Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's Elements, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any positive integer dimension n, which are called Euclidean n-spaces when one wants to specify their dimension. For n equal to one or two, they are commonly called respectively Euclidean lines and Euclidean planes. The qualifier "Euclidean" is used to distinguish Euclidean spaces from other spaces that were later considered in physics and modern mathematics.
Ancient Greek geometers introduced Euclidean space for modeling the physical space. Their work was collected by the ancient Greek mathematician Euclid in his Elements, with the great innovation of proving all properties of the space as theorems, by starting from a few fundamental properties, called postulates, which either were considered as evident, or seemed impossible to prove.
After the introduction at the end of the 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. 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, it is usually possible to work with a specific Euclidean space, denoted or, which can be represented using Cartesian coordinates as the real -space equipped with the standard dot product.
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 these allow reducing geometric problems to algebraic computations with numbers. This reduction of geometry to algebra was a major change in point of view, as, until then, the real numbers were defined in terms of lengths and distances.
Euclidean geometry was not applied in spaces of dimension more than three until the 19th century. Ludwig Schläfli generalized Euclidean geometry to spaces of dimension, using both synthetic and algebraic methods, and discovered all of the regular polytopes that exist in Euclidean spaces of any dimension.
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 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 of the plane should be considered equivalent if one can be transformed into the other by some sequence of translations, rotations and reflections.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 as a set of points on which a real vector space acts – the space of translations which is equipped with an inner product. 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 of -tuples of real numbers equipped with the dot product is a Euclidean space of dimension. 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 and viewed as a Euclidean space.
It follows that everything that can be said about a Euclidean space can also be said about Therefore, many authors, especially at elementary level, call 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 instead of is that it is often preferable to work in a coordinate-free and origin-free manner. Another reason is that there is no standard origin nor any standard basis in the physical world.
Technical definition
A 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 to distinguish them from Euclidean vector spaces.
If is a Euclidean space, its associated vector space is often denoted The dimension of a Euclidean space is the dimension of its associated vector space.
The elements of are called points, and are commonly denoted by capital letters. The elements of are called Euclidean vectors or free vectors. They are also called translations, although, properly speaking, a translation is the geometric transformation resulting from the action 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
Note: The second in the left-hand side is a vector addition; each other denotes an action of a vector on a point. This notation is not ambiguous, as, to distinguish between the two meanings of, it suffices to look at the nature of its left argument.
The fact that the action is free and transitive means that, for every pair of points, there is exactly one displacement vector such that. This vector is denoted or
As previously explained, some of the basic properties of Euclidean spaces result from 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 the associated vector space.A typical case of Euclidean vector space is 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 of dimension, the choice of a point, called an origin and an orthonormal basis of defines an isomorphism of Euclidean spaces from to
As every Euclidean space of dimension is isomorphic to it, the Euclidean space is sometimes called the standard Euclidean space of dimension.
Affine structure
Some basic properties of Euclidean spaces depend only on 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 be a Euclidean space and its associated vector space.A flat, Euclidean subspace or affine subspace of is a subset of such that
as the associated vector space of is a linear subspace of A Euclidean subspace is a Euclidean space with as the associated vector space. This linear subspace is also called the direction of.
If is a point of then
Conversely, if is a point of and is a linear subspace of then
is a Euclidean subspace of direction.
A Euclidean vector space 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 formwhere and are two distinct points of the Euclidean space as a part of the line.
It follows that there is exactly one line that passes through 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
where is an arbitrary point.
In a Euclidean vector space, the zero vector is usually chosen for ; this allows simplifying the preceding formula into
A standard convention allows using this formula in every Euclidean space, see.
The line segment, or simply segment, joining the points and is the subset of points such that in the preceding formulas. It is denoted or ; that is