Euclidean space

A point in three-dimensional Euclidean space can be located by three coordinates.

Euclidean space is the fundamental space of classical geometry. Originally, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension,^[1] including the 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 spaces that were later discovered in physics and modern mathematics.

Ancient Greek geometers introduced Euclidean space for modeling the physical universe. Their great innovation was to prove all properties of the space as theorems by starting from a few fundamental properties, called postulates, which either were considered as evident (for example, there is exactly one straight line passing through two points), or seemed impossible to prove (parallel postulate).

After the introduction at the end of 19th century of non-Euclidean geometries, the old postulates were re-formalized to define Euclidean spaces through axiomatic theory. Another definition of Euclidean spaces by means of vector spaces and linear algebra 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.^[3]

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. Therefore, in many cases, it is possible to work with a specific Euclidean space, which is generally the real n-space ${\displaystyle \mathbb {R} ^{n},}$ equipped with the dot product. An isomorphism from a Euclidean space to ${\displaystyle \mathbb {R} ^{n}}$ associates with each point an n-tuple of real numbers which locate that point in the Euclidean space and are called the Cartesian coordinates of that point.

Definition[]

History of the definition[]

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

In 1637, René Descartes introduced Cartesian coordinates and showed that this allows reducing geometric problems to algebraic computations with numbers. This reduction of geometry to algebra was a major change of point of view, as, until then, the real numbers—that is, rational numbers and non-rational numbers together–were defined in terms of geometry, as lengths and distance.

Euclidean geometry was not applied in spaces of more than three dimensions until the 19th century. Ludwig Schläfli generalized Euclidean geometry to spaces of n dimensions using both synthetic and algebraic methods, and discovered all of the regular polytopes (higher-dimensional analogues of the Platonic solids) that exist in Euclidean spaces of any number of dimensions.^[4]

Despite the wide use of Descartes' approach, which was called analytic geometry, the definition of Euclidean space remained unchanged until the end of 19th century. The introduction of abstract vector spaces 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 points satisfying certain relationships, expressible in terms of distance and angles. For example, there are two fundamental operations (referred to as motions) on the plane. One is translation, which means a shifting of the plane so that every point is shifted in the same direction and by the same distance. The other is rotation 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 subsets) of the plane should be considered equivalent (congruent) if one can be transformed into the other by some sequence of translations, rotations and reflections (see below).

In order to make all of this mathematically precise, the theory must clearly define what is a Euclidean space, and the related notions of distance, angle, translation, and rotation. Even when used in physical theories, Euclidean space is an abstraction detached from actual physical locations, specific reference frames, measurement instruments, and so on. A purely mathematical definition of Euclidean space also ignores questions of units of length and other physical dimensions: the distance in a "mathematical" space is a number, not something expressed in inches or metres.

The standard way to mathematically define a Euclidean space, as carried out in the remainder of this article, is to define a Euclidean space as a set of points on which acts a real vector space, the space of translations which is equipped with an inner product.^[1] The action of translations makes the space an affine space, and this allows defining lines, planes, subspaces, dimension, and parallelism. The inner product allows defining distance and angles.

The set ${\displaystyle \mathbb {R} ^{n}}$ of n-tuples of real numbers equipped with the dot product is a Euclidean space of dimension n. Conversely, the choice of a point called the origin and an orthonormal basis of the space of translations is equivalent with defining an isomorphism between a Euclidean space of dimension n and ${\displaystyle \mathbb {R} ^{n}}$ viewed as a Euclidean space.

It follows that everything that can be said about a Euclidean space can also be said about ${\displaystyle \mathbb {R} ^{n}.}$ Therefore, many authors, specially at elementary level, call ${\displaystyle \mathbb {R} ^{n}}$ the standard Euclidean space of dimension n,^[5] or simply the Euclidean space of dimension n.

A reason for introducing such an abstract definition of Euclidean spaces, and for working with it instead of ${\displaystyle \mathbb {R} ^{n}}$ 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 Euclidean vector space is a finite-dimensional inner product space over the real numbers.

A Euclidean space is an affine space 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.^[6]

If E is a Euclidean space, its associated vector space is often denoted ${\displaystyle {\overrightarrow {E}}.}$ The dimension of a Euclidean space is the dimension of its associated vector space.

The elements of E are called points and are commonly denoted by capital letters. The elements of ${\displaystyle {\overrightarrow {E}}}$ are called Euclidean vectors or free vectors. They are also called translations, although, properly speaking, a translation is the geometric transformation resulting of the action of a Euclidean vector on the Euclidean space.

The action of a translation v on a point P provides a point that is denoted P + v. This action satisfies

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

(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 (P, Q) there is exactly one vector v such that P + v = Q. This vector v is denoted Q − P or ${\displaystyle {\overrightarrow {PQ}}.}$

As previously explained, some of the basic properties of Euclidean spaces result of the structure of affine space. They are described in § Affine structure and its subsections. The properties resulting from the inner product are explained in § Metric structure 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 ${\displaystyle \mathbb {R} ^{n}}$ viewed as a vector space equipped with the dot product as an inner product. The importance of this particular example of Euclidean space lies in the fact that every Euclidean space is isomorphic to it. More precisely, given a Euclidean space E of dimension n, the choice of a point, called an origin and an orthonormal basis of ${\displaystyle {\overrightarrow {E}}}$ defines an isomorphism of Euclidean spaces from E to ${\displaystyle \mathbb {R} ^{n}.}$

As every Euclidean space of dimension n is isomorphic to it, the Euclidean space ${\displaystyle \mathbb {R} ^{n}}$ is sometimes called the standard Euclidean space of dimension n. ^[5]

Affine structure[]

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

Subspaces[]

Let E be a Euclidean space and ${\displaystyle {\overrightarrow {E}}}$ its associated vector space.

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

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

is a linear subspace of ${\displaystyle {\overrightarrow {E}}.}$ A Euclidean subspace F is a Euclidean space with ${\displaystyle {\overrightarrow {F}}}$ as associated vector space. This linear subspace ${\displaystyle {\overrightarrow {F}}}$ is called the direction of F.

If P is a point of F then

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

Conversely, if P is a point of E and V is a linear subspace of ${\displaystyle {\overrightarrow {E}},}$ 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 ${\displaystyle E={\overrightarrow {E}}}$) 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

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

where P and Q 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 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 necessary on the line).

In a Euclidean vector space, the zero vector is usually chosen for O; this allows simplifying the preceding formula into

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

A standard convention allows using this formula in every Euclidean space, see Affine space § Affine combinations and barycenter.

The line segment, or simply segment, joining the points P and Q is the subset of the points such that 0 ≤