Blade (geometry)


In the study of geometric algebras, a blade is a generalization of the concept of scalars and vectors to include simple bivectors, trivectors, etc. Specifically, a -blade is any object that can be expressed as the exterior product of vectors, and is of grade.
In detail:
For an -dimensional space, there are blades of all grades from 0 to inclusive. A vector subspace of finite dimension may be represented by the -blade formed as a wedge product of all the elements of a basis for that subspace.

Examples

For example, in 2-dimensional space scalars are described as 0-blades, vectors are 1-blades, and area elements are 2-blades known as pseudoscalars, in that they are elements of a one-dimensional space distinct from regular scalars.
In three-dimensional space, 0-blades are again scalars and 1-blades are three-dimensional vectors, and 2-blades are oriented area elements. 3-blades represent volume elements and in three-dimensional space; these are scalar-like—i.e., 3-blades in three-dimensions form a one-dimensional vector space.