# 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 *k*-blade is any object that can be expressed as the exterior product (informally *wedge product*) of *k* vectors, and is of *grade* *k*.

In detail:^{[1]}

- A 0-blade is a scalar.
- A 1-blade is a vector. Every vector is simple.
- A 2-blade is a
*simple*bivector. Linear combinations of 2-blades also are bivectors, but need not be simple, and are hence not necessarily 2-blades. A 2-blade may be expressed as the wedge product of two vectors*a*and*b*: - A 3-blade is a simple trivector, that is, it may be expressed as the wedge product of three vectors
*a*,*b*, and*c*: - In a space of dimension
*n*, a blade of grade*n*− 1 is called a*pseudovector*^{[2]}or an*antivector*.^{[3]} - The highest grade element in a space is called a
*pseudoscalar*, and in a space of dimension*n*is an*n*-blade.^{[4]} - In a space of dimension
*n*, there are*k*(*n*−*k*) + 1 dimensions of freedom in choosing a*k*-blade, of which one dimension is an overall scaling multiplier.^{[5]}

For an *n*-dimensional space, there are blades of all grades from 0 to *n* inclusive. A vector subspace of finite dimension *k* may be represented by the *k*-blade formed as a wedge product of all the elements of a basis for that subspace.^{[6]}

## Examples[edit]

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.

## See also[edit]

## Notes[edit]

**^**Marcos A. Rodrigues (2000). "§1.2 Geometric algebra: an outline".*Invariants for pattern recognition and classification*. World Scientific. p. 3*ff*. ISBN 981-02-4278-6.**^**William E Baylis (2004). "§4.2.3 Higher-grade multivectors in Cℓ_{n}: Duals".*Lectures on Clifford (geometric) algebras and applications*. Birkhäuser. p. 100. ISBN 0-8176-3257-3.**^**Lengyel, Eric (2016).*Foundations of Game Engine Development, Volume 1: Mathematics*. Terathon Software LLC. ISBN 978-0-9858117-4-7.**^**John A. Vince (2008).*Geometric algebra for computer graphics*. Springer. p. 85. ISBN 1-84628-996-3.**^**For Grassmannians (including the result about dimension) a good book is: Griffiths, Phillip; Harris, Joseph (1994),*Principles of algebraic geometry*, Wiley Classics Library, New York: John Wiley & Sons, ISBN 978-0-471-05059-9, MR 1288523. The proof of the dimensionality is actually straightforward. Take*k*vectors and wedge them together and perform elementary column operations on these (factoring the pivots out) until the top*k*×*k*block are elementary basis vectors of . The wedge product is then parametrized by the product of the pivots and the lower*k*× (*n*−*k*) block.**^**David Hestenes (1999).*New foundations for classical mechanics: Fundamental Theories of Physics*. Springer. p. 54. ISBN 0-7923-5302-1.

## References[edit]

- David Hestenes; Garret Sobczyk (1987). "Chapter 1: Geometric algebra".
*Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics*. Springer. p. 1*ff*. ISBN 90-277-2561-6. - Chris Doran & Anthony Lasenby (2003).
*Geometric algebra for physicists*. Cambridge University Press. ISBN 0-521-48022-1. - A Lasenby, J Lasenby & R Wareham (2004)
*A covariant approach to geometry using geometric algebra*Technical Report. University of Cambridge Department of Engineering, Cambridge, UK. - R Wareham; J Cameron & J Lasenby (2005). "Applications of conformal geometric algebra to computer vision and graphics". In Hongbo Li; Peter J. Olver & Gerald Sommer (eds.).
*Computer algebra and geometric algebra with applications*. Springer. p. 329*ff*. ISBN 3-540-26296-2.

## External links[edit]

- A Geometric Algebra Primer, especially for computer scientists.