Euclidean Space

Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called " Euclidean Geometry ", which is the study of the relationships between angles and distances in space. Euclid first developed "plane geometry" which dealt with the geometry of two-dimensional objects on a flat surface. He then went on to develop "solid geometry" which analyzed the geometry of three-dimensional objects. All of the axioms of Euclid have been encoded into an abstract mathematical space known as a two- or three-dimensional Euclidean space. These mathematical spaces may be extended to apply to any dimension, and such a space is called an n-dimensional Euclidean space or an '''n-space'''. This article is concerned with such mathematical spaces.

In order to develop these higher dimensional Euclidean spaces, the properties of the familiar Euclidean spaces must be very carefully expressed and then extended to an arbitrary dimension. Although the resulting mathematics is rather abstract, it nevertheless captures the essential nature of the Euclidean spaces we are all familiar with.

An essential property of a Euclidean space is its flatness. Other spaces exist that are not Euclidean. For example, the surface of a sphere is not a Euclidean space, nor is the four-dimensional spacetime described by the Theory Of Relativity when Gravity is present. The geometry of such spaces is called Non-Euclidean Geometry .

In theoretical physics, the term Euclidean space is usually in the context of being compared to Minkowski Space . While a Euclidean space has only Spacelike dimensions, a Minkowski Space has also one Timelike dimension. Therefore the symmetry group of a Euclidean space is the Rotation Group and for a Minkowski Space it is the Lorentz Group .

INTUITIVE OVERVIEW

One way to think of the Euclidean plane is as a Set of Point s satisfying certain relationships, expressible in terms of distance and angle. 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 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, Subset s) of the plane should be considered equivalent ( Congruent ) if one can be transformed into the other by some sequence of translations and rotations. (See Euclidean Group .)

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 Real Vector Space equipped with an Inner Product . For then:

the Vector s in the vector space correspond to the points of the Euclidean plane,

the Addition 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.

Mathematician

A final wrinkle is that Euclidean space is not technically a vector space but rather an Affine Space , on which a vector space Acts . Intuitively, the distinction just says that there is no canonical choice of where the Origin 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 Field of Real Number s. For any non-negative Integer ''n'', the space of all ''n''- Tuple 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

:x = (''x''₁, ''x''₂, …, ''x''_''n''),

where each ''x''_''i'' is a real number. The vector space operations on R^''n'' are defined by

:

\mathbf{x} + \mathbf{y} = (x_1 + y_1, x_2 + y_2, \ldots, x_n + y_n),

a\,\mathbf{x} = (a x_1, a x_2, \ldots, a x_n).

The vector space R^''n'' comes with a Standard Basis :
:

\mathbf{e}_1 = (1, 0, \ldots, 0),

\mathbf{e}_2 = (0, 1, \ldots, 0),

dots

\mathbf{e}_n = (0, 0, \ldots, 1).

An arbitrary vector in R^''n'' can then be written in the form
:

\mathbf{x} = \sum_{i=1}^n x_i \mathbf{e}_i.

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 Basis 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 Product ) on R^''n''. The inner product of any two vectors '''x''' and '''y''' is defined by

:

\mathbf{x}\cdot\mathbf{y} = \sum_{i=1}^n x_iy_i = x_1y_1+x_2y_2+\cdots+x_ny_n.

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

:<math>d(\mathbf{x}, \mathbf{y}) \\mathbf{x} - \mathbf{y}\ = \sqrt{\sum_{i=1}^n (x_i - y_i)^2}</math>
:<math>T\mathbf{x} \mathbf{x}</math>

	CATEGORIES ABOUT EUCLIDEAN SPACE

	euclidean geometry

	linear algebra

	topological spaces

	norms mathematics

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Euclidean Space