In

mathematicsMathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

,

**Euclidean space** is the Euclidean plane and

three-dimensional spaceThree-dimensional space is a geometric 3-parameters model of the physical universe in which we live. These three dimensions are commonly called length, width, and depth , although any three directions can be chosen, provided that they do not lie in the same plane.In physics and mathematics, a...

of

Euclidean geometryEuclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...

, as well as the generalizations of these notions to higher dimensions. The term “Euclidean” distinguishes these spaces from the

curved spaceCurved space often refers to a spatial geometry which is not “flat” where a flat space is described by Euclidean geometry. Curved spaces can generally be described by Riemannian geometry though some simple cases can be described in other ways. Curved spaces play an essential role in General...

s of

non-Euclidean geometryNon-Euclidean geometry is the term used to refer to two specific geometries which are, loosely speaking, obtained by negating the Euclidean parallel postulate, namely hyperbolic and elliptic geometry. This is one term which, for historical reasons, has a meaning in mathematics which is much...

and

Einstein'sAlbert Einstein was a German-born theoretical physicist who developed the theory of general relativity, effecting a revolution in physics. For this achievement, Einstein is often regarded as the father of modern physics and one of the most prolific intellects in human history...

general theory of relativityGeneral relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...

, and is named for the

GreekThe Greeks, also known as the Hellenes , are a nation and ethnic group native to Greece, Cyprus and neighboring regions. They also form a significant diaspora, with Greek communities established around the world....

mathematician

Euclid of AlexandriaEuclid , fl. 300 BC, also known as Euclid of Alexandria, was a Greek mathematician, often referred to as the "Father of Geometry". He was active in Alexandria during the reign of Ptolemy I...

.

Classical Greek geometry defined the Euclidean plane and Euclidean three-dimensional space using certain

postulatesIn traditional logic, an axiom or postulate is a proposition that is not proven or demonstrated but considered either to be self-evident or to define and delimit the realm of analysis. In other words, an axiom is a logical statement that is assumed to be true...

, while the other properties of these spaces were deduced as

theoremIn mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and previously accepted statements, such as axioms...

s. In modern mathematics, it is more common to define Euclidean space using Cartesian coordinates and the ideas of

analytic geometryAnalytic geometry, or analytical geometry has two different meanings in mathematics. The modern and advanced meaning refers to the geometry of analytic varieties...

. This approach brings the tools of

algebraAlgebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...

and

calculusCalculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...

to bear on questions of geometry, and has the advantage that it generalizes easily to Euclidean spaces of more than three

dimensionIn physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it...

s.

From the modern viewpoint, there is essentially only one Euclidean space of each dimension. In dimension one this is the

real lineIn mathematics, the real line, or real number line is the line whose points are the real numbers. That is, the real line is the set of all real numbers, viewed as a geometric space, namely the Euclidean space of dimension one...

; in dimension two it is the Cartesian plane; and in higher dimensions it is the real

coordinate spaceIn mathematics, specifically in linear algebra, the coordinate space, Fn, is the prototypical example of an n-dimensional vector space over a field F. It can be defined as the product space of F over a finite index set.-Definition:...

with three or more

real numberIn mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

coordinates. Thus a

pointIn geometry, topology and related branches of mathematics a spatial point is a primitive notion upon which other concepts may be defined. In geometry, points are zero-dimensional; i.e., they do not have volume, area, length, or any other higher-dimensional analogue. In branches of mathematics...

in Euclidean space is a

tupleIn mathematics and computer science, a tuple is an ordered list of elements. In set theory, an n-tuple is a sequence of n elements, where n is a positive integer. There is also one 0-tuple, an empty sequence. An n-tuple is defined inductively using the construction of an ordered pair...

of real numbers, and distances are defined using the

Euclidean distance formulaIn mathematics, the Euclidean distance or Euclidean metric is the "ordinary" distance between two points that one would measure with a ruler, and is given by the Pythagorean formula. By using this formula as distance, Euclidean space becomes a metric space...

. Mathematicians often denote the

*n*-dimensional Euclidean space by

, or sometimes

if they wish to emphasize its Euclidean nature. Euclidean spaces have finite dimension.

## Intuitive overview

One way to think of the Euclidean plane is as a set of

pointIn geometry, topology and related branches of mathematics a spatial point is a primitive notion upon which other concepts may be defined. In geometry, points are zero-dimensional; i.e., they do not have volume, area, length, or any other higher-dimensional analogue. In branches of mathematics...

s satisfying certain relationships, expressible in terms of distance and angle. For example, there are two fundamental operations on the plane. One is

translationIn Euclidean geometry, a translation moves every point a constant distance in a specified direction. A translation can be described as a rigid motion, other rigid motions include rotations and reflections. A translation can also be interpreted as the addition of a constant vector to every point, or...

, 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

rotationA rotation is a circular movement of an object around a center of rotation. A three-dimensional object rotates always around an imaginary line called a rotation axis. If the axis is within the body, and passes through its center of mass the body is said to rotate upon itself, or spin. A rotation...

about a fixed point in the plane, in which every point in the plane turns about that fixed point through the same angle. One of the basic tenets of Euclidean geometry is that two figures (that is,

subsetIn mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...

s) of the plane should be considered equivalent (

congruentIn geometry, two figures are congruent if they have the same shape and size. This means that either object can be repositioned so as to coincide precisely with the other object...

) if one can be transformed into the other by some sequence of translations, rotations and

reflectionIn mathematics, a reflection is a mapping from a Euclidean space to itself that is an isometry with a hyperplane as set of fixed points; this set is called the axis or plane of reflection. The image of a figure by a reflection is its mirror image in the axis or plane of reflection...

s. (See

Euclidean groupIn mathematics, the Euclidean group E, sometimes called ISO or similar, is the symmetry group of n-dimensional Euclidean space...

.)

In order to make all of this mathematically precise, one must clearly define the notions of distance, angle, translation, and rotation. The standard way to do this, as carried out in the remainder of this article, is to define the Euclidean plane as a two-dimensional

realIn mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

vector spaceA vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...

equipped with an inner product. For then:

- the vector
In linear algebra, a coordinate vector is an explicit representation of a vector in an abstract vector space as an ordered list of numbers or, equivalently, as an element of the coordinate space Fn....

s in the vector space correspond to the points of the Euclidean plane,
- the addition
Addition is a mathematical operation that represents combining collections of objects together into a larger collection. It is signified by the plus sign . For example, in the picture on the right, there are 3 + 2 apples—meaning three apples and two other apples—which is the same as five apples....

operation in the vector space corresponds to translation, and
- the inner product implies notions of angle and distance, which can be used to define rotation.

Once the Euclidean plane has been described in this language, it is actually a simple matter to extend its concept to arbitrary dimensions. For the most part, the vocabulary, formulas, and calculations are not made any more difficult by the presence of more dimensions. (However, rotations are more subtle in high dimensions, and visualizing high-dimensional spaces remains difficult, even for experienced mathematicians.)

A final wrinkle is that Euclidean space is not technically a vector space but rather an

affine spaceIn mathematics, an affine space is a geometric structure that generalizes the affine properties of Euclidean space. In an affine space, one can subtract points to get vectors, or add a vector to a point to get another point, but one cannot add points. In particular, there is no distinguished point...

, on which a vector space

actsIn algebra and geometry, a group action is a way of describing symmetries of objects using groups. The essential elements of the object are described by a set, and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformations of the set...

. Intuitively, the distinction just says that there is no canonical choice of where the

originIn mathematics, the origin of a Euclidean space is a special point, usually denoted by the letter O, used as a fixed point of reference for the geometry of the surrounding space. In a Cartesian coordinate system, the origin is the point where the axes of the system intersect...

should go in the space, because it can be translated anywhere. In this article, this technicality is largely ignored.

## Real coordinate space

Let

**R** denote the

fieldIn abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...

of

real numberIn mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

s. For any positive

integerThe integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...

*n*, the set of all

*n*-

tupleIn mathematics and computer science, a tuple is an ordered list of elements. In set theory, an n-tuple is a sequence of n elements, where n is a positive integer. There is also one 0-tuple, an empty sequence. An n-tuple is defined inductively using the construction of an ordered pair...

s of real numbers forms an

*n*-dimensional vector space over

**R**, which is denoted

**R**^{n} and sometimes called

**real coordinate space**. An element of

**R**^{n} is written

where each

*x*_{i} is a real number. The vector space operations on

**R**^{n} are defined by

The vector space

**R**^{n} comes with a

standard basisIn mathematics, the standard basis for a Euclidean space consists of one unit vector pointing in the direction of each axis of the Cartesian coordinate system...

:

An arbitrary vector in

**R**^{n} can then be written in the form

**R**^{n} is the prototypical example of a real

*n*-dimensional vector space. In fact, every real

*n*-dimensional vector space

*V* is isomorphic to

**R**^{n}. This isomorphism is not canonical, however. A choice of isomorphism is equivalent to a choice of

basisIn linear algebra, a basis is a set of linearly independent vectors that, in a linear combination, can represent every vector in a given vector space or free module, or, more simply put, which define a "coordinate system"...

for

*V* (by looking at the image of the standard basis for

**R**^{n} in

*V*). The reason for working with arbitrary vector spaces instead of

**R**^{n} is that it is often preferable to work in a

*coordinate-free* manner (that is, without choosing a preferred basis).

## Euclidean structure

Euclidean space is more than just a real coordinate space. In order to apply Euclidean geometry one needs to be able to talk about the distances between points and the angles between lines or vectors. The natural way to obtain these quantities is by introducing and using the standard inner product (also known as the

dot productIn mathematics, the dot product or scalar product is an algebraic operation that takes two equal-length sequences of numbers and returns a single number obtained by multiplying corresponding entries and then summing those products...

) on

**R**^{n}. The inner product of any two real

**n**-vectors

**x** and

**y** is defined by

The result is always a real number. Furthermore, the inner product of

**x** with itself is always nonnegative. This product allows us to define the "length" of a vector

*x* as

This length function satisfies the required properties of a

normIn linear algebra, functional analysis and related areas of mathematics, a norm is a function that assigns a strictly positive length or size to all vectors in a vector space, other than the zero vector...

and is called the

**Euclidean norm** on

**R**^{n}.

The

**(non-reflex) angle** θ (0° ≤

*θ* ≤ 180°) between

**x** and

**y** is then given by

where cos

^{−1} is the arccosine function.

Finally, one can use the norm to define a

metricIn mathematics, a metric or distance function is a function which defines a distance between elements of a set. A set with a metric is called a metric space. A metric induces a topology on a set but not all topologies can be generated by a metric...

(or distance function) on

**R**^{n} by

This distance function is called the

**Euclidean metric**In mathematics, the Euclidean distance or Euclidean metric is the "ordinary" distance between two points that one would measure with a ruler, and is given by the Pythagorean formula. By using this formula as distance, Euclidean space becomes a metric space...

. It can be viewed as a form of the

Pythagorean theoremIn mathematics, the Pythagorean theorem or Pythagoras' theorem is a relation in Euclidean geometry among the three sides of a right triangle...

.

Real coordinate space together with this Euclidean structure is called

**Euclidean space** and often denoted

**E**^{n}. (Many authors refer to

**R**^{n} itself as Euclidean space, with the Euclidean structure being understood). The Euclidean structure makes

**E**^{n} an

inner product spaceIn mathematics, an inner product space is a vector space with an additional structure called an inner product. This additional structure associates each pair of vectors in the space with a scalar quantity known as the inner product of the vectors...

(in fact a

Hilbert spaceThe mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...

), a

normed vector spaceIn mathematics, with 2- or 3-dimensional vectors with real-valued entries, the idea of the "length" of a vector is intuitive and can easily be extended to any real vector space Rn. The following properties of "vector length" are crucial....

, and a

metric spaceIn mathematics, a metric space is a set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space...

.

Rotations of Euclidean space are then defined as

orientationIn mathematics, orientation is a notion that in two dimensions allows one to say when a cycle goes around clockwise or counterclockwise, and in three dimensions when a figure is left-handed or right-handed. In linear algebra, the notion of orientation makes sense in arbitrary dimensions...

-preserving

linear transformationIn mathematics, a linear map, linear mapping, linear transformation, or linear operator is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. As a result, it always maps straight lines to straight lines or 0...

s

*T* that preserve angles and lengths:

In the language of

matricesIn mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...

, rotations are special orthogonal matrices.

## Topology of Euclidean space

Since Euclidean space is a

metric spaceIn mathematics, a metric space is a set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space...

it is also a

topological spaceTopological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...

with the natural

topologyTopology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...

induced by the metric. The metric topology on

**E**^{n} is called the

**Euclidean topology**. A set is

openThe concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...

in the Euclidean topology

if and only ifIn logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....

it contains an open ball around each of its points. The Euclidean topology turns out to be equivalent to the

product topologyIn topology and related areas of mathematics, a product space is the cartesian product of a family of topological spaces equipped with a natural topology called the product topology...

on

**R**^{n} considered as a product of

*n* copies of the

real lineIn mathematics, the real line, or real number line is the line whose points are the real numbers. That is, the real line is the set of all real numbers, viewed as a geometric space, namely the Euclidean space of dimension one...

**R** (with its standard topology).

An important result on the topology of

**R**^{n}, that is far from superficial, is Brouwer's

invariance of domainInvariance of domain is a theorem in topology about homeomorphic subsets of Euclidean space Rn. It states:The theorem and its proof are due to L.E.J. Brouwer, published in 1912...

. Any subset of

**R**^{n} (with its

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 natural topology induced from that of X called the subspace topology .- Definition :Given a topological space and a subset S of X, the...

) that is homeomorphic to another open subset of

**R**^{n} is itself open. An immediate consequence of this is that

**R**^{m} is not homeomorphic to

**R**^{n} if

*m* ≠

*n* — an intuitively "obvious" result which is nonetheless difficult to prove.

## Generalizations

In modern mathematics, Euclidean spaces form the prototypes for other, more complicated geometric objects. For example, a smooth manifold is a

HausdorffIn topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" is the most frequently...

topological space that is locally diffeomorphic to Euclidean space. Diffeomorphism does not respect distance and angle, so these key concepts of Euclidean geometry are lost on a smooth manifold. However, if one additionally prescribes a smoothly varying inner product on the manifold's

tangent spaceIn mathematics, the tangent space of a manifold facilitates the generalization of vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other....

s, then the result is what is called a

Riemannian manifoldIn Riemannian geometry and the differential geometry of surfaces, a Riemannian manifold or Riemannian space is a real differentiable manifold M in which each tangent space is equipped with an inner product g, a Riemannian metric, which varies smoothly from point to point...

. Put differently, a Riemannian manifold is a space constructed by deforming and patching together Euclidean spaces. Such a space enjoys notions of distance and angle, but they behave in a

curvedIn mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry. Intuitively, curvature is the amount by which a geometric object deviates from being flat, or straight in the case of a line, but this is defined in different ways depending on the context...

, non-Euclidean manner. The simplest Riemannian manifold, consisting of

**R**^{n} with a constant inner product, is essentially identical to Euclidean

*n*-space itself.

If one alters a Euclidean space so that its inner product becomes negative in one or more directions, then the result is a

pseudo-Euclidean spaceA pseudo-Euclidean space is a finite-dimensional real vector space together with a non-degenerate indefinite quadratic form. Such a quadratic form can, after a change of coordinates, be written as...

. Smooth manifolds built from such spaces are called

pseudo-Riemannian manifoldIn differential geometry, a pseudo-Riemannian manifold is a generalization of a Riemannian manifold. It is one of many mathematical objects named after Bernhard Riemann. The key difference between a Riemannian manifold and a pseudo-Riemannian manifold is that on a pseudo-Riemannian manifold the...

s. Perhaps their most famous application is the

theory of relativityThe theory of relativity, or simply relativity, encompasses two theories of Albert Einstein: special relativity and general relativity. However, the word relativity is sometimes used in reference to Galilean invariance....

, where empty

spacetimeIn physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...

with no

matterMatter is a general term for the substance of which all physical objects consist. Typically, matter includes atoms and other particles which have mass. A common way of defining matter is as anything that has mass and occupies volume...

is represented by the flat pseudo-Euclidean space called

Minkowski spaceIn physics and mathematics, Minkowski space or Minkowski spacetime is the mathematical setting in which Einstein's theory of special relativity is most conveniently formulated...

, spacetimes with matter in them form other pseudo-Riemannian manifolds, and gravity corresponds to the curvature of such a manifold.

Our universe, being subject to relativity, is not Euclidean. This becomes significant in theoretical considerations of

astronomyAstronomy is a natural science that deals with the study of celestial objects and phenomena that originate outside the atmosphere of Earth...

and

cosmologyCosmology is the discipline that deals with the nature of the Universe as a whole. Cosmologists seek to understand the origin, evolution, structure, and ultimate fate of the Universe at large, as well as the natural laws that keep it in order...

, and also in some practical problems such as global positioning and airplane

navigationNavigation is the process of monitoring and controlling the movement of a craft or vehicle from one place to another. It is also the term of art used for the specialized knowledge used by navigators to perform navigation tasks...

. Nonetheless, a Euclidean model of the universe can still be used to solve many other practical problems with sufficient precision.

## See also

- Riemannian geometry
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a Riemannian metric, i.e. with an inner product on the tangent space at each point which varies smoothly from point to point. This gives, in particular, local notions of angle, length...

- Euclidean subspace
In linear algebra, a Euclidean subspace is a set of vectors that is closed under addition and scalar multiplication. Geometrically, a subspace is a flat in n-dimensional Euclidean space that passes through the origin...

- Cartesian coordinate system
A Cartesian coordinate system specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances from the point to two fixed perpendicular directed lines, measured in the same unit of length...

- Polar coordinate system
In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a fixed point and an angle from a fixed direction....

- Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...