Three-dimensional space

In geometry, a three-dimensional space is a mathematical space in which three values are required to determine the position of a point. Alternatively, it can be referred to as 3D space, 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, a solid figure.
Technically, a tuple of numbers can be understood as the Cartesian coordinates of a location in a -dimensional Euclidean space. The set of these -tuples is commonly denoted and can be identified to the pair formed by a -dimensional Euclidean space and a Cartesian coordinate system.
When, this space is called the three-dimensional Euclidean space. 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 compelling and useful 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, 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:
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, 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 which had a vanishing scalar component, that is,.
While not explicitly studied by Hamilton, this work indirectly introduced notions of basis, here given by the quaternion elements, as well as the dot product and cross product, which correspond to 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

In mathematics, analytic 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, and. 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.
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.
Below are images of the above-mentioned systems.

Lines and planes

Two distinct points always determine a line. Three distinct points are either collinear or determine a unique plane. On the other hand, four distinct points can either be collinear, coplanar, or determine the entire space.
Two distinct lines can either intersect, be parallel or be skew. Two parallel lines, or two intersecting lines, lie in a unique plane, so skew lines are lines that do not meet and do not lie in a common plane.
Two distinct planes can either meet in a common line or are parallel. 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.
A line can lie in a given plane, intersect that plane in a unique point, or be parallel to the plane. In the last case, lines can be formed in the plane that are parallel to the given line.
A hyperplane 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 single linear 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.
Varignon's theorem states that the midpoints of any quadrilateral in form a parallelogram, and hence are coplanar.

Spheres and balls

A sphere in 3-space consists of the set of all points in 3-space at a fixed distance from a central point. The solid enclosed by the sphere is called a ball.
The volume of the ball is given by
and the surface area of the sphere is
Another type of sphere arises from a 4-ball, whose three-dimensional surface is the 3-sphere: points equidistant to the origin of the euclidean space. If a point has coordinates,, then characterizes those points on the unit 3-sphere centered at the origin.
This 3-sphere is an example of a 3-manifold: a space which is 'looks locally' like 3-D space. In precise topological terms, each point of the 3-sphere has a neighborhood which is homeomorphic to an open subset of 3-D space.

Polytopes

In three dimensions, there are nine regular polytopes: the five convex Platonic solids and the four nonconvex Kepler-Poinsot polyhedra.

Surfaces of revolution

A surface generated by revolving a plane curve about a fixed line in its plane as an axis is called a surface of revolution. The plane curve is called the generatrix of the surface. A section of the surface, made by intersecting the surface with a plane that is perpendicular to the axis, is a circle.
Simple examples occur when the generatrix is a line. If the generatrix line intersects the axis line, the surface of revolution is a right circular cone with vertex the point of intersection. However, if the generatrix and axis are parallel, then the surface of revolution is a circular cylinder.

Quadric surfaces

In analogy with the conic sections, the set of points whose Cartesian coordinates satisfy the general equation of the second degree, namely,
where and are real numbers and not all of and are zero, is called a quadric surface.
There are six types of non-degenerate quadric surfaces:

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. Elliptic cones are sometimes considered to be degenerate quadric surfaces as well.
Both the hyperboloid of one sheet and the hyperbolic paraboloid are ruled 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. Each family is called a regulus.