that are one-dimensional spaces but are usually referred to by more specific terms. Any field K {\displaystyle K} is a one-dimensional vector space over itself...

three-dimensional Euclidean space, that is, the Euclidean space of dimension three, which models physical space. More general three-dimensional spaces are...

Four-dimensional space (4D) is the mathematical extension of the concept of three-dimensional space (3D). Three-dimensional space is the simplest possible...

A two-dimensional space is a mathematical space with two dimensions, meaning points have two degrees of freedom: their locations can be locally described...

case of metric spaces, (n + 1)-dimensional balls have n-dimensional boundaries, permitting an inductive definition based on the dimension of the boundaries...

finite-dimensional if the dimension of V {\displaystyle V} is finite, and infinite-dimensional if its dimension is infinite. The dimension of the vector space V {\displaystyle...

mathematics, a zero-dimensional topological space (or nildimensional space) is a topological space that has dimension zero with respect to one of several inequivalent...

A five-dimensional (5D) space is a space with five dimensions. 5D Euclidean geometry designated by the mathematical sign: E {\displaystyle \mathbb {E}...

Although the space we live in is considered three-dimensional, there are practical applications for four-dimensional space. Quaternions, one of the ways...

Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's Elements, it was the three-dimensional space...

also refer to a seven-dimensional manifold such as a 7-sphere, or a variety of other geometric constructions. Seven-dimensional spaces have a number of special...

in n-dimensional space. When n = 8, the set of all such locations is called 8-dimensional space. Often such spaces are studied as vector spaces, without...

half-space is either of the two parts into which a plane divides the three-dimensional Euclidean space. If the space is two-dimensional, then a half-space...

Euclidean space of dimension n, En (Euclidean line, E; Euclidean plane, E2; Euclidean three-dimensional space, E3) form a real coordinate space of dimension n...

describes a one-dimensional space curve. At a point r(t = c) = a = (a1, a2, ..., an) for some constant t = c, the equations of the one-dimensional tangent...

The set of all one-dimensional linear subspaces of a (n+1)-dimensional linear space is, by definition, a n-dimensional projective space. And the affine...

covering dimension or topological dimension of a topological space is one of several different ways of defining the dimension of the space in a topologically...

1D and 2D space that preserves locality fairly well. This means that two data points which are close to each other in one-dimensional space are also close...

dimension is an infinite cardinal. Finite-dimensional vector spaces occur naturally in geometry and related areas. Infinite-dimensional vector spaces...

generally an n-dimensional unit hypercube). Because Giuseppe Peano (1858–1932) was the first to discover one, space-filling curves in the 2-dimensional plane are...

circle. In Euclidean 3-space, a ball is taken to be the region of space bounded by a 2-dimensional sphere. In a one-dimensional space, a ball is a line segment...

the span of this vector as a one dimensional subspace of Rn, then the complement is an (n − 1)-dimensional vector space that is invariant under an orthogonal...

Dimensionality reduction, or dimension reduction, is the transformation of data from a high-dimensional space into a low-dimensional space so that the...

−3. Likewise, in genus 1, there is a one-dimensional space of curves, but every such curve has a one-dimensional group of automorphisms. Hence, the stack...

covered) and continuously, so that a one-dimensional object completely fills up a higher-dimensional object. Every space-filling curve hits some points multiple...

for more details. dimension The projective space, being the "space" of all one-dimensional linear subspaces of a given vector space V is generalized to...

second-order control system. It models the dynamics of a simple mass in one-dimensional space under the effect of a time-varying force input u {\displaystyle...

in the space without any size or shape: zero-dimensional. Through any pair of points an infinite straight line can be drawn, a one-dimensional set of...

plane is a flat two-dimensional surface that extends indefinitely. Euclidean planes often arise as subspaces of three-dimensional space R 3 {\displaystyle...

reflections, each of them having an entire (n − 1)-dimensional flat of fixed points in a n-dimensional space. Mathematically, a rotation is a map. All rotations...