Dimension

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