In geometry, a three-dimensional (3D) space is a mathematical space in which three values (termed coordinates) are required to determine the position of a point. Alternatively, it can be referred to as 3-space or, rarely, tri-dimensional space. Most commonly, it means 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 refer colloquially to a subset of space, a three-dimensional region (or 3D domain), 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). 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. While this space remains the most widely used way to model the world as it is experienced, 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 same plane. Furthermore, if these directions are pairwise perpendicular, the three values are often labeled by the terms width/breadth, height/depth, and length.
History The philosopher Aristotle 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. Books XI to XIII of Euclid's Elements dealt with three-dimensional geometry. Book XI develops notions of perpendicularity, parallelism, and orthogonality of lines and planes, the construction and properties of angles, and parallelepiped solids. Book XII discusses infinitesimals and the method of exhaustion for finding the area of a circle or the volume of a pyramid, cone, cylinder, or sphere. Book XIII describes the construction of the five regular Platonic solids in a sphere, covering the cube, octahedra, icosahedra and dodecahedra. In the 17th century, three-dimensional space was described with Cartesian coordinates, with the advent of analytic geometry developed by René Descartes in his work La Géométrie. Pierre de Fermat independently developed similar ideas in the manuscript Ad locos planos et solidos isagoge (Introduction to Plane and Solid Loci), which was unpublished during Fermat's lifetime. Fermat's work on seeking extrema of a curve would lay the groundwork for differential calculus. Isaac Newton introduced the polar coordinate system as an alternative non-Cartesian system that is useful for certain geometries. The 18th century, Alexis Clairaut studied algebraic curves in space, the concept of tangent space and curvature, and the use of calculus for this purpose. Leonhard Euler studied the notion of a geodesic on a surface deriving the first analytical geodesic equation, and later introduced the first set of intrinsic coordinate systems on a surface, beginning the theory of intrinsic 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, known as Euler's theorem. Later in the century, Gaspard Monge made important contributions to the study of curves and surfaces in space. The work of Euler and Monge laid the foundations for differential geometry. In the 19th century, developments of the geometry of three-dimensional space came with William Rowan Hamilton's development of the quaternions, a hypercomplex number system. For this purpose, Hamilton coined the terms scalar and vector, and they were first defined in the three dimensional sense 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 a vanishing scalar component, that is, a = 0 {\displaystyle a=0} . While not explicitly studied by Hamilton, this work 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. It was not until Josiah Willard Gibbs that these two products were identified in their own right, and the modern notation for the dot and cross product were introduced in his classroom teaching notes, found also in the 1901 textbook Vector Analysis written by Edwin Bidwell Wilson based on Gibbs' lectures. Further development came in the abstract formalism of vector spaces, with the work of Hermann Grassmann and Giuseppe Peano, the latter of whom first gave the modern definition of vector spaces as an algebraic structure. The development of matrix mathematics and its application to n-dimensional geometry was made by Arthur Cayley.
In Euclidean geometry
Coordinate systems
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