mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, a plane is a two-dimensional space or flat surface that extends indefinitely. A plane is the two-dimensional analogue of a point (zero dimensions), a line (one dimension) and

three-dimensional space In geometry, a three-dimensional space (3D space, 3-space or, rarely, tri-dimensional space) is a mathematical space in which three values ('' coordinates'') are required to determine the position of a point. Most commonly, it is the three- ...

. When working exclusively in two-dimensional

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 ...

, the definite article is used, so ''the''

Euclidean plane In mathematics, a Euclidean plane is a Euclidean space of Two-dimensional space, dimension two, denoted \textbf^2 or \mathbb^2. It is a geometric space in which two real numbers are required to determine the position (geometry), position of eac ...

refers to the whole space. Several notions of a plane may be defined. The Euclidean plane follows

Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry, ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small set ...

, and in particular the

parallel postulate In geometry, the parallel postulate is the fifth postulate in Euclid's ''Elements'' and a distinctive axiom in Euclidean geometry. It states that, in two-dimensional geometry: If a line segment intersects two straight lines forming two interior ...

. A projective plane may be constructed by adding "points at infinity" where two otherwise parallel lines would intersect, so that every pair of lines intersects in exactly one point. The elliptic plane may be further defined by adding a metric to the real projective plane. One may also conceive of a hyperbolic plane, which obeys

hyperbolic geometry In mathematics, hyperbolic geometry (also called Lobachevskian geometry or János Bolyai, Bolyai–Nikolai Lobachevsky, Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For a ...

and has a negative

curvature In mathematics, curvature is any of several strongly related concepts in geometry that intuitively measure the amount by which a curve deviates from being a straight line or by which a surface deviates from being a plane. If a curve or su ...

. Abstractly, one may forget all structure except the topology, producing the topological plane, which is homeomorphic to an open disk. Viewing the plane as an

affine space In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties relat ...

produces the affine plane, which lacks a notion of distance but preserves the notion of collinearity. Conversely, in adding more structure, one may view the plane as a 1-dimensional

complex manifold In differential geometry and complex geometry, a complex manifold is a manifold with a ''complex structure'', that is an atlas (topology), atlas of chart (topology), charts to the open unit disc in the complex coordinate space \mathbb^n, such th ...

, called the complex line. Many fundamental tasks in mathematics,

geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

trigonometry Trigonometry () is a branch of mathematics concerned with relationships between angles and side lengths of triangles. In particular, the trigonometric functions relate the angles of a right triangle with ratios of its side lengths. The fiel ...

graph theory In mathematics and computer science, graph theory is the study of ''graph (discrete mathematics), graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of ''Vertex (graph ...

, and graphing are performed in a two-dimensional or ''planar'' space.

Euclidean plane

Embedding in three-dimensional space

Elliptic plane

Projective plane

Further generalizations

In addition to its familiar geometric structure, with

isomorphism In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...

s that are isometries with respect to the usual inner product, the plane may be viewed at various other levels of

abstraction Abstraction is a process where general rules and concepts are derived from the use and classifying of specific examples, literal (reality, real or Abstract and concrete, concrete) signifiers, first principles, or other methods. "An abstraction" ...

. Each level of abstraction corresponds to a specific category. At one extreme, all geometrical and metric concepts may be dropped to leave the topological plane, which may be thought of as an idealized homotopically trivial infinite rubber sheet, which retains a notion of proximity, but has no distances. The topological plane has a concept of a linear path, but no concept of a straight line. The topological plane, or its equivalent the open disc, is the basic topological neighborhood used to construct surfaces (or 2-manifolds) classified in low-dimensional topology. Isomorphisms of the topological plane are all continuous

bijection In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Equival ...

s. The topological plane is the natural context for the branch of

that deals with planar graphs, and results such as the four color theorem. The plane may also be viewed as an

, whose isomorphisms are combinations of translations and non-singular linear maps. From this viewpoint there are no distances, but collinearity and ratios of distances on any line are preserved.

Differential geometry Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of Calculus, single variable calculus, vector calculus, lin ...

views a plane as a 2-dimensional real

manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a N ...

, a topological plane which is provided with a differential structure. Again in this case, there is no notion of distance, but there is now a concept of smoothness of maps, for example a differentiable or smooth path (depending on the type of differential structure applied). The isomorphisms in this case are bijections with the chosen degree of differentiability. In the opposite direction of abstraction, we may apply a compatible field structure to the geometric plane, giving rise to the

complex plane In mathematics, the complex plane is the plane (geometry), plane formed by the complex numbers, with a Cartesian coordinate system such that the horizontal -axis, called the real axis, is formed by the real numbers, and the vertical -axis, call ...

and the major area of

complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebraic ...

. The complex field has only two isomorphisms that leave the real line fixed, the identity and

conjugation Conjugation or conjugate may refer to: Linguistics *Grammatical conjugation, the modification of a verb from its basic form *Emotive conjugation or Russell's conjugation, the use of loaded language Mathematics *Complex conjugation, the change o ...

. In the same way as in the real case, the plane may also be viewed as the simplest, one-dimensional (in terms of

complex dimension In mathematics, complex dimension usually refers to the dimension of a complex manifold or a complex algebraic variety. These are spaces in which the local neighborhoods of points (or of non-singular points in the case of a variety) are modeled on ...

, over the complex numbers)

, sometimes called the complex line. However, this viewpoint contrasts sharply with the case of the plane as a 2-dimensional real manifold. The isomorphisms are all conformal bijections of the complex plane, but the only possibilities are maps that correspond to the composition of a multiplication by a complex number and a translation. In addition, the Euclidean geometry (which has zero

everywhere) is not the only geometry that the plane may have. The plane may be given a spherical geometry by using the

stereographic projection In mathematics, a stereographic projection is a perspective transform, perspective projection of the sphere, through a specific point (geometry), point on the sphere (the ''pole'' or ''center of projection''), onto a plane (geometry), plane (th ...

. This can be thought of as placing a sphere tangent to the plane (just like a ball on the floor), removing the top point, and projecting the sphere onto the plane from this point. This is one of the projections that may be used in making a flat map of part of the Earth's surface. The resulting geometry has constant positive curvature. Alternatively, the plane can also be given a metric which gives it constant negative curvature giving the hyperbolic plane. The latter possibility finds an application in the theory of

special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory of the relationship between Spacetime, space and time. In Albert Einstein's 1905 paper, Annus Mirabilis papers#Special relativity, "On the Ele ...

in the simplified case where there are two spatial dimensions and one time dimension. (The hyperbolic plane is a timelike hypersurface in three-dimensional Minkowski space.)

Topological and differential geometric notions

The one-point compactification of the plane is homeomorphic to a

sphere A sphere (from Ancient Greek, Greek , ) is a surface (mathematics), surface analogous to the circle, a curve. In solid geometry, a sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...

(see

); the open disk is homeomorphic to a sphere with the "north pole" missing; adding that point completes the (compact) sphere. The result of this compactification is a

referred to as the Riemann sphere or the

complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...

projective line In projective geometry and mathematics more generally, a projective line is, roughly speaking, the extension of a usual line by a point called a '' point at infinity''. The statement and the proof of many theorems of geometry are simplified by the ...

. The projection from the Euclidean plane to a sphere without a point is a

diffeomorphism In mathematics, a diffeomorphism is an isomorphism of differentiable manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are continuously differentiable. Definit ...

and even a

conformal map In mathematics, a conformal map is a function (mathematics), function that locally preserves angles, but not necessarily lengths. More formally, let U and V be open subsets of \mathbb^n. A function f:U\to V is called conformal (or angle-prese ...

. The plane itself is homeomorphic (and diffeomorphic) to an open disk. For the hyperbolic plane such diffeomorphism is conformal, but for the Euclidean plane it is not.

References

{{reflist Geometry Surfaces