Threedimensional space (mathematics)
Geometry  

History  
Geometers  
by name


by period


Threedimensional space (also: tridimensional space) is a geometric threeparameter model of the physical universe (without considering time) in which all known matter exists. These three dimensions can be labeled by a combination of three chosen from the terms length, width, height, depth, and breadth. Any three directions can be chosen, provided that they do not all lie in the same plane.
In physics and mathematics, a sequence of n numbers can be understood as a location in ndimensional space. When n = 3, the set of all such locations is called threedimensional Euclidean space. It is commonly represented by the symbol . This space is only one example of a great variety of spaces in three dimensions called 3manifolds.
Contents
In geometry
Coordinate systems
In mathematics, analytic geometry (also called Cartesian geometry) describes every point in threedimensional 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 threedimensional 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 threedimensional space include cylindrical coordinates and spherical coordinates, though there is an infinite number of possible methods. See Euclidean space.
Below are images of the abovementioned systems.
Polytopes
In three dimensions, there are nine regular polytopes: the five convex Platonic solids and the four nonconvex KeplerPoinsot polyhedra.
Class  Platonic solids  KeplerPoinsot polyhedra  

Symmetry  T_{d}  O_{h}  I_{h}  
Coxeter group  A_{3}, [3,3]  B_{3}, [4,3]  H_{3}, [5,3]  
Order  24  48  120  
Regular polyhedron 
{3,3} 
{4,3} 
{3,4} 
{5,3} 
{3,5} 
{5/2,5} 
{5,5/2} 
{5/2,3} 
{3,5/2} 
Sphere
A sphere in 3space (also called a 2sphere because its surface is 2dimensional) consists of the set of all points in 3space at a fixed distance r from a central point P. The volume enclosed by this surface is:
Another type of sphere, but having a threedimensional surface is the 3sphere: points equidistant to the origin of the euclidean space at distance one. If any position is , then characterize a point in the 3sphere.
Orthogonality
In the familiar 3dimensional space that we live in, there are three pairs of cardinal directions: north/south (latitude), east/west (longitude) and up/down (altitude). These pairs of directions are mutually orthogonal: They are at right angles to each other. Movement along one axis does not change the coordinate value of the other two axes. In mathematical terms, they lie on three coordinate axes, usually labelled x, y, and z. The zbuffer in computer graphics refers to this zaxis, representing depth in the 2dimensional imagery displayed on the computer screen.
In linear algebra
Another mathematical way of viewing threedimensional 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 threedimensional because every point in space can be described by a linear combination of three independent vectors.
Dot product, angle, and length
The dot product of two vectors A = [A_{1}, A_{2},A_{3}] and B = [B_{1}, B_{2},B_{3}] is defined as:^{[1]}
A vector can be pictured as an arrow. Its magnitude is its length, and its direction is the direction the arrow points. The magnitude of a vector A is denoted by . In this viewpoint, the dot product of two Euclidean vectors A and B is defined by^{[2]}
where θ is the angle between A and B.
The dot product of a vector A by itself is
which gives
the formula for the Euclidean length of the vector.
Cross product
The cross product or vector product is a binary operation on two vectors in threedimensional 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 neither commutative nor associative, but is a Lie algebra with the cross product being the Lie bracket.
One can in n dimensions take the product of n − 1 vectors to produce a vector perpendicular to all of them. But if the product is limited to nontrivial binary products with vector results, it exists only in three and seven dimensions.^{[3]}
In calculus
Gradient, divergence and curl
In a rectangular coordinate system, the gradient is given by
The divergence of a continuously differentiable vector field F = U i + V j + W k is equal to the scalarvalued function:
Expanded in Cartesian coordinates (see Del in cylindrical and spherical coordinates for spherical and cylindrical coordinate representations), the curl ∇ × F is, for F composed of [F_{x}, F_{y}, F_{z}]:
where i, j, and k are the unit vectors for the x, y, and zaxes, respectively. This expands as follows:^{[4]}
Line integrals, surface integrals, and volume integrals
For some scalar field f : U ⊆ R^{n} → R, the line integral along a piecewise smooth curve C ⊂ U is defined as
where r: [a, b] → C is an arbitrary bijective parametrization of the curve C such that r(a) and r(b) give the endpoints of C and .
For a vector field F : U ⊆ R^{n} → R^{n}, the line integral along a piecewise smooth curve C ⊂ U, in the direction of r, is defined as
where · is the dot product and r: [a, b] → C is a bijective parametrization of the curve C such that r(a) and r(b) give the endpoints of C.
A surface integral is a generalization of multiple integrals to integration over surfaces. It can be thought of as the double integral analog of the line integral. To find an explicit formula for the surface integral, we need to parameterize the surface of interest, S, by considering a system of curvilinear coordinates on S, like the latitude and longitude on a sphere. Let such a parameterization be x(s, t), where (s, t) varies in some region T in the plane. Then, the surface integral is given by
where the expression between bars on the righthand side is the magnitude of the cross product of the partial derivatives of x(s, t), and is known as the surface element. Given a vector field v on S, that is a function that assigns to each x in S a vector v(x), the surface integral can be defined componentwise according to the definition of the surface integral of a scalar field; the result is a vector.
A volume integral refers to an integral over a 3dimensional domain.
It can also mean a triple integral within a region D in R^{3} of a function and is usually written as:
Fundamental theorem of line integrals
The fundamental theorem of line integrals, says that a line integral through a gradient field can be evaluated by evaluating the original scalar field at the endpoints of the curve.
Let . Then
Stokes' theorem
Stokes' theorem relates the surface integral of the curl of a vector field F over a surface Σ in Euclidean threespace to the line integral of the vector field over its boundary ∂Σ:
Divergence theorem
Suppose V is a subset of (in the case of n = 3, V represents a volume in 3D space) which is compact and has a piecewise smooth boundary S (also indicated with ∂V = S ). If F is a continuously differentiable vector field defined on a neighborhood of V, then the divergence theorem says:^{[5]}
The left side is a volume integral over the volume V, the right side is the surface integral over the boundary of the volume V. The closed manifold ∂V is quite generally the boundary of V oriented by outwardpointing normals, and n is the outward pointing unit normal field of the boundary ∂V. (dS may be used as a shorthand for ndS.)
In topology
Threedimensional 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.^{[6]}
With the space , the topologists locally model all other 3manifolds.
See also
 3D science and technology (disambiguation)
 3manifolds
 Dimensional analysis
 Distance from a point to a plane
 Skew lines#Distance
 Space
 Threedimensional graph
 Twodimensional space
References
 ↑ S. Lipschutz, M. Lipson (2009). Linear Algebra (Schaum’s Outlines) (4th ed.). McGraw Hill. ISBN 9780071543521.
 ↑ M.R. Spiegel, S. Lipschutz, D. Spellman (2009). Vector Analysis (Schaum’s Outlines) (2nd ed.). McGraw Hill. ISBN 9780071615457.
 ↑ WS Massey (1983). "Cross products of vectors in higher dimensional Euclidean spaces". The American Mathematical Monthly. 90 (10): 697–701. JSTOR 2323537. doi:10.2307/2323537.
If one requires only three basic properties of the cross product ... it turns out that a cross product of vectors exists only in 3dimensional and 7dimensional Euclidean space.
 ↑ Arfken, p. 43.
 ↑ M. R. Spiegel; S. Lipschutz; D. Spellman (2009). Vector Analysis. Schaum’s Outlines (2nd ed.). USA: McGraw Hill. ISBN 9780071615457.
 ↑ Rolfsen, Dale (1976). Knots and Links. Berkeley, California: Publish or Perish. ISBN 0914098160.
External links
Wikiquote has quotations related to: Threedimensional space (mathematics) 
Wikimedia Commons has media related to 3D. 
 The dictionary definition of threedimensional at Wiktionary
 Weisstein, Eric W., "FourDimensional Geometry", MathWorld.
 Elementary Linear Algebra  Chapter 8: Threedimensional Geometry Keith Matthews from University of Queensland, 1991