dimensionsthree

dimensionsthree refers to the three-dimensional Euclidean space, commonly denoted R^3. It comprises all points with three real coordinates and is the standard mathematical model of physical space in classical physics and geometry.

In dimensionsthree, the distance between two points p = (x1, y1, z1) and q = (x2, y2, z2) is sqrt((x1 − x2)^2 + (y1 − y2)^2 + (z1 − z2)^2). As a real vector space, dimensionsthree has dimension three; the standard basis e1 = (1, 0, 0), e2 = (0, 1, 0), e3 = (0, 0, 1) allows any vector v to be expressed as (x, y, z). The dot product defines lengths and angles; the cross product produces a vector orthogonal to a pair of vectors; the triple scalar product gives a signed volume. The space is equipped with an inner product and a norm.

Common coordinate systems include Cartesian coordinates (x, y, z) and alternative systems such as cylindrical (r, θ, z) and spherical (ρ, φ, θ).

Topologically, dimensionsthree carries the Euclidean topology; it is simply connected and locally flat. In topology, a 3-manifold is a space where every point has a neighborhood homeomorphic to dimensionsthree.

History and significance: The explicit use of three-dimensional coordinates by Descartes and Fermat laid the groundwork for analytic geometry. Today dimensionsthree is central to physics, engineering, computer graphics, architecture, and many sciences, even though physical theories describe higher-dimensional spaces when needed (for example, spacetime in relativity is four-dimensional).

Related notions include 3D space in computer graphics and robotics, 3D printing, and the study of 3-manifolds in topology.