Еуклидов простор

Тачка у тродимензионалном еуклидском простору може се лоцирати по три координате.

Еуклидов простор је фундаментални простор класичне геометрије чија се својства описују аксиомама апсолутне геометрије и Еуклидовим постулатом (аксиомом) о паралелним правама. Originally, that is, 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, including the three-dimensional space and the Euclidean plane (dimension two). 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 were collected by the ancient Greek mathematician Euclid in its 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 (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.

Општије речено, Еуклидов простор се назива m-димензионални метрички простор, у којем је могуће увести Декартов координатни систем и тада се метрика дефинише на следећи начин: растојање између тачке M са координатама $(x_{1},x_{2},...,x_{m})$ и тачке M'$(x_{1}',x_{2}',...,x_{m}')$ израчунава се по формули:

$\rho ={\overline {MM'}}={\sqrt {(x_{1}-x_{1}')^{2}+(x_{2}-x_{2}')^{2}}}.$

Дефиниција

Историја дефиниције

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 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 n, 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 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 (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. 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 $\mathbb {R} ^{n}$ 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 $\mathbb {R} ^{n}$ viewed as a Euclidean space.

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

Види још

• Простор
• Геометрија

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Спољашње везе

• Медији везани за чланак Еуклидов простор на Викимедијиној остави