage; that is $\langle x,y \rangle$ will be denoted in the remainder of this article. The Euclidean norm of a vector is $\, x\, = \sqrt .$ The inner product and the norm allows expressing and proving all
metric METRIC (Mapping EvapoTranspiration at high Resolution with Internalized Calibration) is a computer model Computer simulation is the process of mathematical modelling, performed on a computer, which is designed to predict the behaviour of or th ...
and
topological s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structu ...

properties of
Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandria Alexandria ( or ; ar, الإسكندرية ; arz, اسكندرية ; Coptic Coptic may refer to: Afro-Asia * Copts, an ethnoreligious group mainly in the area of modern ...
. The next subsection describe the most fundamental ones. ''In these subsections,'' ''denotes an arbitrary Euclidean space, and $\overrightarrow E$ denotes its vector space of translations.''

## Distance and length

The ''distance'' (more precisely the ''Euclidean distance'') between two points of a Euclidean space is the norm of the translation vector that maps one point to the other; that is $d(P,Q) = \left\, \overrightarrow \right\, .$ The ''length'' of a segment is the distance between its endpoints. It is often denoted $, PQ,$. The distance is a
metric METRIC (Mapping EvapoTranspiration at high Resolution with Internalized Calibration) is a computer model Computer simulation is the process of mathematical modelling, performed on a computer, which is designed to predict the behaviour of or th ...
, as it is positive definite, symmetric, and satisfies the
triangle inequality In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

$d(P,Q)\le d(P,R) + d(R, Q).$ Moreover, the equality is true if and only if belongs to the segment . This inequality means that the length of any edge of a
triangle A triangle is a polygon In geometry, a polygon () is a plane (mathematics), plane Shape, figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The ...

is smaller than the sum of the lengths of the other edges. This is the origin of the term ''triangle inequality''. With the Euclidean distance, every Euclidean space is a
complete metric space In mathematical analysis Analysis is the branch of mathematics dealing with Limit (mathematics), limits and related theories, such as Derivative, differentiation, Integral, integration, Measure (mathematics), measure, sequences, Series (mathema ...
.

## Orthogonality

Two nonzero vectors and of $\overrightarrow E$ are ''perpendicular'' or ''orthogonal'' if their inner product is zero: $u \cdot v =0$ Two linear subspaces of $\overrightarrow E$ are orthogonal if every nonzero vector of the first one is perpendicular to every nonzero vector of the second one. This implies that the intersection of the linear subspace is reduced to the zero vector. Two lines, and more generally two Euclidean subspaces are orthogonal if their direction are orthogonal. Two orthogonal lines that intersect are said ''perpendicular''. Two segments and that share a common endpoint are ''perpendicular'' or ''form a
right angle In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position ...

'' if the vectors $\overrightarrow$ and $\overrightarrow$ are orthogonal. If and form a right angle, one has $, BC, ^2 = , AB, ^2 + , AC, ^2.$ This is the
Pythagorean theorem In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...

. Its proof is easy in this context, as, expressing this in terms of the inner product, one has, using bilinearity and symmetry of the inner product: $\begin , BC, ^2 &= \overrightarrow \cdot \overrightarrow \\ &=\left(\overrightarrow +\overrightarrow \right ) \cdot \left(\overrightarrow +\overrightarrow \right)\\ &=\overrightarrow \cdot \overrightarrow + \overrightarrow \cdot \overrightarrow -2 \overrightarrow \cdot \overrightarrow \\ &=\overrightarrow \cdot \overrightarrow + \overrightarrow \cdot\overrightarrow \\ &=, AB, ^2 + , AC, ^2. \end$

## Angle

The (non-oriented) ''angle'' between two nonzero vectors and in $\overrightarrow E$ is $\theta = \arccos\left(\frac\right)$ where is the
principal value In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
of the
arccosine In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...

function. By
Cauchy–Schwarz inequality In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
, the argument of the arccosine is in the interval . Therefore is real, and (or if angles are measured in degrees). Angles are not useful in a Euclidean line, as they can be only 0 or . In an
oriented is non-orientable In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical a ...
Euclidean plane, one can define the ''oriented angle'' of two vectors. The oriented angle of two vectors and is then the opposite of the oriented angle of and . In this case, the angle of two vectors can have any value modulo an integer multiple of . In particular, a
reflex angle In Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandria ) , name = Alexandria ( or ; ar, الإسكندرية ; arz, اسكندرية ; Coptic language, Coptic: Rakod ...

equals the negative angle . The angle of two vectors does not change if they are by positive numbers. More precisely, if and are two vectors, and and are real numbers, then $\operatorname(\lambda x, \mu y)= \begin \operatorname(x, y) \qquad\qquad \text \lambda \text \mu \text\\ \pi - \operatorname(x, y)\qquad \text. \end$ If , , and are three points in a Euclidean space, the angle of the segments and is the angle of the vectors $\overrightarrow$ and $\overrightarrow .$ As the multiplication of vectors by positive numbers do not change the angle, the angle of two
half-line 290px, A representation of one line segment. In geometry, the notion of line or straight line was introduced by ancient mathematicians to represent straight objects (i.e., having no curvature In mathematics, curvature is any of several str ...
s with initial point can be defined: it is the angle of the segments and , where and are arbitrary points, one on each half-line. Although this is less used, one can define similarly the angle of segments or half-lines that do not share an initial points. The angle of two lines is defined as follows. If is the angle of two segments, one on each line, the angle of any two other segments, one on each line, is either or . One of these angles is in the interval , and the other being in . The ''non-oriented angle'' of the two lines is the one in the interval . In an oriented Euclidean plane, the ''oriented angle'' of two lines belongs to the interval .

## Cartesian coordinates

Every Euclidean vector space has an
orthonormal basisIn linear algebra, two vectors in an inner product space are orthonormal if they are orthogonal (or perpendicular along a line) unit vectors. A set of vectors form an orthonormal set if all vectors in the set are mutually orthogonal and all of u ...
(in fact, infinitely many in dimension higher than one, and two in dimension one), that is a
basis Basis may refer to: Finance and accounting *Adjusted basisIn tax accounting, adjusted basis is the net cost of an asset after adjusting for various tax-related items. Adjusted Basis or Adjusted Tax Basis refers to the original cost or other b ...
$\left(e_1, \dots, e_n\right)$ of
unit vector In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
s ($\, e_i\, = 1$) that are pairwise orthogonal ($e_i\cdot e_j = 0$ for ). More precisely, given any
basis Basis may refer to: Finance and accounting *Adjusted basisIn tax accounting, adjusted basis is the net cost of an asset after adjusting for various tax-related items. Adjusted Basis or Adjusted Tax Basis refers to the original cost or other b ...
$\left(b_1, \dots, b_n\right),$ the
Gram–Schmidt process In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

computes an orthonormal basis such that, for every , the
linear span In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
s of $\left(e_1, \dots, e_i\right)$ and $\left(b_1, \dots, b_i\right)$ are equal. Given a Euclidean space , a ''Cartesian frame'' is a set of data consisting of an orthonormal basis of $\overrightarrow E,$ and a point of , called the ''origin'' and often denoted . A Cartesian frame $\left(O, e_1, \dots, e_n\right)$ allows defining Cartesian coordinates for both and $\overrightarrow E$ in the following way. The Cartesian coordinates of a vector are the coefficients of on the basis $e_1, \dots, e_n.$ As the basis is orthonormal, the th coefficient is the dot product $v\cdot e_i.$ The Cartesian coordinates of a point of are the Cartesian coordinates of the vector $\overrightarrow .$

## Other coordinates

As a Euclidean space is an
affine space In mathematics, an affine space is a geometric Structure (mathematics), structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping on ...
, one can consider an affine frame on it, which is the same as a Euclidean frame, except that the basis is not required to be orthonormal. This define
affine coordinates In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
, sometimes called ''skew coordinates'' for emphasizing that the basis vectors are not pairwise orthogonal. An affine basis of a Euclidean space of dimension is a set of points that are not contained in a hyperplane. An affine basis define barycentric coordinates for every point. Many other coordinates systems can be defined on a Euclidean space of dimension , in the following way. Let be a
homeomorphism In the mathematical Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantiti ...
(or, more often, a
diffeomorphism In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
) from a
dense The density (more precisely, the volumetric mass density; also known as specific mass), of a substance is its mass Mass is both a property Property (''latin: Res Privata'') in the Abstract and concrete, abstract is what belongs to or ...
open subset Open or OPEN may refer to: Music * Open (band) Open is a band. Background Drummer Pete Neville has been involved in the Sydney/Australian music scene for a number of years. He has recently completed a Masters in screen music at the Australian ...
of to an open subset of $\R^n.$ The ''coordinates'' of a point of are the components of . The
polar coordinate system In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...

(dimension 2) and the
spherical of a sphere A sphere (from Greek language, Greek —, "globe, ball") is a geometrical object in three-dimensional space Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values ...

and
cylindrical A cylinder (from ) has traditionally been a Solid geometry, three-dimensional solid, one of the most basic of curvilinear geometric shapes. Geometrically, it can be considered as a Prism (geometry), prism with a circle as its base. This traditi ...

coordinate systems (dimension 3) are defined this way. For points that are outside the domain of , coordinates may sometimes be defined as the limit of coordinates of neighbour points, but these coordinates may be not uniquely defined, and may be not continuous in the neighborhood of the point. For example, for the spherical coordinate system, the longitude is not defined at the pole, and on the
antimeridian The International Date Line zigzags around the 180th Meridian. The 180th meridian or antimeridian is the meridian 180° both east and west of the Prime Meridian A prime meridian is the meridian (geography), meridian (a line of longitude) i ...
, the longitude passes discontinuously from –180° to +180°. This way of defining coordinates extends easily to other mathematical structures, and in particular to
manifold The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's surface. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of su ...

s.

# Isometries

An
isometry In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
between two
metric space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
s is a bijection preserving the distance, that is $d(f(x), f(y))= d(x,y).$ In the case of a Euclidean vector space, an isometry that maps the origin to the origin preserves the norm $\, f(x)\, = \, x\, ,$ since the norm of a vector is its distance from the zero vector. It preserves also the inner product $f(x)\cdot f(y)=x\cdot y,$ since $x \cdot y=\frac 1 2 \left(\, x+y\, ^2-\, x\, ^2-\, y\, ^2\right).$ An isometry of Euclidean vector spaces is a
linear isomorphism In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
. An isometry $f\colon E\to F$ of Euclidean spaces defines an isometry $\overrightarrow f \colon \overrightarrow E \to \overrightarrow F$ of the associated Euclidean vector spaces. This implies that two isometric Euclidean spaces have the same dimension. Conversely, if and are Euclidean spaces, , , and $\overrightarrow f\colon \overrightarrow E\to \overrightarrow F$ is an isometry, then the map $f\colon E\to F$ defined by $f(P)=O' + \overrightarrow f\left(\overrightarrow\right)$ is an isometry of Euclidean spaces. It follows from the preceding results that an isometry of Euclidean spaces maps lines to lines, and, more generally Euclidean subspaces to Euclidean subspaces of the same dimension, and that the restriction of the isometry on these subspaces are isometries of these subspaces.

## Isometry with prototypical examples

If is a Euclidean space, its associated vector space $\overrightarrow E$ can be considered as a Euclidean space. Every point defines an isometry of Euclidean spaces $P\mapsto \overrightarrow ,$ which maps to the zero vector and has the identity as associated linear map. The inverse isometry is the map $v\mapsto O+v.$ A Euclidean frame allows defining the map $\begin E&\to \R^n\\ P&\mapsto \left(e_1\cdot \overrightarrow , \dots, e_n\cdot\overrightarrow \right), \end$ which is an isometry of Euclidean spaces. The inverse isometry is $\begin \R^n&\to E \\ (x_1\dots, x_n)&\mapsto \left(O+x_1e_1+ \dots + x_ne_n\right). \end$ ''This means that, up to an isomorphism, there is exactly one Euclidean space of a given dimension.'' This justifies that many authors talk of $\R^n$ as ''the'' Euclidean space of dimension .

## Euclidean group

An isometry from a Euclidean space onto itself is called ''Euclidean isometry'', ''Euclidean transformation'' or ''rigid transformation''. The rigid transformations of a Euclidean space form a group (under
composition Composition or Compositions may refer to: Arts * Composition (dance), practice and teaching of choreography * Composition (music), an original piece of music and its creation *Composition (visual arts) The term composition means "putting togethe ...
), called the ''Euclidean group'' and often denoted of . The simplest Euclidean transformations are
translations Translation is the communication of the meaning of a source-language text by means of an equivalent target-language text. The English language draws a terminological distinction (which does not exist in every language) between ''transla ...
$P \to P+v.$ They are in bijective correspondence with vectors. This is a reason for calling ''space of translations'' the vector space associated to a Euclidean space. The translations form a
normal subgroup In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), ...
of the Euclidean group. A Euclidean isometry of a Euclidean space defines a linear isometry $\overrightarrow f$ of the associated vector space (by ''linear isometry'', it is meant an isometry that is also a
linear map In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

) in the following way: denoting by the vector $\overrightarrow$, if is an arbitrary point of , one has $\overrightarrow f(\overrightarrow )= f(P)-f(O).$ It is straightforward to prove that this is a linear map that does not depend from the choice of The map $f \to \overrightarrow f$ is a
group homomorphism In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...

from the Euclidean group onto the group of linear isometries, called the
orthogonal group In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
. The kernel of this homomorphism is the translation group, showing that it is a normal subgroup of the Euclidean group. The isometries that fix a given point form the
stabilizer subgroup In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
of the Euclidean group with respect to . The restriction to this stabilizer of above group homomorphism is an isomorphism. So the isometries that fix a given point form a group isomorphic to the orthogonal group. Let be a point, an isometry, and the translation that maps to . The isometry $g=t^\circ f$ fixes . So $f= t\circ g,$ and ''the Euclidean group is the
semidirect product In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
of the translation group and the orthogonal group.'' The
special orthogonal group In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
is the normal subgroup of the orthogonal group that preserves
handedness In human biology Human biology is an interdisciplinary area of academic study that examines humans through the influences and interplay of many diverse fields such as human genetics, genetics, human evolution, evolution, human physiology, physio ...
. It is a subgroup of
index Index may refer to: Arts, entertainment, and media Fictional entities * Index (''A Certain Magical Index''), a character in the light novel series ''A Certain Magical Index'' * The Index, an item on a Halo megastructure in the ''Halo'' series ...
two of the orthogonal group. Its inverse image by the group homomorphism $f \to \overrightarrow f$ is a normal subgroup of index two of the Euclidean group, which is called the ''special Euclidean group'' or the ''displacement group''. Its elements are called ''rigid motions'' or ''displacements''. Rigid motions include the
identity Identity may refer to: Social sciences * Identity (social science), personhood or group affiliation in psychology and sociology Group expression and affiliation * Cultural identity, a person's self-affiliation (or categorization by others ...

, translations,
rotation A rotation is a circular movement of an object around a center (or point) of rotation. The plane (geometry), geometric plane along which the rotation occurs is called the ''rotation plane'', and the imaginary line extending from the center an ...

s (the rigid motions that fix at least a point), and also screw motions. Typical examples of rigid transformations that are not rigid motions are
reflectionReflection or reflexion may refer to: Philosophy * Self-reflection Science * Reflection (physics), a common wave phenomenon ** Specular reflection, reflection from a smooth surface *** Mirror image, a reflection in a mirror or in water ** Signal r ...
s, which are rigid transformations that fix a hyperplane and are not the identity. They are also the transformations consisting in changing the sign of one coordinate over some Euclidean frame. As the special Euclidean group is a subgroup of index two of the Euclidean group, given a reflection , every rigid transformation that is not a rigid motion is the product of and a rigid motion. A
glide reflection In 2-dimensional geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with propert ...

is an example of a rigid transformation that is not a rigid motion or a reflection. All groups that have been considered in this section are
Lie group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
s and
algebraic group In algebraic geometry, an algebraic group (or group variety) is a Group (mathematics), group that is an algebraic variety, such that the multiplication and inversion operations are given by regular map (algebraic geometry), regular maps on the varie ...
s.

# Topology

The Euclidean distance makes a Euclidean space a
metric space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
, and thus a
topological space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
. This topology is called the
Euclidean topologyIn mathematics, and especially general topology, the Euclidean topology is the natural topology induced on n-dimensional Euclidean space \R^n by the Euclidean distance, Euclidean metric. Definition In any metric space, the Ball (mathematics), ope ...
. In the case of $\mathbb R^n,$ this topology is also the
product topology Product may refer to: Business * Product (business) In marketing, a product is an object or system made available for consumer use; it is anything that can be offered to a Market (economics), market to satisfy the desire or need of a customer ...
. The
open set In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s are the subsets that contains an
open ball In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
around each of their points. In other words, open balls form a base of the topology. The
topological dimension In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
of a Euclidean space equals its dimension. This implies that Euclidean spaces of different dimensions are not
homeomorphic In the mathematical Mathematics (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its populat ...
. Moreover, the theorem of invariance of domain asserts that a subset of a Euclidean space is open (for the
subspace topologyIn topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a Topological_space#Definitions, topology induced from that of ''X'' called the subspace topology (or the relative ...

) if and only if it is homeomorphic to an open subset of a Euclidean space of the same dimension. Euclidean spaces are complete and
locally compact In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...
. That is, a closed subset of a Euclidean space is compact if it is bounded (that is, contained in a ball). In particular, closed balls are compact.

# Axiomatic definitions

The definition of Euclidean spaces that has been described in this article differs fundamentally of
Euclid Euclid (; grc-gre, Εὐκλείδης Euclid (; grc, Εὐκλείδης – ''Eukleídēs'', ; fl. 300 BC), sometimes called Euclid of Alexandria to distinguish him from Euclid of Megara, was a Greek mathematician, often referre ...

's one. 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 need of a formal definition appeared only at the end of 19th century, with the introduction of
non-Euclidean geometries In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
. Two different approaches have been used.
Felix Klein Christian Felix Klein (; 25 April 1849 – 22 June 1925) was a German mathematician and mathematics educator, known for his work with group theory, complex analysis, non-Euclidean geometry, and on the associations between geometry and group ...
suggested to define geometries through their
symmetries Symmetry (from Ancient Greek, Greek συμμετρία ''symmetria'' "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" ...
. The presentation of Euclidean spaces given in this article, is essentially issued from his
Erlangen program In mathematics, the Erlangen program is a method of characterizing geometries based on group theory The popular puzzle Rubik's cube invented in 1974 by Ernő Rubik has been used as an illustration of permutation group">Ernő_Rubik.html" ;"titl ...
, with the emphasis given on the groups of translations and isometries. On the other hand,
David Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician This is a List of German mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, G ...
proposed a set of
axioms An axiom, postulate or assumption is a statement that is taken to be truth, true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Greek ''axíōma'' () 'that which is thought worthy or fit' or ...
, inspired by . They belong to
synthetic geometry Synthetic geometry (sometimes referred to as axiomatic geometry or even pure geometry) is the study of geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is conce ...
, as they do not involve any definition of
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s. Later G. D. Birkhoff and Alfred Tarski proposed simpler sets of axioms, which use
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s (see Birkhoff's axioms and Tarski's axioms). In ''Geometric Algebra (book), Geometric Algebra'', Emil Artin has proved that all these definitions of a Euclidean space are equivalent. It is rather easy to prove that all definitions of Euclidean spaces satisfy Hilbert's axioms, and that those involving real numbers (including the above given definition) are equivalent. The difficult part of Artin's proof is the following. In Hilbert's axioms, congruence (geometry), congruence is an equivalence relation on segments. One can thus define the ''length'' of a segment as its equivalence class. One must thus prove that this length satisfies properties that characterize nonnegative real numbers. Artin proved this with axioms equivalent to those of Hilbert.

# Usage

Since
ancient Greeks Ancient Greece ( el, Ἑλλάς, Hellás) was a civilization belonging to a period of History of Greece, Greek history from the Greek Dark Ages of the 12th–9th centuries BC to the end of Classical Antiquity, antiquity ( AD 600). This era was ...
, Euclidean space is used for modeling shapes in the physical world. It is thus used in many sciences such as
physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical scie ...

, mechanics, and astronomy. It is also widely used in all technical areas that are concerned with shapes, figure, location and position, such as architecture, geodesy, topography, navigation, industrial design, or technical drawing. Space of dimensions higher than three occurs in several modern theories of physics; see Higher dimension. They occur also in configuration space (physics), configuration spaces of physical systems. Beside
Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandria Alexandria ( or ; ar, الإسكندرية ; arz, اسكندرية ; Coptic Coptic may refer to: Afro-Asia * Copts, an ethnoreligious group mainly in the area of modern ...
, Euclidean spaces are also widely used in other areas of mathematics. Tangent spaces of differentiable manifolds are Euclidean vector spaces. More generally, a
manifold The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's surface. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of su ...

is a space that is locally approximated by Euclidean spaces. Most
non-Euclidean geometries In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
can be modeled by a manifold, and embedding, embedded in a Euclidean space of higher dimension. For example, an elliptic space can be modeled by an ellipsoid. It is common to represent in a Euclidean space mathematics objects that are ''a priori'' not of a geometrical nature. An example among many is the usual representation of Graph (discrete mathematics), graphs.

# Other geometric spaces

Since the introduction, at the end of 19th century, of Non-Euclidean geometries, many sorts of spaces have been considered, about which one can do geometric reasoning in the same way as with Euclidean spaces. In general, they share some properties with Euclidean spaces, but may also have properties that could appear as rather strange. Some of these spaces use Euclidean geometry for their definition, or can be modeled as subspaces of a Euclidean space of higher dimension. When such a space is defined by geometrical
axiom An axiom, postulate or assumption is a statement that is taken to be truth, true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Greek ''axíōma'' () 'that which is thought worthy or fit' o ...

s, embedding the space in a Euclidean space is a standard way for proving consistency of its definition, or, more precisely for proving that its theory is consistent, if
Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandria Alexandria ( or ; ar, الإسكندرية ; arz, اسكندرية ; Coptic Coptic may refer to: Afro-Asia * Copts, an ethnoreligious group mainly in the area of modern ...
is consistent (which cannot be proved).

## Affine space

A Euclidean space is an affine space equipped with a
metric METRIC (Mapping EvapoTranspiration at high Resolution with Internalized Calibration) is a computer model Computer simulation is the process of mathematical modelling, performed on a computer, which is designed to predict the behaviour of or th ...
. Affine spaces have many other uses in mathematics. In particular, as they are defined over any field (mathematics), field, they allow doing geometry in other contexts. As soon as non-linear questions are considered, it is generally useful to consider affine spaces over the complex numbers as an extension of Euclidean spaces. For example, a circle and a line (geometry), line have always two intersection points (possibly not distinct) in the complex affine space. Therefore, most of algebraic geometry is built in complex affine spaces and affine spaces over algebraically closed fields. The shapes that are studied in algebraic geometry in these affine spaces are therefore called affine algebraic variety, affine algebraic varieties. Affine spaces over the rational numbers and more generally over algebraic number fields provide a link between (algebraic) geometry and number theory. For example, the Fermat's Last Theorem can be stated "a Fermat curve of degree higher than two has no point in the affine plane over the rationals." Geometry in affine spaces over a finite fields has also been widely studied. For example, elliptic curves over finite fields are widely used in cryptography.

## Projective space

Originally, projective spaces have been introduced by adding "points at infinity" to Euclidean spaces, and, more generally to affine spaces, in order to make true the assertion "two coplanar lines meet in exactly one point". Projective space share with Euclidean and affine spaces the property of being isotropic, that is, there is no property of the space that allows distinguishing between two points or two lines. Therefore, a more isotropic definition is commonly used, which consists as defining a projective space as the set of the vector lines in a
vector space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
of dimension one more. As for affine spaces, projective spaces are defined over any field (mathematics), field, and are fundamental spaces of algebraic geometry.

## Non-Euclidean geometries

''Non-Euclidean geometry'' refers usually to geometrical spaces where the
parallel postulate In geometry, the parallel postulate, also called Euclid's fifth postulate because it is the fifth postulate in Euclid's Elements, Euclid's ''Elements'', is a distinctive axiom in Euclidean geometry. It states that, in two-dimensional geometry: ' ...

is false. They include elliptic geometry, where the sum of the angles of a triangle is more than 180°, and hyperbolic geometry, where this sum is less than 180°. Their introduction in the second half of 19th century, and the proof that their theory is consistency, 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 theory, axiomatic theories in mathematics.

## Curved spaces

A
manifold The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's surface. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of su ...

is a space that in the neighborhood of each point resembles a Euclidean space. In technical terms, a manifold is a
topological space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
, such that each point has a neighborhood that is
homeomorphic In the mathematical Mathematics (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its populat ...
to an
open subset Open or OPEN may refer to: Music * Open (band) Open is a band. Background Drummer Pete Neville has been involved in the Sydney/Australian music scene for a number of years. He has recently completed a Masters in screen music at the Australian ...
of a Euclidean space. Manifold can be classified by increasing degree of this "resemblance" into topological manifolds, differentiable manifolds, smooth manifolds, and analytic manifolds. However, none of these types of "resemblance" respect distances and angles, even approximately. Distances and angles can be defined on a smooth manifold by providing a smooth function, smoothly varying Euclidean metric on the tangent spaces at the points of the manifold (these tangent spaces are thus Euclidean vector spaces). This results in a Riemannian manifold. Generally,
straight line 290px, A representation of one line segment. In geometry, the notion of line or straight line was introduced by ancient mathematicians to represent straight objects (i.e., having no curvature In mathematics, curvature is any of several str ...

s do not exist in a Riemannian manifold, but their role is played by geodesics, which are the "shortest paths" between two points. This allows defining distances, which are measured along geodesics, and angles between geodesics, which are the angle of their tangents in the tangent space at their intersection. So, Riemannian manifolds behave locally like a Euclidean space that has been bent. Euclidean spaces are trivially Riemannian manifolds. An example illustrating this well is the surface of a sphere. In this case, geodesics are great circle, arcs of great circle, which are called orthodromes in the context of navigation. More generally, the spaces of
non-Euclidean geometries In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
can be realized as Riemannian manifolds.

## Pseudo-Euclidean space

An
inner product In mathematics, an inner product space or a Hausdorff space, Hausdorff pre-Hilbert space is a vector space with a binary operation called an inner product. This operation associates each pair of vectors in the space with a Scalar (mathematics), ...
of a real vector space is a positive definite bilinear form, and so characterized by a Bilinear form#Derived quadratic form, positive definite quadratic form. A pseudo-Euclidean space is an affine space with an associated real vector space equipped with a non-degenerate quadratic form (that may be indefinite quadratic form, indefinite). A fundamental example of such a space is the Minkowski space, which is the space-time of Albert Einstein, Einstein's special relativity. It is a four-dimensional space, where the metric is defined by the quadratic form $x^2+y^2+z^2-t^2,$ where the last coordinate (''t'') is temporal, and the other three (''x'', ''y'', ''z'') are spatial. To take gravity into account, general relativity uses a pseudo-Riemannian manifold that has Minkowski spaces as tangent spaces. The Curvature of Riemannian manifolds, curvature of this manifold at a point is a function of the value of the gravitational field at this point.