Dimension

dimensionsdimensionalone-dimensional
A surface such as a plane or the surface of a cylinder or sphere has a dimension of two because two coordinates are needed to specify a point on itfor example, both a latitude and longitude are required to locate a point on the surface of a sphere. The inside of a cube, a cylinder or a sphere is three-dimensional because three coordinates are needed to locate a point within these spaces. In classical mechanics, space and time are different categories and refer to absolute space and time. That conception of the world is a four-dimensional space but not the one that was found necessary to describe electromagnetism.

Line (geometry)

linestraight linelines
The vector equation of the line through points A and B is given by (where λ is a scalar). If a is vector OA and b is vector OB, then the equation of the line can be written:. A ray starting at point A is described by limiting λ. One ray is obtained if λ ≥ 0, and the opposite ray comes from λ ≤ 0. In three-dimensional space, a first degree equation in the variables x, y, and z defines a plane, so two such equations, provided the planes they give rise to are not parallel, define a line which is the intersection of the planes. More generally, in n-dimensional space n-1 first-degree equations in the n coordinate variables define a line under suitable conditions.

Euclidean vector

vectorvectorsvector addition
In physics and engineering, a vector is typically regarded as a geometric entity characterized by a magnitude and a direction. It is formally defined as a directed line segment, or arrow, in a Euclidean space. In pure mathematics, a vector is defined more generally as any element of a vector space. In this context, vectors are abstract entities which may or may not be characterized by a magnitude and a direction. This generalized definition implies that the above-mentioned geometric entities are a special kind of vectors, as they are elements of a special kind of vector space called Euclidean space. This article is about vectors strictly defined as arrows in Euclidean space.

Euclidean geometry

plane geometryEuclideanEuclidean plane geometry
Starting with Moritz Pasch in 1882, many improved axiomatic systems for geometry have been proposed, the best known being those of Hilbert, George Birkhoff, and Tarski. Einstein's theory of special relativity involves a four-dimensional space-time, the Minkowski space, which is non-Euclidean. This shows that non-Euclidean geometries, which were introduced a few years earlier for showing that the parallel postulate cannot be proved, are also useful for describing the physical world. However, the three-dimensional "space part" of the Minkowski space remains the space of Euclidean geometry.

Dot product

scalar productdotinner product
The dot product of two vectors = and = is defined as: where Σ denotes summation and n is the dimension of the vector space. For instance, in three-dimensional space, the dot product of vectors and is: If vectors are identified with row matrices, the dot product can also be written as a matrix product Expressing the above example in this way, a 1 × 3 matrix (row vector) is multiplied by a 3 × 1 matrix (column vector) to get a 1 × 1 matrix that is identified with its unique entry: :. In Euclidean space, a Euclidean vector is a geometric object that possesses both a magnitude and a direction. A vector can be pictured as an arrow.

Real number

realrealsreal-valued
In French mathematics, the positive real numbers and negative real numbers commonly include zero, and these sets are noted respectively ℝ + and ℝ −. In this understanding, the respective sets without zero are called strictly positive real numbers and strictly negative real numbers, and are noted ℝ + * and ℝ − *. The notation R n refers to the Cartesian product of n copies of R, which is an n-dimensional vector space over the field of the real numbers; this vector space may be identified to the n-dimensional space of Euclidean geometry as soon as a coordinate system has been chosen in the latter.

Cartesian coordinate system

Cartesian coordinatesCartesian coordinateCartesian
Cartesian coordinates are the foundation of analytic geometry, and provide enlightening geometric interpretations for many other branches of mathematics, such as linear algebra, complex analysis, differential geometry, multivariate calculus, group theory and more. A familiar example is the concept of the graph of a function. Cartesian coordinates are also essential tools for most applied disciplines that deal with geometry, including astronomy, physics, engineering and many more. They are the most common coordinate system used in computer graphics, computer-aided geometric design and other geometry-related data processing.

Two-dimensional space

Euclidean planetwo-dimensional2D
Later, the plane was thought of as a field, where any two points could be multiplied and, except for 0, divided. This was known as the complex plane. The complex plane is sometimes called the Argand plane because it is used in Argand diagrams. These are named after Jean-Robert Argand (1768–1822), although they were first described by Danish-Norwegian land surveyor and mathematician Caspar Wessel (1745–1818). Argand diagrams are frequently used to plot the positions of the poles and zeroes of a function in the complex plane. In mathematics, analytic geometry (also called Cartesian geometry) describes every point in two-dimensional space by means of two coordinates.

Elementary mathematics

Basic Mathematicselementarypure mathematics
Analytic geometry is the study of geometry using a coordinate system. This contrasts with synthetic geometry. Usually the Cartesian coordinate system is applied to manipulate equations for planes, straight lines, and squares, often in two and sometimes in three dimensions. Geometrically, one studies the Euclidean plane (2 dimensions) and Euclidean space (3 dimensions). As taught in school books, analytic geometry can be explained more simply: it is concerned with defining and representing geometrical shapes in a numerical way and extracting numerical information from shapes' numerical definitions and representations.

Discrete geometry

combinatorial geometrydiscretecombinatorial
The following are some of the aspects of polytopes studied in discrete geometry: Packings, coverings, and tilings are all ways of arranging uniform objects (typically circles, spheres, or tiles) in a regular way on a surface or manifold. A sphere packing is an arrangement of non-overlapping spheres within a containing space. The spheres considered are usually all of identical size, and the space is usually three-dimensional Euclidean space.

Axiom

axiomspostulateaxiomatic
Structuralist mathematics goes further, and develops theories and axioms (e.g. field theory, group theory, topology, vector spaces) without any particular application in mind. The distinction between an "axiom" and a "postulate" disappears. The postulates of Euclid are profitably motivated by saying that they lead to a great wealth of geometric facts. The truth of these complicated facts rests on the acceptance of the basic hypotheses. However, by throwing out Euclid's fifth postulate, one can get theories that have meaning in wider contexts (e.g., hyperbolic geometry). As such, one must simply be prepared to use labels such as "line" and "parallel" with greater flexibility.

Linear algebra

linearlinear algebraiclinear-algebraic
There is a strong relationship between linear algebra and geometry, which started with the introduction by René Descartes, in 1637, of Cartesian coordinates. In this new (at that time) geometry, now called Cartesian geometry, points are represented by Cartesian coordinates, which are sequences of three real numbers (in the case of the usual three-dimensional space). The basic objects of geometry, which are lines and planes are represented by linear equations. Thus, computing intersections of lines and planes amounts to solving systems of linear equations. This was one of the main motivations for developing linear algebra.

Orthogonality

orthogonalorthogonallyorthogonal vector
The orthogonal complement of a subspace is the space of all vectors that are orthogonal to every vector in the subspace. In a three-dimensional Euclidean vector space, the orthogonal complement of a line through the origin is the plane through the origin perpendicular to it, and vice versa. Note that the geometric concept of two planes being perpendicular does not correspond to the orthogonal complement, since in three dimensions a pair of vectors, one from each of a pair of perpendicular planes, might meet at any angle. In four-dimensional Euclidean space, the orthogonal complement of a line is a hyperplane and vice versa, and that of a plane is a plane.

Metric space

metricmetric spacesmetric geometry
The Euclidean metric defines the distance between two points as the length of the straight line segment connecting them. Other metric spaces occur for example in elliptic geometry and hyperbolic geometry, where distance on a sphere measured by angle is a metric, and the hyperboloid model of hyperbolic geometry is used by special relativity as a metric space of velocities. In 1906 Maurice Fréchet introduced metric spaces in his work Sur quelques points du calcul fonctionnel. However the name is due to Felix Hausdorff.

Cross product

vector cross productvector productcross-product
In mathematics, the cross product or vector product (occasionally directed area product to emphasize the geometric significance) is a binary operation on two vectors in three-dimensional space and is denoted by the symbol \times. Given two linearly independent vectors \mathbf{a} and \mathbf{b}, the cross product, (read "a cross b"), is a vector that is perpendicular to both \mathbf{a} and \mathbf{b} and thus normal to the plane containing them. It has many applications in mathematics, physics, engineering, and computer programming. It should not be confused with the dot product (projection product).

Rotation (mathematics)

rotationrotationsrotate
In general (even for vectors equipped with a non-Euclidean Minkowski quadratic form) the rotation of a vector space can be expressed as a bivector. This formalism is used in geometric algebra and, more generally, in the Clifford algebra representation of Lie groups. In the case of a positive-definite Euclidean quadratic form, the double covering group of the isometry group is known as the Spin group,. It can be conveniently described in terms of a Clifford algebra. Unit quaternions give the group. In spherical geometry, a direct motion of the n-sphere (an example of the elliptic geometry) is the same as a rotation of (n + 1) -dimensional Euclidean space about the origin ( SO(n + 1) ).

Angle

acute angleobtuse angleoblique
In plane geometry, an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle. Angles formed by two rays lie in a plane, but this plane does not have to be a Euclidean plane. Angles are also formed by the intersection of two planes in Euclidean and other spaces. These are called dihedral angles. Angles formed by the intersection of two curves in a plane are defined as the angle determined by the tangent rays at the point of intersection. Similar statements hold in space, for example, the spherical angle formed by two great circles on a sphere is the dihedral angle between the planes determined by the great circles.

Curvature

curvednegative curvatureextrinsic curvature
For a curve drawn on a surface (embedded in three-dimensional Euclidean space), several curvatures are defined, which relates the direction of curvature to the surface's unit normal vector. These are the normal curvature, geodesic curvature and geodesic torsion. Any non-singular curve on a smooth surface has its tangent vector T contained in the tangent plane of the surface.

Mathematical analysis

analysisclassical analysisanalytic
The idea of normed vector space was in the air, and in the 1920s Banach created functional analysis. In mathematics, a metric space is a set where a notion of distance (called a metric) between elements of the set is defined. Much of analysis happens in some metric space; the most commonly used are the real line, the complex plane, Euclidean space, other vector spaces, and the integers. Examples of analysis without a metric include measure theory (which describes size rather than distance) and functional analysis (which studies topological vector spaces that need not have any sense of distance).

Differential geometry

differentialdifferential geometerdifferential geometry and topology
Differential geometry is a mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry. The theory of plane and space curves and surfaces in the three-dimensional Euclidean space formed the basis for development of differential geometry during the 18th century and the 19th century. Since the late 19th century, differential geometry has grown into a field concerned more generally with the geometric structures on differentiable manifolds. Differential geometry is closely related to differential topology and the geometric aspects of the theory of differential equations.

David Hilbert

HilbertHilbert, DavidD. Hilbert
Geometry may treat things, about which we have powerful intuitions, but it is not necessary to assign any explicit meaning to the undefined concepts. The elements, such as point, line, plane, and others, could be substituted, as Hilbert is reported to have said to Schoenflies and Kötter, by tables, chairs, glasses of beer and other such objects. It is their defined relationships that are discussed. Hilbert first enumerates the undefined concepts: point, line, plane, lying on (a relation between points and lines, points and planes, and lines and planes), betweenness, congruence of pairs of points (line segments), and congruence of angles.

Affine space

affine subspaceaffineaffine line
Euclidean spaces (including the one-dimensional line, two-dimensional plane, and three-dimensional space commonly studied in elementary geometry, as well as higher-dimensional analogues) are affine spaces. Indeed, in most modern definitions, a Euclidean space is defined to be an affine space, such that the associated vector space is a real inner product space of finite dimension, that is a vector space over the reals with a positive-definite quadratic form q(x) .

Manifold

manifoldsboundarymanifold with boundary
Examples include the plane, the sphere, and the torus, which can all be embedded (formed without self-intersections) in three dimensional real space, but also the Klein bottle and real projective plane, which will always self-intersect when immersed in three-dimensional real space. Although a manifold locally resembles Euclidean space, meaning that every point has a neighbourhood homeomorphic to an open subset of Euclidean space, globally it may be not homeomorphic to Euclidean space.

Topology

topologicaltopologicallytopologist
The cube and the sphere are homeomorphic, as are the coffee cup and the doughnut. But the circle is not homeomorphic to the doughnut. While topological spaces can be extremely varied and exotic, many areas of topology focus on the more familiar class of spaces known as manifolds. A manifold is a topological space that resembles Euclidean space near each point. More precisely, each point of an n-dimensional manifold has a neighborhood that is homeomorphic to the Euclidean space of dimension n. Lines and circles, but not figure eights, are one-dimensional manifolds. Two-dimensional manifolds are also called surfaces, although not all surfaces are manifolds.

Real coordinate space

standard topologyreal spaceR''' 3
With various numbers of dimensions (sometimes unspecified), R n is used in many areas of pure and applied mathematics, as well as in physics. With component-wise addition and scalar multiplication, it is the prototypical real vector space and is a frequently used [[#Euclidean space|representation of Euclidean n -space]]. Due to the latter fact, geometric metaphors are widely used for R n, namely a plane for R 2 and three-dimensional space for R 3 . For any natural number n, the set R n consists of all n-tuples of real numbers ( R ). It is called (the) "n-dimensional real space".