Ingeometry, athree-dimensional space is amathematical space in which three values (termedcoordinates) are required to determine theposition of apoint. Alternatively, it can be referred to as3D space,3-space or, rarely,tri-dimensional space. Most commonly, it means thethree-dimensional Euclidean space, that is, theEuclidean space ofdimension three, which modelsphysical space. More general three-dimensional spaces are called3-manifolds. The term may refer colloquially to a subset of space, athree-dimensional region (or 3Ddomain),[1] asolid figure.
Technically, atuple ofnnumbers can be understood as theCartesian coordinates of a location in an-dimensional Euclidean space. The set of thesen-tuples is commonly denoted and can be identified to the pair formed by an-dimensional Euclidean space and aCartesian coordinate system.Whenn = 3, this space is called thethree-dimensional Euclidean space (or simply "Euclidean space" when the context is clear).[2] Inclassical physics, it serves as a model of the physicaluniverse, in which all knownmatter exists. Whenrelativity theory is considered, it can be considered a local subspace ofspace-time.[3] While this space remains the most compelling and useful way to model the world as it is experienced,[4] it is only one example of a 3-manifold. In this classical example, when the three values refer to measurements in different directions (coordinates), any three directions can be chosen, provided that these directions do not lie in the sameplane. Furthermore, if these directions are pairwiseperpendicular, the three values are often labeled by the termswidth/breadth,height/depth, andlength.
The philosopherAristotle recognised the existence of three dimensions:
A magnitude if divisible one way is a line, if two ways a surface, and if three a body. Beyond these there is no other magnitude, because the three dimensions are all that there are, and that which is divisible in three directions is divisible in all.[5]
The 18th century,Alexis Clairaut studied algebraic curves in space, the concept oftangent space and curvature, and the use of calculus for this purpose.[12][13]Leonhard Euler studied the notion of ageodesic on a surface deriving the first analyticalgeodesic equation,[14] and later introduced the first set of intrinsic coordinate systems on a surface,[13] beginning the theory ofintrinsic geometry upon which modern geometric ideas are based. In 1760, Euler proved a theorem expressing the curvature of a space curve on a surface in terms of the principal curvatures,[15] known asEuler's theorem. Later in the century,Gaspard Monge made important contributions to the study of curves and surfaces in space.[13] The work of Euler and Monge laid the foundations fordifferential geometry.
In the 19th century, developments of the geometry of three-dimensional space came withWilliam Rowan Hamilton's development of thequaternions, ahypercomplex number system. For this purpose, Hamilton coined the termsscalar andvector, and they were first defined in the three dimensional sense withinhis geometric framework for quaternions.[16] Three dimensional space could then be described by quaternions which had a vanishing scalar component, that is,.[17]
While not explicitly studied by Hamilton, this work indirectly introduced notions of basis, here given by the quaternion elements, as well as thedot product andcross product, which correspond to (the negative of) the scalar part and the vector part of the product of two vector quaternions. It was not untilJosiah Willard Gibbs that these two products were identified in their own right,[17] and the modern notation for the dot and cross product were introduced in his classroom teaching notes, found also in the 1901 textbookVector Analysis written byEdwin Bidwell Wilson based on Gibbs' lectures.[18]
In mathematics,analytic geometry (also called Cartesian geometry) describes every point in three-dimensional space by means of three coordinates. Threecoordinate axes are given, each perpendicular to the other two at theorigin, the point at which they cross. They are usually labeledx,y, andz. Relative to these axes, the position of any point in three-dimensional space is given by an ordered triple ofreal numbers, each number giving the distance of that point from theorigin measured along the given axis, which is equal to the distance of that point from the plane determined by the other two axes.[21]
Two distinct points always determine a (straight)line. Three distinct points are eithercollinear or determine a uniqueplane. On the other hand, four distinct points can either be collinear,coplanar, or determine the entire space.[24]
Two distinct lines can either intersect, beparallel or beskew. Two parallel lines, ortwo intersecting lines, lie in a unique plane, so skew lines are lines that do not meet and do not lie in a common plane.[25]
Relations between up to three planes; only in example 12 do three planes meet to form a point
Two distinct planes can either meet in a common line or are parallel (i.e., do not meet).[25] Three distinct planes, no pair of which are parallel, can either meet in a common line, meet in a unique common point, or have no point in common. In the last case, the three lines of intersection of each pair of planes are mutually parallel.[26]
A line can lie in a given plane, intersect that plane in a unique point, or be parallel to the plane.[25] In the last case, lines can be formed in the plane that are parallel to the given line.
Ahyperplane is a subspace of one dimension less than the dimension of the full space. The hyperplanes of a three-dimensional space are the two-dimensional subspaces, that is, the planes. In terms of Cartesian coordinates, the points of a hyperplane satisfy a singlelinear equation, so planes in this 3-space are described by linear equations. A line can be described by a pair of independent linear equations—each representing a plane having this line as a common intersection.[27]
Asphere in 3-space (also called a2-sphere because it is a 2-dimensional object) consists of the set of all points in 3-space at a fixed distancer from a central pointP. The solid enclosed by the sphere is called aball (or, more precisely a3-ball).[29]
The volume of the ball is given by[30]and the surface area of the sphere is[30]
Another type of sphere arises from a 4-ball, whose three-dimensional surface is the3-sphere: points equidistant to the origin of the euclidean spaceR4. If a point has coordinates,P(x,y,z,w), thenx2 +y2 +z2 +w2 = 1 characterizes those points on the unit 3-sphere centered at the origin.[31]
This 3-sphere is an example of a3-manifold: aspace which is 'looks locally' like 3-D space.[32] In precise topological terms, each point of the 3-sphere has a neighborhood which ishomeomorphic to an opensubset of 3-D space.
Asurface generated by revolving a planecurve about a fixed line in its plane as an axis is called asurface of revolution. The plane curve is called thegeneratrix of the surface. A section of the surface, made by intersecting the surface with a plane that is perpendicular (orthogonal) to the axis, is a circle.[34][35]
Simple examples occur when the generatrix is a line. If the generatrix line intersects the axis line, the surface of revolution is a right circularcone with vertex (apex) the point of intersection. However, if the generatrix and axis are parallel, then the surface of revolution is a circularcylinder.[34][35]
In analogy with theconic sections, the set of points whose Cartesian coordinates satisfy the general equation of the second degree, namely,whereA,B,C,F,G,H,J,K,L andM are real numbers and not all ofA,B,C,F,G andH are zero, is called aquadric surface.[36]
The degenerate quadric surfaces are the empty set, a single point, a single line, a single plane, a pair of planes or a quadratic cylinder (a surface consisting of a non-degenerate conic section in a planeπ and all the lines ofR3 through that conic that are normal toπ).[36] Elliptic cones are sometimes considered to be degenerate quadric surfaces as well.[citation needed]
Both the hyperboloid of one sheet and the hyperbolic paraboloid areruled surfaces, meaning that they can be made up from a family of straight lines. In fact, each has two families of generating lines, the members of each family are disjoint and each member of one family intersects, with just one exception, every member of the other family.[36] Each family is called aregulus.[37]
Inlinear algebra, the perspective of three-dimensional space is crucially dependent on the concept of independence. Space has three dimensions because the length of abox is independent of its width or breadth. In the technical language of linear algebra, space is three-dimensional because every point in space can be described by a linear combination of three independentvectors.[38]
A vector can be pictured as an arrow. The vector's magnitude is its length, and its direction is the direction the arrow points. A vector in can be represented by an ordered triple of real numbers. These numbers are called thecomponents of the vector.
The dot product of two vectorsA = [A1,A2,A3] andB = [B1,B2,B3] is defined as:[39]
The magnitude of a vectorA is denoted by||A||. The dot product of a vectorA = [A1,A2,A3] with itself is
For a physical example, consider a block on aninclined plane that is being pulled downward by agravitational force. The dot product can be used to compute thework performed by the constantforce vector that is applied at an angle to the downslope direction of motion. That is:[40]
In function language, the cross product is a function.[43]
The cross-product in respect to a right-handed coordinate system
The components of the cross product are, and can also be written in components, using Einstein summation convention as where is theLevi-Civita symbol.[44] It has the property that.[41]
Its magnitude is related to the angle between and by the identity[41]
One can inn dimensions take the product ofn − 1 vectors to produce a vector perpendicular to all of them. But if the product is limited to non-trivial binary products with vector results, it exists only in three andseven dimensions.[46]
It can be useful to describe three-dimensional space as a three-dimensional vector space over the real numbers. This differs from in a subtle way. By definition, there exists a basis for. This corresponds to anisomorphism between and:[38] the construction for the isomorphism is foundhere. However, there is no 'preferred' or 'canonical basis' for.
On the other hand, there is a preferred basis for, which is due to its description as aCartesian product of copies of, that is,, the three-dimensional Euclidean space.[47] This allows the definition of canonical projections,, where. For example,. This then allows the definition of thestandard basis defined bywhere is theKronecker delta. Written out in full, the standard basis is[48]
Therefore can be viewed as the abstract vector space, together with the additional structure of a choice of basis. Conversely, can be obtained by starting with and 'forgetting' the Cartesian product structure, or equivalently the standard choice of basis.
As opposed to a general vector space, the space is sometimes referred to as a coordinate space.[49]
Physically, it is conceptually desirable to use the abstract formalism in order to assume as little structure as possible if it is not given by the parameters of a particular problem. For example, in a problem with rotational symmetry, working with the more concrete description of three-dimensional space assumes a choice of basis, corresponding to a set of axes. But in rotational symmetry, there is no reason why one set of axes is preferred to say, the same set of axes which has been rotated arbitrarily. Stated another way, a preferred choice of axes breaks the rotational symmetry of physical space.
Computationally, it is necessary to work with the more concrete description in order to do concrete computations.
A more abstract description still is to model physical space as a three-dimensional affine space over the real numbers. This is unique up to affine isomorphism. It is sometimes referred to as three-dimensional Euclidean space.[50] Just as the vector space description came from 'forgetting the preferred basis' of, the affine space description comes from 'forgetting the origin' of the vector space. Euclidean spaces are sometimes calledEuclidean affine spaces for distinguishing them from Euclidean vector spaces.[51]
This is physically appealing as it makes the translation invariance of physical space manifest. A preferred origin breaks the translational invariance.[50]
The above discussion does not involve thedot product. The dot product is an example of aninner product. Physical space can be modelled as a vector space which additionally has the structure of an inner product. The inner product defines notions of length and angle (and therefore in particular the notion of orthogonality).[52] For any inner product, there exist bases under which the inner product agrees with the dot product,[citation needed] but again, there are many different possible bases, none of which are preferred. They differ from one another by a rotation, an element of the group of rotationsSO(3).
Thegradient indicates the direction of greatest increase of a function, and its magnitude. An example is a flow of particles, with the gradient being the magnitude and direction of the flow at a location.[53] In a rectangular coordinate system, the gradient of a differentiable function is given by[54]
Thedivergence indicates the netflux of a vector field around a point, such as an increase or decrease of particle density. That is, whether the location is asource or sink.[56] The divergence of a (differentiable)vector fieldF =Ui +Vj +Wk, that is, a function, is equal to thescalar-valued function:[54]
Illustration of a line integral along curve C in a vector field F
Aline integral of a function along acurve can be thought of as a continuous summation of the function value along every infinitesimal increment of that curve. For somescalar fieldf :U ⊆Rn →R, theline integral along apiecewise smooth curveC ⊂U is defined as[58]
wherer: [a, b] →C is an arbitrarybijective (one-to-one correspondance) parametrization of the curveC such thatr(a) andr(b) give the endpoints ofC and.
For avector fieldF :U ⊆Rn →Rn, the line integral along a piecewise smoothcurveC ⊂U, in the direction ofr, is defined as[58]
where is thedot product andr: [a, b] →C is a bijectiveparametrization of the curveC such thatr(a) andr(b) give the endpoints ofC. A subtype of line integral found in physics is the plane closed loop, which determines the circulation of the function around the loop[59]
where the expression between bars on the right-hand side is themagnitude of thecross product of thepartial derivatives ofx(s,t), and is known as the surfaceelement. Given a vector fieldv onS, that is a function that assigns to eachx inS a vectorv(x), the surface integral can be defined component-wise according to the definition of the surface integral of a scalar field; the result is a vector.
SupposeV is a subset of (in the case ofn = 3,V represents a volume in 3D space) which iscompact and has a piecewisesmooth boundaryS (also indicated with∂V =S). IfF is a continuously differentiable vector field defined on a neighborhood ofV, then thedivergence theorem says:[63]
The left side is avolume integral over the volumeV, the right side is thesurface integral over the boundary of the volumeV. The closed manifold∂V is quite generally the boundary ofV oriented by outward-pointingnormals, andn is the outward pointing unit normal field of the boundary∂V. (dS may be used as a shorthand forndS.)
Three-dimensional space has a number of topological properties that distinguish it from spaces of other dimension numbers. For example, at least three dimensions are required to tie aknot in a piece of string.[64][page needed]
^Hiriart-Urrety, J.-B. (2012)."Mathematical Faits Divers". In Ponstein, Jacob (ed.).Convexity and Duality in Optimization: Proceedings of the Symposium on Convexity and Duality in Optimization Held at the University of Groningen, The Netherlands June 22, 1984. Lecture Notes in Economics and Mathematical Systems. Springer Science & Business Media. p. 3.ISBN978-3-642-45610-7.
^Boyer, C. B. (February 1949). "Newton as an Originator of Polar Coördinates".The American Mathematical Monthly.56 (2). Taylor & Francis, Ltd.:73–78.doi:10.2307/2306162.JSTOR2306162.
^O'Connor, J. J.; Robertson, E. F. (November 2014)."Arthur Cayley".MacTutor. School of Mathematics and Statistics, University of St Andrews, Scotland. Retrieved2025-11-05.
^Massey, W. S. (1983). "Cross products of vectors in higher dimensional Euclidean spaces".The American Mathematical Monthly.90 (10):697–701.doi:10.2307/2323537.JSTOR2323537.If one requires only three basic properties of the cross product ... it turns out that a cross product of vectors exists only in 3-dimensional and 7-dimensional Euclidean space.
^Cameron, P. J.; et al., eds. (1981).Introduction. Finite Geometries and Designs: Proceedings of the Second Isle of Thorns Conference 1980. Lecture note series, London Mathematical Society. Vol. 49. Cambridge University Press. p. 1.ISBN978-0-521-28378-6.
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