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Article: 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.

Dimension in Manifolds

  • 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.

Dimension in 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.

Applications of 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.

Dimension Data Sources

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