Higher dimension as a term in mathematics most commonly refers to any number of spatial dimensions greater than three.

The three standard dimensions are length, width, and breadth (or height). The first higher dimension required is often time, and space-time is the most common example of a four-dimensional space.

In more abstract branches of pure mathematics, however, any independent variable represents a dimension, (in a vector space, vector in a set of linearly independent vectors). There are higher-dimensional spaces for any number larger than three. In Hilbert space there are infinitely many dimensions. These definitions go beyond any immediate geometrical intuitions, but have proved invaluable in extending the range of mathematical methods. For example, in statistics, factor analysis postulates a dimension per factor under consideration.

In geometric topology, the nature of the difficulties in the subject has turned out to be such that dimensions 3 and 4 are the most resistant (see for example Whitney disc). Therefore in that context higher dimension usually means dimension ≥ 5.

See also

• Fourth dimension
• Hyperspace
• Planes of existence
• Dimensions in science fiction
• Multiverse