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A hyperplane is an affine subspace of an n-dimensional space that is (n  − 1)-dimensional. In other words, it is 1 dimension lower than its ambient dimension.

The word "hyperplane" is sometimes used to mean an affine subspace of any dimension, thus generalizing the word for a plane analogously to how a hypercube generalizes a cube, or a hypersphere generalizes a sphere. However, this usage of the term isn't standard and should be avoided.

For polytopes, a finite enclosure of a hyperplane that lives in a hyperplane is commonly called a facet.