nLab hyperplane

Idea

By a hyperplane one usually means an affine or linear subspace of an affine space or linear space, respectively, typically required to be a positive dimension and codimension and often required to have codimension = 1.

Examples

In the archetypical example the ambient space is the time-honored Euclidean/Cartesian space $\mathbb{R}^3$ and a hyperplane of codimension 1 is an ordinary plane.

More sophisticated examples arise for instance in the discussion of configuration spaces of points in a Cartesian space $E = \mathbb{R}^d$, where hyperplanes in $E^n$ given by $\big\{ (\vec x_1, \cdots, \vec x_n) \,\in\, E^n \;\big\vert\; \vec x_i = \vec x_j \big\}$ reflect the subspaces where positions of a given pair of points coincide. This leads to discussion of arrangements of hyperplanes and their complements, see also at Knizhnik-Zamolodchikov equation the references on twisted cohomology of configuration spaces of points.

Related concepts

• plane

• hypercube

• hyperplane line bundle

• arrangements of hyperplanes

• distribution of subspaces

References

See also:

• Wikipedia, Hyperplane