Three-dimensional space

Three-dimensional space is a geometric 3-parameters model of the physical universe (without considering time) in which we live. These three dimensions are commonly called length, width, and depth (or height), although any three directions can be chosen, provided that they do not lie in the same plane.

In physics and mathematics, a sequence of n numbers can be understood as a location in n-dimensional space. When n = 3, the set of all such locations is called 3-dimensional Euclidean space. As commonly the mathematician symbolizise, \scriptstyle{\mathbb{R}}^3, this space is only one example of a great variety of spaces in three dimensions called 3-manifolds.

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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, usually 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 is 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 dimension numbers. For example, at least 3 dimensions are required 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.

With the space \scriptstyle{\mathbb{R}}^3, the topologists locally model all other 3-manifolds.

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 hyperbolic-orthogonal to each of the three spatial dimensions.

Geometry

Polytopes

Main article: Polyhedron

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

Tetrahedron  
Cube  
Octahedron  
Dodecahedron  
Icosahedron  
Small stellated dodecahedron  
Great dodecahedron  
Great stellated dodecahedron  
Great icosahedron  

Hypersphere

Main article: Sphere

A hypersphere in 3-space (also called a 2-sphere because its surface is 2-dimensional) consists of the set of all points in 3-space at a fixed distance r from a central point P. The volume enclosed by this surface is:

V = \frac{4}{3}\pi r^{3}

Another hypersphere, but having three dimensions is the 3-sphere: points equidistant to the origin of the euclidean space \mathbb{R}^4 at distance one. If any position is P=(x,y,z,t), then x^2%2By^2%2Bz^2%2Bt^2=1 characterize a point in the 3-sphere.

Orthogonality

In the familiar 3-dimensional space that we live in, there are three pairs of cardinal directions: up/down (altitude), north/south (latitude), and east/west (longitude). These pairs of directions are mutually orthogonal: They are at right angles to each other. In mathematical terms, they lie on three coordinate axes, usually labelled x, y, and z. The z-buffer in computer graphics refers to this z-axis, representing depth in the 2-dimensional imagery displayed on the computer screen.

Coordinate systems

Main article: Coordinate system

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.

Other popular methods of describing the location of a point in three-dimensional 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.

See also

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