Dimension

Dimension is the number of degrees of freedom or independent parameters needed to define the position of a point on an object.
A point has zero dimensions, a line has one dimension, and a plane has two dimensions.
The dimension of an object is independent of the dimension of the space it is embedded in.
The Minkowski dimension and Hausdorff dimension are used to define dimension in spaces other than Euclidean space.
Development of higher-dimensional geometry began in the 19th century.

Dimension in Vector Spaces

The dimension of a vector space is the number of vectors in any basis for the space.
It is also referred to as the 'Hamel dimension' or 'algebraic dimension.'
For non-free cases, the dimension generalizes to the length of a module.
The dimension of a vector space determines the number of coordinates needed to specify any vector.
The concept of dimension in vector spaces is fundamental in linear algebra.

The dimension of a connected topological manifold is the dimension of Euclidean space to which it is locally homeomorphic.
For connected differentiable manifolds, the dimension is also the dimension of the tangent vector space at any point.
Geometric topology distinguishes dimensions 1 and 2 as relatively elementary, while dimensions 3 and 4 are considered more difficult.
The Poincaré conjecture, which involves different dimensions, required multiple proof methods.
Complex dimension is half the real dimension, and it is useful in the study of complex manifolds and algebraic varieties.

The dimension of an algebraic variety can be defined as the dimension of the tangent space at any regular point.
Another definition is the number of hyperplanes needed to intersect the variety at a finite number of points.
These definitions are equivalent and intuitive ways to determine the dimension of an algebraic variety.
Algebraic varieties may have different dimensions depending on the base field used.
Complex coordinate systems can be applied to objects with two real dimensions, resulting in a single complex dimension.

Dimension is not restricted to physical objects and is frequently used in mathematics and sciences.
High-dimensional spaces are common in various mathematical fields, such as Lagrangian or Hamiltonian mechanics.
Superstring theory uses 10 dimensions, while supergravity and M-theory use 11 dimensions.
The state-space of quantum mechanics is an infinite-dimensional function space.
Dimension is an abstract concept that plays a crucial role in understanding and describing various mathematical and physical phenomena.

Reference	URL
Glossary	https://harryandcojewellery.com.au/blogs/glossary/dimension
Wikipedia	http://en.wikipedia.org/wiki/Dimension
Wikidata	https://www.wikidata.org/wiki/Q4440864
Knowledge Graph	https://www.google.com/search?kgmid=/m/02bht