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Geometric algebra

In mathematics, a geometric algebra (also known as a Clifford algebra) is an algebra that can represent and manipulate geometrical objects such as vectors. Geometric algebra is built out of two fundamental operations, addition and the geometric product. Multiplication of vectors results in higher-dimensional objects called multivectors. Compared to other…

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Overview

Definition and notation

Modeling geometries

Geometric interpretation in the vector space model

Examples and applications

Geometric calculus

History

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Geometric algebra

Nodes177
Edges176
Triples113
Avg. degree1.99
Density0.011299
Components1

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Geometric algebra

Top relations

related to Before the 20th century · 29
Geometric algebra → Although, Clifford, Clifford's, Edwin Bidwell Wilson, Euclid's Elements, Euclidean, Following Grassmann, From, GA, Gibbs, Grassmann, Grassmann's, Greek, Hermann Grassmann, His, In, James Clerk Maxwell's, Josiah Willard Gibbs, Later, Nevertheless
related to 20th century and present · 14
Geometric algebra → Bayro, Cartan, Claude Chevalley, Clifford, David Hestenes, Dirac, Emil Artin's Geometric Algebra, For, GA, Hermann Weyl, In, Pauli, Progress, The
related to Unit pseudoscalars · 10
Geometric algebra → GA, Given, If, It, Orthonormality, Plücker, Suppose, The, Unit, Using
related to Definition and notation · 8
Geometric algebra → Cl, Clifford, Euclidean, Given, Hestenes's, Lorentzian, The Clifford, There
related to Linear functions · 5
Geometric algebra → Although, However, If, The, With
related to Representation of subspaces · 5
Geometric algebra → Blades, Geometric, The, There, These
related to Spacetime model · 5
Geometric algebra → APS, In, Minkowski, STA, While
related to Blades, grades, and basis · 4
Geometric algebra → Consider, From, Gramian, With
related to Homogeneous models · 3
Geometric algebra → Geometric, Homogeneous, In
is a · 2
Geometric algebra → filtered algebra.A multivector A, sum of blades.Consider a set of r

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Important terminology

displaystyle algebra geometric product vectors vector elements space basis exterior mathcal form clifford two number called rotations blades inner may

Entity relationships Subject–Predicate–Object triples

SubjectPredicateObjectConfidenceSrc
Geometric algebrais asum of blades.Consider a set of r0.90text
Geometric algebrais afiltered algebra.A multivector A0.90text
vectorsinstance ofis an algebra that can represent and manipulate geometrical objects0.80text
in relativity.Examples of geometric algebras applied in physics include the spacetime algebrainstance ofa geometric algebra naturally accommodates any number of dimensions and any quadratic form0.80text
complex analysisinstance ofcan be used to formulate other theories0.80text
differential geometryinstance ofcan be used to formulate other theories0.80text
e.g. by using the Clifford algebra instead of differential formsinstance ofcan be used to formulate other theories0.80text
rotationsinstance ofalthough they may represent geometric quantities0.80text
rotating a rotor or reflecting a spinor always provided that some geometrical or physical significance can be attached to such operations.By the Cartaninstance ofSince both operators and operand are versors there is potential for alternative examples0.80text
projectionsinstance ofand the negative multiples the opposite orientation.Blades are important since geometric operations0.80text
rotationsinstance ofand the negative multiples the opposite orientation.Blades are important since geometric operations0.80text
reflections depend on the factorability via the exterior product thatinstance ofand the negative multiples the opposite orientation.Blades are important since geometric operations0.80text

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