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'Three-dimensional space' is the physical universe we live in. The three dimensions are commonly called length, width, and breadth, although any three mutually perpendicular directions can serve as the three dimensions. Pictures are commonly two dimensional, they lack depth. A three-dimensional picture, as one sees with a stereopticon or a View Master, tricks the human eye into experiencing the illusion of depth.
In physics, our three-dimensional space is viewed as embedded in 4-dimensional space-time, called Minkowski space (see special relativity). The idea behind space-time is that time is perpendicular to each of the three spacial dimensions.
In mathematics, Cartesian geometry (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 ''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.
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. See Euclidean space.
Another mathematical 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. In this view, space-time is four dimensional because the location of a point in time is independent of its location in space.
Three-dimensional space has a number of properties that distinguish it from spaces of other dimensions. It is, for example, the only dimension in which it is possible to tie a knot in a piece of string ^[1]. Many of the laws of physics, such as the various inverse square laws, depend on dimension three ^[2].
The understanding of three-dimensional space in humans is thought to be learned during infancy using unconscious inference, and is closely related to hand-eye coordination. The visual ability to perceive the world in three dimensions is called depth perception.

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References

1. Dale Rolfsen, ''Knots and Links'', Publish or Perish, Berkeley, 1976, ISBN 0-914098-16-0
2. Brian Greene, ''The Fabric of the Cosmos'', Random House, New York, 2003, ISBN 0-375-72720-5