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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-dimensional Euclidean space, that is, the Euclidean space of dimension three, which models physical space. More general three-dimensional spaces are called 3-manifolds. The term may also refer colloquially to a subset of space, a three-dimensional region (or 3D domain),[1] a solid figure.

Technically, a tuple of n numbers can be understood as the Cartesian coordinates of a location in a n-dimensional Euclidean space. The set of these n-tuples is commonly denoted R n , {\displaystyle \mathbb {R} ^{n},} and can be identified to the pair formed by a n-dimensional Euclidean space and a Cartesian coordinate system.When n = 3, this space is called the three-dimensional Euclidean space (or simply "Euclidean space" when the context is clear).[2] In classical physics, it serves as a model of the physical universe, in which all known matter exists. When relativity theory is considered, it can be considered a local subspace of space-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 large variety of spaces in three dimensions called 3-manifolds. 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 same plane. Furthermore, if these directions are pairwise perpendicular, the three values are often labeled by the terms width/breadth, height/depth, and length.

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Books XI to XIII of Euclid's Elements dealt with three-dimensional geometry. Book XI develops notions of orthogonality and parallelism of lines and planes, and defines solids including parallelpipeds, pyramids, prisms, spheres, octahedra, icosahedra and dodecahedra. Book XII develops notions of similarity of solids. Book XIII describes the construction of the five regular Platonic solids in a sphere.

In the 17th century, three-dimensional space was described with Cartesian coordinates, with the advent of analytic geometry developed by Ren Descartes in his work La Gomtrie and Pierre de Fermat in the manuscript Ad locos planos et solidos isagoge (Introduction to Plane and Solid Loci), which was unpublished during Fermat's lifetime. However, only Fermat's work dealt with three-dimensional space.

In the 19th century, developments of the geometry of three-dimensional space came with William Rowan Hamilton's development of the quaternions. In fact, it was Hamilton who coined the terms scalar and vector, and they were first defined within his geometric framework for quaternions. Three dimensional space could then be described by quaternions q = a + u i + v j + w k {\displaystyle q=a+ui+vj+wk} which had vanishing scalar component, that is, a = 0 {\displaystyle a=0} . While not explicitly studied by Hamilton, this indirectly introduced notions of basis, here given by the quaternion elements i , j , k {\displaystyle i,j,k} , as well as the dot product and cross product, which correspond to (the negative of) the scalar part and the vector part of the product of two vector quaternions.

In mathematics, analytic geometry (also called Cartesian geometry) describes every point in three-dimensional space by means of three coordinates. Three coordinate axes are given, each perpendicular to the other two at the origin, the point at which they cross. They are usually labeled x, y, and z. Relative to these axes, the position of any point in three-dimensional space is given by an ordered triple of real numbers, each number giving the distance of that point from the origin measured along the given axis, which is equal to the distance of that point from the plane determined by the other two axes.[5]

Other popular methods of describing the location of a point in three-dimensional space include cylindrical coordinates and spherical coordinates, though there are an infinite number of possible methods. For more, see Euclidean space.

Two distinct planes can either meet in a common line or are parallel (i.e., do not meet). 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.

Another way of viewing three-dimensional space is found in linear algebra, where the idea of independence is crucial. Space has three dimensions because the length of a box 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 independent vectors.

The cross product or vector product is a binary operation on two vectors in three-dimensional space and is denoted by the symbol . The cross product A B of the vectors A and B is a vector that is perpendicular to both and therefore normal to the plane containing them. It has many applications in mathematics, physics, and engineering.

The space and product form an algebra over a field, which is not commutative nor associative, but is a Lie algebra with the cross product being the Lie bracket. Specifically, the space together with the product, ( R 3 , ) {\displaystyle (\mathbb {R} ^{3},\times )} is isomorphic to the Lie algebra of three-dimensional rotations, denoted s o ( 3 ) {\displaystyle {\mathfrak {so}}(3)} . In order to satisfy the axioms of a Lie algebra, instead of associativity the cross product satisfies the Jacobi identity. For any three vectors A , B {\displaystyle \mathbf {A} ,\mathbf {B} } and C {\displaystyle \mathbf {C} }

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 R 3 {\displaystyle \mathbb {R} ^{3}} 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.

A more abstract description still is to model physical space as a three-dimensional affine space E ( 3 ) {\displaystyle E(3)} over the real numbers. This is unique up to affine isomorphism. It is sometimes referred to as three-dimensional Euclidean space. Just as the vector space description came from 'forgetting the preferred basis' of R 3 {\displaystyle \mathbb {R} ^{3}} , the affine space description comes from 'forgetting the origin' of the vector space. Euclidean spaces are sometimes called Euclidean affine spaces for distinguishing them from Euclidean vector spaces.[12]

A volume integral is an integral over a three-dimensional domain or region.When the integrand is trivial (unity), the volume integral is simply the region's volume.[14][1]It can also mean a triple integral within a region D in R3 of a function f ( x , y , z ) , {\displaystyle f(x,y,z),} and is usually written as:

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 a knot in a piece of string.[16]

Many ideas of dimension can be tested with finite geometry. The simplest instance is PG(3,2), which has Fano planes as its 2-dimensional subspaces. It is an instance of Galois geometry, a study of projective geometry using finite fields. Thus, for any Galois field GF(q), there is a projective space PG(3,q) of three dimensions. For example, any three skew lines in PG(3,q) are contained in exactly one regulus.[17]

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