Euclidean space is the fundamental space of classical geometry . Originally, this was the three-dimensional space of Euclidean geometry , but in modern mathematics there are Euclidean spaces of any non-negative integer dimension , [1] including three-dimensional space and the Euclidean plane (dimension two). It was introduced by the ancient Greek mathematician Euclid of Alexandria , [2] and the qualifier Euclidean is used to distinguish it from other realms that would later be used in physics .and was discovered in modern mathematics.

Ancient Greek geometry introduced Euclidean space for modeling the physical universe . His great innovation was to prove all properties of space in the form of theorems, starting with some fundamental properties , called postulates , which were assumed to be either categorical (for example, there is a straight line passing through two points ) , or prove that seemed impossible ( parallel postulate ).

After the introduction of non-Euclidean geometry in the late 19th century , the old postulates were re-formalized to define Euclidean spaces through axioms . Another definition of Euclidean spaces has been shown to be equivalent to the axiomatic definition by means of vector spaces and linear algebra . It is this definition that is more commonly used in modern mathematics, and is detailed in this article.

In all definitions, Euclidean spaces contain points, which are defined only by the properties they must have in order to form a Euclidean space.

Each dimension has essentially only one Euclidean space; That is, all Euclidean spaces of a given dimension are isomorphic . Therefore, in many cases, it is possible to work with a specific Euclidean space, which is usually a real n – space equipped with a dot product . A symmetry from Euclidean space to each point is associated with an n -tuple of real numbers that locates that point in Euclidean space and is called the Cartesian coordinates of that point .

\mathbb {r} ^{n},\mathbb {r} ^{n}

**Definition**

**History of definition**

Euclidean space was introduced by the ancient Greeks as an abstraction of our physical space. His great innovation, visible in Euclid’s Elements , was to construct and prove all geometry, starting with some very basic properties , which are abstracted from the physical world, and cannot be proved mathematically due to lack of more basic tools. Is. These properties are called axioms or axioms in modern language. This method of defining Euclidean space is still in use under the name of synthetic geometry.

In 1637, René Descartes introduced Cartesian coordinates and showed that it allowed geometric problems to be reduced to algebraic calculations with numbers. This lack of geometry for algebra was a major change of approach, because until then, the real numbers—that is, the rational numbers and the non-rational numbers together—were defined, in the context of geometry, as lengths and distances.

Euclidean geometry was not applied to spaces of more than three dimensions until the 19th century. *Ludwig Schlfli generalized Euclidean geometry to spaces of n* dimensions using both synthetic and algebraic methods , and discovered all regular polytopes (high-dimensional analogs of Platonic solids) that could exist in Euclidean spaces of any dimension. are present. ^{[4]}

Despite the widespread use of Descartes’ approach, which was called analytic geometry, the definition of Euclidean space remained unchanged until the late 19th century. The introduction of abstract vector spaces allowed their use in defining Euclidean spaces with a purely algebraic definition. This new definition is shown to be equivalent to the classical definition in terms of geometric axioms. It is this algebraic definition that is now most often used to introduce Euclidean spaces.

**Modern definition of inspiration**

One way to think of the Euclidean plane is as a set of points satisfying some relationship, which can be expressed in terms of distances and angles. For example, there are two fundamental operations (called motion) on the plane. There is a translation, which means the shifting of the plane so that each point moves in the same direction and by the same distance. The second is a rotation around a fixed point in the plane, in which all points on the plane rotate around that fixed point through the same angle. One of the fundamental principles of Euclidean geometry is that two figures of the plane (usually treated as subsets) must be considered to be congruent if one is transformed into the other by some sequence of translation, rotation, and reflection. can be done (see below).

To make all this mathematically accurate, the theory must clearly define what a Euclidean space is, and the related notions of distance, angle, translation, and rotation. Even when used in physical theories, Euclidean space is an abstraction distinct from actual physical spaces, specific reference frames, measurement instruments, etc. A purely mathematical definition of Euclidean space also ignores questions of units of length and other physical dimensions: distance in “mathematical” space is a number, not something expressed in inches or meters.

The standard way to define Euclidean space mathematically, as done in the remainder of this article, is to define Euclidean space as a set of points on which a real vector space acts, *translating* The *space* of which is equipped with an internal product. , ^{[1]} The act of translation makes the space an affine space, and it allows to define lines, planes, subspaces, dimension, and parallelism. The inner product allows distances and angles to be defined.^{}

*The n* -tuples of real numbers equipped with the set dot product have a Euclidean space of dimension *n . *In contrast, the substituent of a point called the *origin* and an orthonormal basis is equivalent with defining a space of translation between an Euclidean space of dimension *N* andViewed as Euclidean space.

^{It follows that whatever} can be said about Euclidean space can *also* be said *about* Euclidean space of dimension *n* .^{}

One reason for introducing such an abstract definition of Euclidean spaces and working with it instead is that it is often preferable to work in a *coordinate-free* and *origin-free* way (that is, without choosing a preferred basis and a preferred origin). . The second reason is that there is no origin and no basis in the material world.

**Technical definition**

A**The Euclidean vector space** is a finite-dimensional inner product space over the real numbers.

An **Euclidean space** is an affine space greater than reals such that the associated vector space is not an Euclidean vector space. Euclidean spaces are sometimes called *Euclidean affine spaces* to distinguish them from Euclidean vector spaces .

If *E* is a Euclidean space, then its associated vector space is often denoted as the *dimension* of an Euclidean space is the dimension of its corresponding vector space.

The elements of *e are **called* numerals *and* are usually denoted by capital letters. The elements are called *Euclidean vectors* or *free vectors* . They are also called *translation* , although, to put it properly, translation is the action resulting from the geometric transformation of an Euclidean vector over Euclidean space.

The action of translation *v* at a point *P* gives a point which is denoted *P* + *v . *This action satisfies

{\displaystyle P+(v+w)=(P+v)+w.}

(The second + on the left is a vector addition; all other + denotes an action of a vector at a point. This notation is not ambiguous, because, to distinguish between the two meanings of + , it is sufficient to see The nature of its left argument.)

The fact that the action is independent and transitive means that for every pair of points ( *p* , *q* ) there is exactly a vector *v* such that *p* + *v* = *q* . This vector *v* is called *Q* – *P* or . is denoted by

{\displaystyle {\overrightarrow {PQ}}.}

As mentioned earlier, some of the basic properties of Euclidean spaces are a consequence of the structure of affine spaces. They are described in the affine structure and its subdivisions. The properties resulting from the inner product are explained in the metric structure and its subsections.

**Prototypical example**

For any vector space, the sum acts independently and transitively on the vector space itself. Thus a Euclidean vector space can be viewed as a Euclidean space with itself as a corresponding vector space.

A specific case of Euclidean vector space is seen as a vector space equipped with a dot product as an inner product. The importance of this particular example of Euclidean space lies in the fact that every Euclidean space is homogeneous to it. More precisely, given a Euclidean space *E of dimension n* , the choice of a point, which *is* called the *origin* and has an orthonormal basisDefines a isomerism of Euclidean spaces from *E to*

As every Euclidean space of dimension *n* is homogeneous to it, the Euclidean space is sometimes called the *standard Euclidean space* of dimension *n* . ^{[5]}^{}^{}

**Affine structure**

Some basic properties of Euclidean space depend only on the fact that Euclidean space is an affine space. They are called affine properties and include the concepts of lines, subspaces, and parallelism, which are described in the next subsections.

**Subspace**

Let *E* be a Euclidean space and its corresponding vector space.

A *flat* , *Euclidean subspace* or *affine subspace* of *E* is a subset of *F* of *E* such that

{\displaystyle {\overrightarrow {F}}=\{{\overrightarrow {PQ}}\mid P\in F,Q\in F\}}

is a linear subspace of a Euclidean subspace *F* is a Euclidean space with the corresponding vector space. This linear subspace is called the *direction* of *f* .

{\displaystyle {\overrightarrow {E}}.}\overrightarrow F\overrightarrow F

If *P* is a point of F *then*

{\displaystyle F=\{P+v\mid v\in {\overrightarrow {F}}\}.}

Conversely, if P *is* a point on *E* and is a linear subspace of *V* , then

{\displaystyle P+V=\{P+v\mid v\in V\}}

is a Euclidean subspace of direction *V.*

A Euclidean vector space (that is, a Euclidean space such that ) has two types of subspaces: its Euclidean subspace and its linear subspace. Linear subspaces are Euclidean subspaces and a Euclidean subspace is a linear subspace if and only if it has a zero vector.

**lines and segments**

In Euclidean space, a *line* is the Euclidean subspace of dimension one. Since a vector space of dimension is spanned by any non-zero vector, a line is a set of the form

{\displaystyle \{P+\lambda {\overrightarrow {PQ}}\mid \lambda \in \mathbb {R} \},}

Where *P* and *Q* are two different points.

It follows that there is *exactly one line that passes through two different points. *This means that two distinct lines intersect at at most one point.

The more symmetric representation of the line passing through *P* and *Q is*

{\displaystyle \{O+(1-\lambda ){\overrightarrow {OP}}+\lambda {\overrightarrow {OQ}}\mid \lambda \in \mathbb {R} \},}

where *O* is an arbitrary point (not necessarily on the line).

In Euclidean vector space, the zero vector is usually chosen for *O* ; This allows to simplify the previous formula

{\displaystyle \{(1-\lambda )P+\lambda Q\mid \lambda \in \mathbb {R} \}.}

A standard convention allows this formula to be used in every Euclidean space, see Affine Space Affine Combination and Barycenter.

*The line segment* , or simply the *segment* , joining the points *P* and *Q* is the subset of points that are 0 1 in the previous *formulas* . It is denoted by *PQ* or *QP ; *ie

{\displaystyle PQ=QP=\{P+\lambda {\overrightarrow {PQ}}\mid 0\leq \lambda \leq 1\}.}

**Equality**

Two subspaces *S* and *T* of the same dimension in Euclidean space are *parallel if they have* the same direction. ^{[a]} Equivalently, they are parallel, if there is a translation *V* vector that maps from one to the other:

{\displaystyle T=S+v.}

Given a point *P* and a subspace *S* , there exists exactly one subspace that contains *P* and is parallel to *S* , that is, in the case where *S* is a line (subspace of one dimension), it The property is owned by Playfair.

{\displaystyle P+{\overrightarrow {S}}.}

It follows that in a Euclidean plane, two lines either meet at a 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 lies in the direction of the other.

**Metric structure**

The vector space *E* associated with a Euclidean space is an inner product space. It implies a symmetric bilinear form

{\displaystyle {\begin{aligned}{\overrightarrow {E}}\times {\overrightarrow {E}}&\to \mathbb {R} \\(x,y)&\mapsto \langle x,y\rangle \end{aligned}}}

It is definite positive (i.e. *x* is always positive for 0 ).

The inner product of an Euclidean space is often called the *dot product **and* denoted *x* y . This is especially the case when a Cartesian coordinate system has been chosen, in which case, the inner product of two vectors is the dot product of their coordinate vectors. For this reason, and for historical reasons, dot notation is more commonly used than bracket notation for the inner product of Euclidean spaces. This article will follow this experiment; i.e. will be denoted by *x* y in the *remainder* of this article.

A vector *x of the ***Euclidean ideal** is

{\displaystyle \|x\|={\sqrt {x\cdot x}}.}

The inner product and norm allow all metric and topological properties of Euclidean geometry to be expressed and proved. ^{[ citation needed ]} The next subsection describes the most fundamental ones. *In these subdivisions, **e **denotes an arbitrary Euclidean space, and denotes its vector space of translation.*

**Distance and length**

*The distance* between two points in Euclidean space (more precisely *Euclidean distance* ) is the norm of the translation vector that maps one point to another; ie

{\displaystyle d(P,Q)=\|{\overrightarrow {PQ}}\|.}

The *length* of a segment *PQ* is the distance *d* ( *P* , *Q* ) between its end points . it is often denoted.

Distance is a metric because it is positive, definite, symmetric, and satisfies the triangle inequality.

{\displaystyle d(P,Q)\leq d(P,R)+d(R,Q).}

Furthermore, the equality is true if and only if *R **is related to the* segment PQ . This inequality means that the length of any side of a triangle is smaller than the sum of the lengths of the other sides. This is the origin of the term *triangle inequality .*

As with Euclidean distance, every Euclidean space is a complete metric space.

**Orthogonality**

Two non-zero vectors *u* and *v* of are *perpendicular* or *orthogonal* if the product within them is zero:

{\displaystyle u\cdot v=0}

Two linear subspaces of k are orthogonal if each non-zero vector of the first one is perpendicular to each non-zero vector of the second. This means that the intersection of the linear subspace is reduced to the zero vector.

Two lines, and more commonly two Euclidean subspaces, are orthogonal if their direction is orthogonal. Two orthogonal lines intersecting *are called perpendiculars* .

Two segments *AB* and *AC* which share a common endpoint are *perpendicular* or *form **a right angle* if the vectors and are orthogonal.

If *AB* and *AC* make a right angle, then one has

{\displaystyle |BC|^{2}=|AB|^{2}+|AC|^{2}.}

This is the Pythagorean theorem. The proof of this in this context is easy, e.g., expressing it in terms of the inner product, one has to use the bilinearity and symmetry of the inner product:

{\displaystyle {\begin{aligned}|BC|^{2}&={\overrightarrow {BC}}\cdot {\overrightarrow {BC}}\\&=\left({\overrightarrow {BA}}+{\overrightarrow {AC}}\right)\cdot \left({\overrightarrow {BA}}+{\overrightarrow {AC}}\right)\\&={\overrightarrow {BA}}\cdot {\overrightarrow {BA}}+{\overrightarrow {AC}}\cdot {\overrightarrow {AC}}-2{\overrightarrow {AB}}\cdot {\overrightarrow {AC}}\\&={\overrightarrow {AB}}\cdot {\overrightarrow {AB}}+{\overrightarrow {AC}}\cdot {\overrightarrow {AC}}\\&=|AB|^{2}+|AC|^{2}.\end{aligned}}}

**Who**

(non-oriented) *angle* two non-zero vectors *x and* y *in* . is between

{\displaystyle \theta =\arccos \left({\frac {x\cdot y}{\|x\|\|y\|}}\right)}

where ARccOS is the quotient function of the principal value. By the Cauchy–Schwarz inequality, the argument of coticosine is in the interval [-1, 1] . Therefore *is* real, and 0 (or *0* 180 *angles* are *measured* in degrees ).

Angles are not useful in a Euclidean line, as they can only be 0 *or* .

In an oriented Euclidean plane, one can define the *orientation angle of two vectors. *The angles facing the two vectors *x* and *y* are then opposite to the angles facing *y* and *x . *In this case, the angle of two vectors can be any value relative to an integer multiple *of* 2 . In particular, a reflex angle *<* < 2 equals *the* negative *angle* – *<* – 2 *< **0* .

The angle of two vectors does not change if they are multiplied by positive numbers. Indeed, if *x* and *y* are two vectors, and and *μ are* real numbers, then

{\displaystyle \operatorname {angle} (\lambda x,\mu y)={\begin{cases}\operatorname {angle} (x,y)\qquad \qquad {\text{if }}\lambda {\text{ and }}\mu {\text{ have the same sign}}\\\pi -\operatorname {angle} (x,y)\qquad {\text{otherwise}}.\end{cases}}}

*If A* , *B* , and *C* are three points in Euclidean space , then the angle of the segments *AB* and *AC* is the angle of the vectors and since the multiplication of the vectors by positive numbers does not change the angle, the angles of the two semi-lines with *the starting point A* can be defined as: This is the angle of the segments *AB* and *AC** , where B* and *C* are arbitrary points, one on each semi-line. Although it is less used, one can uniformly define the angle of segments or semi-lines that do not share starting points.

The angle of two lines is defined as. So is an *angle* on two segments, each line, the angle of any two other segments, one on each line , is *either* or *– **. *One of these angles is in the interval *[* 0, /2] , and the other is in [ */* 2, *]* . *The non-oriented angle* is one of the intervals of two lines [0, */* 2] . In an oriented Euclidean plane, the *orientation angle* of two lines is related to the interval [- */* 2, */* 2] .

**Cartesian coordinates**

Every Euclidean vector space has an orthonormal basis (indeed, infinitely many in more than one dimension, and two in one dimension), which is a basis for unit vectors ( ) that are pairwise orthogonal ( for *i **J* ). More precisely, on any given basis the Gram–Schmidt procedure calculates an orthonormal basis such that, for the denominators *i* , of the linear spread and are equal. ^{[7]}^{}

Given a Euclidean space *E , a **Cartesian frame* is a set of data that has an orthonormal basis and a point of *E , called the **origin* and often denoted *O. *A Cartesian frame allows to define Cartesian coordinates for both *e* and accordingly.

The cartesian coordinates of the vector *v* are the coefficients of the base *v* . Since the base is perpendicular, the *i* th coefficient is the dot product

The cartesian coordinates of the point *P of **E are* the cartesian coordinates of the vector

**Other coordinates**

As Euclidean space is an affine space, one can consider it an affine frame, which is similar to Euclidean frame, except that the basis does not need to be orthonormal. It defines affine coordinates, sometimes called *oblique coordinates* , to emphasize that the basis vectors are not pairwise orthogonal.

An affine basis of a Euclidean space of dimension *n is a set of **n* +1 points that do not lie in the hyperplane. An affine basis defines the barycentric coordinates for each point.

Many other coordinate systems can be defined on Euclidean space *E of **n dimension in the following way. *Let *f* be a homeomorphism (or more often, a diffeomorphism) of a dense open subset of *e* to an open subset of e *The coordinates* of a point *x* of *e* are the components of *f* ( *x* ) . The polar coordinate system (dimension 2) and the spherical and cylindrical coordinate system (dimension 3) are defined in this way.

For points that are outside the domain of *f* , the coordinates can sometimes be defined as the range of coordinates of neighboring points, but these coordinates may not be uniquely defined, and the point’s neighborhood has a constant Can’t be. For example, for the spherical coordinate system, longitude is not defined at the pole, and at the antimeridian, the longitude passes discontinuously from -180° to +180°.

This way of defining coordinates extends readily to other mathematical structures, and in particular to manifolds.

**Isometry**

There is a bijection preserving an isometric distance between two metric spaces, ^{[b]} that is

{\displaystyle 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 retains the ideal.

{\displaystyle \|f(x)\|=\|x\|,}

Since the value of a vector is its distance from the zero vector. It also protects the inner product

{\displaystyle f(x)\cdot f(y)=x\cdot y,}

since

{\displaystyle x\cdot y={\frac {1}{2}}\left(\|x+y\|^{2}-\|x\|^{2}-\|y\|^{2}\right).}

A symmetry of Euclidean vector spaces is a linear symmetry. ^{[C] }^{[8]}

An isometry of Euclidean spaces defines an isometry of related Euclidean vector spaces. This implies that two isometric Euclidean spaces have the same dimension. Conversely, if *E* and *F are *Euclidean spaces, *O E* , *O *‘ *F* , and have an isometry , then the map defined by

{\displaystyle f\colon E\to F}{\displaystyle {\overrightarrow {f}}\colon {\overrightarrow {E}}\to {\overrightarrow {F}}}{\displaystyle {\overrightarrow {f}}\colon {\overrightarrow {E}}\to {\overrightarrow {F}}}

{\displaystyle f\colon E\to F}

{\displaystyle f(P)=O'+{\overrightarrow {f}}\left({\overrightarrow {OP}}\right)}

There 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 isometry of these subspaces The restriction is the isometry of these subspaces.

**Isometry with prototypical examples**

If *E* is a Euclidean space, then the associated vector space can be thought of as a Euclidean space. Each point, *O E* , defines an isometry of Euclidean spaces .

{\displaystyle {\overrightarrow {E}}}

{\displaystyle P\mapsto {\overrightarrow {OP}},}

which maps to the zero vector and *is* identified as the associated linear map. The inverse isometry map is

{\displaystyle v\mapsto O+v.}

A Euclidean frame allows the map to be defined as

{\displaystyle {\begin{aligned}E&\to \mathbb {R} ^{n}\\P&\mapsto \left(e_{1}\cdot {\overrightarrow {OP}},\dots ,e_{n}\cdot {\overrightarrow {OP}}\right),\end{aligned}}}

which is an isometry of Euclidean spaces. is inverse isometry

{\displaystyle {\begin{aligned}\mathbb {R} ^{n}&\to E\\(x_{1}\dots ,x_{n})&\mapsto \left(O+x_{1}e_{1}+\dots +x_{n}e_{n}\right).\end{aligned}}}

*This means that, up to a symmetry, a given dimension has exactly one Euclidean space.*

This justifies that many authors talk of dimension *n **as **the* Euclidean space .

**Euclidean group**

An isometry in itself from Euclidean space is called *Euclidean isometry* , *Euclidean transformation* or *Rigid transformation* . A group in an Euclidean space (below the rigid transformations of the composition) is called the *Euclidean group* and is often denoted E( *n* ) of iso( *n* ) .

The simplest Euclidean transformation translations are

{\displaystyle P\to P+v.}

They are in adjective correspondence with vectors. This is one reason to call the vector space associated with the Euclidean space *the translation space . *Translations form a general subset of the Euclidean group.

A Euclidean isometry of a Euclidean space *E* defines a linear isometry *f*The corresponding vector space ( by *linear isometry* , it means an isometry that is also a linear map) in the following way: denotes the vector by *Q* – *P* , if *O* is an arbitrary point of *E* , then one is

{\displaystyle {\overrightarrow {f}}({\overrightarrow {OP}})=f(P)-f(O).}

It is easy to prove that it is a linear map that does not depend on the choice *of **O.*

That map is a group symmetry from the Euclidean group to a group of linear symmetries, called orthogonal groups. The core of this symmetry is the translation group, which shows that it is a general subset of the Euclidean group.

The symmetries that stabilize a given point *P* form a stabilizer subgroup of the Euclidean group *with respect to P. *The restriction on this stabilizer of the above group symmetry is a isomer. So the symmetries that fix a given point make a group isomorphic to the orthogonal group.

Suppose *P* is a point, *f* is an isometry, and *t* is the translation that maps *P* to *f* ( *P* ) . The isometry fixes *P. *Therefore and the *Euclidean group is the semi-direct product of the translation group and the orthogonal group.*

The special orthogonal group is the general subgroup of the orthogonal group that preserves arbitraryness. It is a subset of the index two of the orthogonal group. Its inverse reflection by the symmetry group is a general subset of the index two of the Euclidean group, called the *special Euclidean group* or *displacement group* . Its elements are called *rigid motion* or *displacement* .

Rigid motions also include detection, translation, rotation (rigid motions that fix at least one point), and screw motions.

Typical examples of rigid transformations that are not rigid motions are reflections, which are rigid transformations that fix a hyperplane and are not identifiable. They are also the transformations involved in changing the sign of a coordinate on some Euclidean frame.

As the particular Euclidean group is a subset of the index two of the Euclidean group, given an image *r* , every rigid transformation that is not a rigid motion is the product of *r* and a rigid motion. A glide reflection is an example of a rigid change that is not a rigid motion or reflection.

All the groups considered in this section are Lie groups and Algebraic groups.

**Topology**

Euclidean distance makes an Euclidean space a metric space, and thus a topological space. This topology is called Euclidean topology. In the case of this topology is also the product topology.

Open sets are subsets that have an open ball around each of their points. In other words, open balls form the basis of the topology.

The topological dimension of Euclidean space is equal to its dimension. This implies that Euclidean spaces of different dimensions are not homeomorphic. Furthermore, the theorem of domain invariance asserts that a subset of Euclidean space is open (for subspace topology) if and only if it is homeomorphic to an open subset of Euclidean space of the same dimension.

Euclidean spaces are complete and locally compact. 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 described in this article is fundamentally different from that of Euclid. In fact, Euclid did not formally define space, as it was regarded as a description of the physical world that exists independently of the human mind. The need for a formal definition only appeared in the late 19th century with the introduction of non-Euclidean geometry.

Two different methods have been used. Felix Klein suggested defining geometry through their symmetries. The presentation of Euclidean spaces given in this article is essentially a release from his Erlangen program, with an emphasis on translation and groups of isometry.

David Hilbert, on the other hand, proposed a set of axioms inspired by Euclid’s postulates. They belong to synthetic geometry, as they do not contain any definition of the real numbers. Later GD Birkhoff and Alfred Tarski proposed simpler sets of axioms, which use the real numbers (see Birkhoff’s axioms and Tarski’s axioms).

In *Geometrical Algebra* , Emile Artin proved that all these definitions of a Euclidean space are equivalent. ^{[9]} It is easy to prove that all definitions of Euclidean spaces satisfy Hilbert’s axioms, and that the real numbers involved (including the definition above) are equivalent. Following is the hard part of Artin’s proof. In Hilbert’s axioms, there is an equivalence relation on congruence clauses. Thus the *length* of a segment can be defined as its equivalence square. Thus it is necessary to prove that this length satisfies the properties characteristic of non-negative real numbers. Artin proved this with Hilbert’s equivalent axioms.

**Experiment**

Since the ancient Greeks, Euclidean space is used for modeling shapes in the physical world. Thus it is used in many sciences such as physics, mechanics and astronomy. It is widely used in all technical fields that are concerned with size, shape, location and position, such as architecture, geodesy, topography, navigation, industrial design or technical drawing.

The space of more than three dimensions is found in many modern theories of physics; See higher dimensions. They also occur in configurational spaces of physical systems.

In addition to Euclidean geometry, Euclidean spaces are also widely used in other areas of mathematics. Differentiable manifolds of tangent spaces are Euclidean vector spaces. More generally, a manifold is a space that is locally approximated by Euclidean spaces. Most non-Euclidean geometry can be modeled by manifolds, and embedded in Euclidean space of higher dimension. For example, an elliptical space can be represented by an ellipse. Euclidean space is common in mathematics to represent objects that are *a priori* that are not of a geometric nature. One example among many is the general representation of graphs.

**Other geometric spaces**

Since the introduction of non-Euclidean geometry, in the late 19th century, several types of spaces have been considered, which can be reasoned geometrically like Euclidean spaces. In general, they share some properties with Euclidean spaces, but they can also have properties that can seem quite strange. Some of these spaces use Euclidean geometry for their definition, or can be modeled as subspaces of Euclidean spaces of higher dimension. When such a space is defined by a geometric axiom, embedding the space in Euclidean space is a standard way of proving the consistency of its definition, or more precisely, to prove that its axioms are consistent. is, if Euclidean geometry is consistent (which cannot be proved)

**Affine space**

A Euclidean space is an affine space equipped with a metric. Affine spaces have many other uses in mathematics. In particular, as they are defined in any field, they allow to do geometry in other contexts.

As soon as non-linear questions are considered, it is generally useful to consider affine spaces over complex numbers as an extension of Euclidean spaces. For example, a circle and a line always have two points of intersection (possibly not differentiable) in complex affine space. Therefore, most algebraic geometry is built into complex affine spaces and affine spaces over algebraically closed fields. The shapes that are studied in algebraic geometry in these affine spaces are called affine algebraic varieties.

Affine spaces on rational numbers and, more generally, algebraic number fields, provide a link between (algebraic) geometry and number theory. For example, Fermat’s last theorem can be stated as “The Fermat curve of more than two degrees has no point in the affine plane on the rational.”

Geometry in affine spaces over 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 connecting spaces, so that “two coplanar lines meet at exactly one point” to prove true. For . Projective space shares with Euclidean and affine space the property of being isotropic, i.e. there is no property of space that allows to distinguish between two points or two lines. Therefore, a more isotropic definition is usually used, which involves defining a projective space as the set of vector lines in a vector space of another dimension.

For affine spaces, projective spaces are defined in any field, and are fundamental spaces of algebraic geometry.

**Non-Euclidean Geometry**

*Non-Euclidean geometry* usually refers to geometric spaces where the parallel postulate is false. These include elliptic geometry, where the sum of the angles of a triangle is greater than 180 degrees, and hyperbolic geometry, where the sum is less than 180 degrees. His introduction in the late 19th century, and the proof that his theory is consistent (if not contradictory to Euclidean geometry) is one of the paradoxes that lie at the core of the fundamental crisis in mathematics of the early 20th century, and axioms in mathematics. inspired the systematization.

**Curved space**

A space of a manifold resembles an Euclidean space in the neighborhood of each point. In technical terms, a manifold is a topological space, each point is a neighborhood such that it is homeomorphic to an open subset of an Euclidean space. Manifolds can be classified by increasing the degree of this “similarity” into topological manifolds, differential manifolds, smooth manifolds and analytic manifolds. However, none of these “similarities” respect distances and angles, even approx.

Distances and angles can be defined on a smooth manifold by providing a smoothly varying Euclidean metric on the tangent spaces at the points of the manifold (these tangents are thus Euclidean vector spaces). This results in a Riemannian manifold. Normally, straight lines do not exist in Riemannian manifolds, but their role is played by geodesics, which is the “shortest path” between two points. This allows to define distances, which are measured with geodesics, and angles between geodesics, which are the angles of their tangents in the tangent space at their intersection. So, a Riemannian manifold behaves locally like a Euclidean that has been tilted.

Euclidean spaces are trivially Riemannian manifolds. An example illustrating this well is the surface of a sphere. In this case, the geodesics are the arcs of the larger circle, which in navigation terms are called orthodromes. More generally, spaces of non-Euclidean geometry can be realized as Riemannian manifolds.

**Pseudo-Euclidean Space**

An inner product of a real vector space has a positive definite bilinear form, and is therefore characterized by a 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 (which can be indefinite).

A fundamental example of such a space is the Minkowski space, the space-time of Einstein’s special relativity. It is a four-dimensional space, where the metric is defined by the quadratic form

{\displaystyle 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 spaces as Minkowski has tangent spaces. The curvature of this manifold at a point is a function of the value of the gravitational field at this point.