Editing Geometric algebra (section)

{{short description|Algebraic structure designed for geometry}}
{{for-multi|the general algebraic structure|Clifford algebra|other uses}}{{Not to be confused with|Algebraic geometry}}
In [[mathematics]], a '''geometric algebra''' (also known as a [[Clifford algebra]]) is an [[Algebra over a field|algebra]] that can represent and manipulate geometrical objects such as [[Vector (mathematics and physics)|vector]]s. Geometric algebra is built out of two fundamental operations, addition and the geometric product. Multiplication of vectors results in higher-dimensional objects called [[multivector]]s. Compared to other formalisms for manipulating geometric objects, geometric algebra is noteworthy for supporting vector division (though generally not by all elements) and addition of objects of different dimensions.

The geometric product was first briefly mentioned by [[Hermann Grassmann]],{{sfn|ps=|Hestenes|1986|loc=p. 6}} who was chiefly interested in developing the closely related [[exterior algebra]]. In 1878, [[William Kingdon Clifford]] greatly expanded on Grassmann's work to form what are now usually called Clifford algebras in his honor (although Clifford himself chose to call them "geometric algebras"). Clifford defined the Clifford algebra and its product as a unification of the [[exterior algebra|Grassmann algebra]] and Hamilton's [[quaternion algebra]]. Adding the [[Duality (mathematics)|dual]] of the Grassmann exterior product allows the use of the [[Grassmann–Cayley algebra]]. In the late 1990s, [[plane-based geometric algebra]] and [[conformal geometric algebra]] (CGA) respectively provided a framework for euclidean geometry and [[Klein geometry|classical geometries]].{{sfn|ps=|Li|2008|p=411}} In practice, these and several derived operations allow a correspondence of elements, [[vector subspace|subspaces]] and operations of the algebra with geometric interpretations. For several decades, geometric algebras went somewhat ignored, greatly eclipsed by the [[vector calculus]] then newly developed to describe electromagnetism. The term "geometric algebra" was repopularized in the 1960s by [[David Hestenes]], who advocated its importance to relativistic physics.{{sfn|ps=|Hestenes|1966}}

The scalars and vectors have their usual interpretation and make up distinct subspaces of a geometric algebra. [[Bivector]]s provide a more natural representation of the [[pseudovector]] quantities of 3D vector calculus that are derived as a [[cross product]], such as oriented area, oriented angle of rotation, torque, angular momentum and the [[magnetic field]]. A [[trivector]] can represent an oriented volume, and so on. An element called a [[Blade (geometry)|blade]] may be used to represent a subspace and [[orthogonal projection]]s onto that subspace. Rotations and reflections are represented as elements. Unlike a vector algebra, a geometric algebra naturally accommodates any number of dimensions and any quadratic form such as in [[Theory of relativity|relativity]].

Examples of geometric algebras applied in physics include the [[spacetime algebra]] (and the less common [[algebra of physical space]]). [[Geometric calculus]], an extension of GA that incorporates [[differentiation (mathematics)|differentiation]] and [[integral|integration]], can be used to formulate other theories such as [[complex analysis]] and [[differential geometry]], e.g. by using the Clifford algebra instead of [[differential form]]s. Geometric algebra has been advocated, most notably by [[David Hestenes]]{{sfn|ps=|Hestenes|2003}} and [[Chris J. L. Doran|Chris Doran]],{{sfn|ps=|Doran|1994}} as the preferred mathematical framework for [[physics]]. Proponents claim that it provides compact and intuitive descriptions in many areas including [[classical mechanics|classical]] and [[quantum mechanics]], [[electromagnetic theory]], and [[theory of relativity|relativity]].{{sfn|ps=|Lasenby|Lasenby|Doran|2000}} GA has also found use as a computational tool in [[computer graphics]]{{sfn|ps=|Hildenbrand|Fontijne|Perwass|Dorst|2004}} and [[robotics]].