Euclidean space

Around 300 BC, the Greek mathematician Euclid undertook a study of relationships among distances and angles, first in a plane (an idealized flat surface) and then in space. Events By place Egypt Pyrrhus, the King of Epirus, is taken as a hostage to Egypt after the Battle of Ipsus The term ancient Greece refers to the period of Greek history lasting from the Greek Dark Ages ca A mathematician is a person whose primary area of study and research is the field of Mathematics. Euclid ( Greek:.) fl 300 BC also known as Euclid of Alexandria, is often referred to as the Father of Geometry Distance is a numerical description of how far apart objects are In Geometry and Trigonometry, an angle (in full plane angle) is the figure formed by two rays sharing a common Endpoint, called An example of such a relationship is that the sum of the angles in a triangle is always 180 degrees. A triangle is one of the basic Shapes of Geometry: a Polygon with three corners or vertices and three sides or edges which are Line In Mathematics, there are several meanings of degree depending on the subject Today these relationships are known as two- and three-dimensional Euclidean geometry. In mathematics the dimension of a Space is roughly defined as the minimum number of Coordinates needed to specify every point within it Euclidean geometry is a mathematical system attributed to the Greek Mathematician Euclid of Alexandria.

In modern mathematical language, distance and angle can be generalized easily to 4-dimensional, 5-dimensional, and even higher-dimensional spaces. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and An n-dimensional space with notions of distance and angle that obey the Euclidean relationships is called an n-dimensional Euclidean space. Most of this article is devoted to developing the modern language necessary for the conceptual leap to higher dimensions.

An essential property of a Euclidean space is its flatness. Other spaces exist in geometry that are not Euclidean. Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position For example, the surface of a sphere is not; a triangle on a sphere (suitably defined) will have angles that sum to something greater than 180 degrees. In fact, there is essentially only one Euclidean space of each dimension, while there are many non-Euclidean spaces of each dimension. Often these other spaces are constructed by systematically deforming Euclidean space.

1 Intuitive overview
2 Real coordinate space
3 Euclidean structure
4 Topology of Euclidean space
5 Generalizations
6 See also
7 References

Intuitive overview

One way to think of the Euclidean plane is as a set of points satisfying certain relationships, expressible in terms of distance and angle. In Geometry, Topology and related branches of mathematics a spatial point describes a specific point within a given space that consists of neither Volume 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. In Euclidean geometry, a translation is moving every point a constant distance in a specified direction 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. A rotation is a movement of an object in a circular motion A two- Dimensional object rotates around a center (or point) of rotation One of the basic tenets of Euclidean geometry is that two figures (that is, subsets) 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 Mathematics, the Euclidean group E ( n) sometimes called ISO( n) or similar is the Symmetry group of n -dimensional )

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. In Mathematics, the real numbers may be described informally in several different ways In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added In Mathematics, an inner product space is a Vector space with the additional Structure of inner product. For then:

the vectors 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. 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 Addition is the mathematical process of putting things together

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 space, on which a vector space acts. In Mathematics, an affine space is an abstract structure that generalises the affine-geometric properties of Euclidean space. In Algebra and Geometry, a group action is a way of describing symmetries of objects using groups. 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 Mathematics, the origin of a Euclidean space is a special point, usually denoted by the letter O, used as a fixed point of reference In this article, this technicality is largely ignored.

Real coordinate space

Let R denote the field of real numbers. In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division In Mathematics, the real numbers may be described informally in several different ways For any non-negative integer n, the space of all n-tuples of real numbers forms an n-dimensional vector space over R, which is denoted Rⁿ and sometimes called real coordinate space. The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French In Mathematics, a tuple is a Sequence (also known as an "ordered list" of values called the components of the tuple An element of Rⁿ is written

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

where each x_i is a real number. The vector space operations on Rⁿ 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 Rⁿ comes with a standard basis:

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

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

$\vdots$

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

An arbitrary vector in Rⁿ can then be written in the form

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

Rⁿ is the prototypical example of a real n-dimensional vector space. In Mathematics, the standard basis (also called natural basis or canonical basis) of the n- dimensional Euclidean space In fact, every real n-dimensional vector space V is isomorphic to Rⁿ. In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective This isomorphism is not canonical, however. Canonical is an Adjective derived from canon. Canon comes from the Greek word kanon, "rule" (perhaps originally from A choice of isomorphism is equivalent to a choice of basis for V (by looking at the image of the standard basis for Rⁿ in V). Basis vector redirects here For basis vector in the context of crystals see Crystal structure. The reason for working with arbitrary vector spaces instead of Rⁿ 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 Rⁿ. In Mathematics, the dot product, also known as the scalar product, is an operation which takes two vectors over the Real numbers R 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

$\|\mathbf{x}\| = \sqrt{\mathbf{x}\cdot\mathbf{x}} = \sqrt{\sum_{i=1}^{n}(x_i)^2}.$

This length function satisfies the required properties of a norm and is called the Euclidean norm on Rⁿ. In Linear algebra, Functional analysis and related areas of Mathematics, a norm is a function that assigns a strictly positive length

The (non-reflex) angle θ (0° ≤ θ ≤ 180°) between x and y is then given by

$\theta = \cos^{-1}\left(\frac{\mathbf{x}\cdot\mathbf{y}}{\|\mathbf{x}\|\|\mathbf{y}\|}\right)$

where cos⁻¹ is the arccosine function.

Finally, one can use the norm to define a metric (or distance function) on Rⁿ by

$d(\mathbf{x}, \mathbf{y}) = \|\mathbf{x} - \mathbf{y}\| = \sqrt{\sum_{i=1}^n (x_i - y_i)^2}.$

This distance function is called the Euclidean metric. In Mathematics, a metric or distance function is a function which defines a Distance between elements of a set. In Mathematics, the Euclidean distance or Euclidean metric is the "ordinary" Distance between two points that one would measure with a ruler It can be viewed as a form of the Pythagorean theorem. In Mathematics, the Pythagorean theorem ( American English) or Pythagoras' theorem ( British English) is a relation in Euclidean geometry

Real coordinate space together with this Euclidean structure is called Euclidean space and often denoted Eⁿ. (Many authors refer to Rⁿ itself as Euclidean space, with the Euclidean structure being understood). The Euclidean structure makes Eⁿ an inner product space (in fact a Hilbert space), a normed vector space, and a metric space. In Mathematics, an inner product space is a Vector space with the additional Structure of inner product. This article assumes some familiarity with Analytic geometry and the concept of a limit. In 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 In Mathematics, a metric space is a set where a notion of Distance (called a metric) between elements of the set is defined

Rotations of Euclidean space are then defined as orientation-preserving linear transformations T that preserve angles and lengths:

$T\mathbf{x} \cdot T\mathbf{y} = \mathbf{x} \cdot \mathbf{y},$

$|T\mathbf{x}| = |\mathbf{x}|.$

In the language of matrices, rotations are special orthogonal matrices. See also Orientation (geometry. In Mathematics, an orientation on a real Vector space is a choice of which In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally In Matrix theory, a real orthogonal matrix is a square matrix Q whose Transpose is its inverse: Q^T

Topology of Euclidean space

Since Euclidean space is a metric space it is also a topological space with the natural topology induced by the metric. In Mathematics, a metric space is a set where a notion of Distance (called a metric) between elements of the set is defined Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of In Mathematics, a metric space is a set where a notion of Distance (called a metric) between elements of the set is defined The metric topology on Eⁿ is called the Euclidean topology. A set is open in the Euclidean topology if and only if it contains an open ball around each of its points. In Metric topology and related fields of Mathematics, a set U is called open if intuitively speaking starting from any point x in ↔ In Mathematics, a ball is the inside of a Sphere; both concepts apply not only in the three-dimensional space but also for lower and higher dimensions and for metric The Euclidean topology turns out to be equivalent to the product topology on Rⁿ considered as a product of n copies of the real line R (with its standard topology). In Topology and related areas of Mathematics, a product space is the Cartesian product of a family of Topological spaces equipped with a natural In Mathematics, the real line is simply the set R of singleton Real numbers However this term is usually used when R is to be treated as a

An important result on the topology of Rⁿ, that is far from superficial, is Brouwer's invariance of domain. Luitzen Egbertus Jan Brouwer ɛxˈbɛʁtəs jɑn ˈbʁʌuəʁ ( February 27 1881, Overschie – December 2 1966, Blaricum Invariance of domain is a theorem in Topology about Homeomorphic Subsets of Euclidean space R n. Any subset of Rⁿ (with its subspace topology) that is homeomorphic to another open subset of Rⁿ is itself open. In Topology and related areas of Mathematics, a subspace of a Topological space X is a Subset S of X which is Topological equivalence redirects here see also Topological equivalence (dynamical systems. An immediate consequence of this is that R^m is not homeomorphic to Rⁿ 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 Hausdorff topological space that is locally diffeomorphic to Euclidean space. A differentiable manifold is a type of Manifold that is locally similar enough to Euclidean space to allow one to do Calculus. In Topology and related branches of Mathematics, a Hausdorff space, separated space or T2 space is a Topological space In Mathematics, a diffeomorphism is an Isomorphism of Smooth manifolds It is an Invertible function that maps one Differentiable 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 spaces, then the result is what is called a Riemannian manifold. In Mathematics, the tangent space of a Manifold is a concept which facilitates the generalization of vectors from Affine spaces to general manifolds since In Riemannian geometry, a Riemannian manifold ( M, g) (with Riemannian metric g) is a real Differentiable manifold M 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 curved, non-Euclidean manner. In Mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry The simplest Riemannian manifold, consisting of Rⁿ 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 space. A pseudo-Euclidean space is a finite- Dimensional real Vector space together with a non-degenerate indefinite Quadratic form Smooth manifolds built from such spaces are called pseudo-Riemannian manifolds. In Differential geometry, a pseudo-Riemannian manifold (also called a semi-Riemannian manifold) is a generalization of a Riemannian manifold. Perhaps their most famous application is the theory of relativity, where empty spacetime with no matter is represented by the flat pseudo-Euclidean space called Minkowski space, spacetimes with matter in them form other pseudo-Riemannian manifolds, and gravity corresponds to the curvature of such a manifold. This page is about the scientific concept of relativity for philosophical or sociological theories about relativity see Relativism. SpaceTime is a patent-pending three dimensional graphical user interface that allows end users to search their content such as Google Google Images Yahoo! YouTube eBay Amazon and RSS Matter is commonly defined as being anything that has mass and that takes up space. In Physics and Mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einstein's theory of Special relativity Gravitation is a natural Phenomenon by which objects with Mass attract one another

Our universe, being subject to relativity, is not Euclidean. This becomes significant in theoretical considerations of astronomy and cosmology, and also in some practical problems such as global positioning and airplane navigation. Astronomy (from the Greek words astron (ἄστρον "star" and nomos (νόμος "law" is the scientific study Cosmology (from Greek grc κοσμολογία - grc κόσμος kosmos, "universe" and grc -λογία -logia) is study Basic concept of GPS operation A GPS receiver calculates its position by carefully timing the signals sent by the constellation of GPS Satellites high above the Earth Overview Fixed-wing aircraft range from small training and recreational aircraft to Wide-body aircraft and military cargo aircraft. Navigation is the process of reading and controlling the movement of a craft or vehicle from one place to another Nonetheless, a Euclidean model of the universe can still be used to solve many other practical problems with sufficient precision.

References

Kelley, John L. Elliptic geometry is also sometimes called Riemannian geometry. In Linear algebra, an Euclidean subspace (or subspace of R n) is a set of vectors that is closed under addition In Mathematics, the Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane In Mathematics, the polar coordinate system is a two-dimensional Coordinate system in which each point on a plane is determined by (1975). General Topology. Springer-Verlag. ISBN 0-387-90125-6.
Munkres, James (1999). Topology. Prentice-Hall. ISBN 0-13-181629-2.

Dictionary

-noun

Ordinary two- or three-dimensional space, characterised by an infinite extent along each dimension and a constant distance between any pair of parallel lines.
(mathematics) any vector space on which a real-valued inner product is defined.

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