The Genesis of Geometric Algebra: A Personal Retrospective

Hestenes, David

doi:10.1007/s00006-016-0664-z

Cited by 16 publications

(14 citation statements)

References 48 publications

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“…"But, if geometric algebra is so good, why is it not more widely used?" As Hestenes replied in [11]: "Its time will come!" The published geometric algebra literature is more than sufficient to support instruction with geometric algebra at intermediate and advanced levels in physics, mathematics, engineering, and computer science.…”

Section: Discussionmentioning

confidence: 99%

“…Comparing (11) with (4), we find the two algebras are isomorphic in sense of linear algebra, but their definitions of multiplication rules are different. The Grassmann products have clear geometrical meaning, but the Clifford product is isomorphic to matrix algebra and the multiplication of physical variables is Clifford product.…”

Section: Application Of Clifford Algebra In Differential Geometrymentioning

confidence: 98%

“…Although his initial motivation was to gain insight into the nature of quantum mechanics, he quickly realized that Clifford algebra was a unified language and tool for mathematics, physics, and engineering. He published "space-time algebra" in 1966 and has been working on the promotion of Clifford algebra in teaching and research [9][10][11][12]. Because representation and algorithm in geometric algebra are seemingly as ordinary as arithmetic, his work has been neglected by the scientific community for more than 20 years.…”

Section: A Brief Historical Reviewmentioning

confidence: 99%

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Some Applications of Clifford Algebra in Geometry

Gu¹

2021

Structure Topology and Symplectic Geometry

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In this chapter, we provide some enlightening examples of the application of Clifford algebra in geometry, which show the concise representation, simple calculation, and profound insight of this algebra. The definition of Clifford algebra implies geometric concepts such as vector, length, angle, area, and volume and unifies the calculus of scalar, spinor, vector, and tensor, so that it is able to naturally describe all variables and calculus in geometry and physics. Clifford algebra unifies and generalizes real number, complex, quaternion, and vector algebra and converts complicated relations and operations into intuitive matrix algebra independent of coordinate systems. By localizing the basis or frame of space-time and introducing differential and connection operators, Clifford algebra also contains Riemann geometry. Clifford algebra provides a unified, standard, elegant, and open language and tools for numerous complicated mathematical and physical theories. Clifford algebra calculus is an arithmetic-like operation that can be well understood by everyone. This feature is very useful for teaching purposes, and popularizing Clifford algebra in high schools and universities will greatly improve the efficiency of students to learn fundamental knowledge of mathematics and physics. So, Clifford algebra can be expected to complete a new big synthesis of scientific knowledge.

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Section: Discussionmentioning

confidence: 99%

Section: Application Of Clifford Algebra In Differential Geometrymentioning

confidence: 98%

Section: A Brief Historical Reviewmentioning

confidence: 99%

See 1 more Smart Citation

Some Applications of Clifford Algebra in Geometry

Gu¹

2021

Structure Topology and Symplectic Geometry

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show abstract

“…"But, if geometric algebra is so good, why is it not more widely used?" As D. Hestenes replied in [29]: "Its time will come!" The published geometric algebra literature is more than sufficient to support instruction with geometric algebra at intermediate and advanced levels in physics, mathematics, engineering and computer science.…”

Section: Transformation Of Clifford Algebramentioning

confidence: 99%

Some Applications of Clifford Algebra in Geometry

Gu¹

2020

Preprint

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In this paper, we provide some enlightening examples of the application of Clifford algebra in geometry, which show the concise representation, simple calculation and profound insight of this algebra. The definition of Clifford algebra implies geometric concepts such as vector, length, angle, area and volume, and unifies the calculus of scalar, spinor, vector and tensor, so that it is able to naturally describe all variables and calculus in geometry and physics. Clifford algebra unifies and generalizes real number, complex, quaternion and vector algebra, converts complicated relations and operations into intuitive matrix algebra independent of coordinate systems. By localizing the basis or frame of space-time and introducing differential and connection operators, Clifford algebra also contains Riemann geometry. Clifford algebra provides a unified, standard, elegant and open language and tools for numerous complicated mathematical and physical theories. Clifford algebra calculus is an arithmetic-like operation that can be well understood by everyone. This feature is very useful for teaching purposes, and popularizing Clifford algebra in high schools and universities will greatly improve the efficiency of students to learn fundamental knowledge of mathematics and physics. So Clifford algebra can be expected to complete a new big synthesis of scientific knowledge.

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“…Videlicet, giving a geometric meaning to the algebraic operations from the outset. This, in turn, lends naturally to describe time, space and other properties in the physical world [10]. The set of elliptic scators is akin to split-quaternions in the sense that it contains zero divisors and nilpotent elements.…”

Section: Introductionmentioning

confidence: 99%

Powers of Elliptic Scator Numbers

Fernández‐Guasti

2021

Axioms

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Elliptic scator algebra is possible in 1+n dimensions, n∈N. It is isomorphic to complex algebra in 1 + 1 dimensions, when the real part and any one hypercomplex component are considered. It is endowed with two representations: an additive one, where the scator components are represented as a sum; and a polar representation, where the scator components are represented as products of exponentials. Within the scator framework, De Moivre’s formula is generalized to 1+n dimensions in the so called Victoria equation. This novel formula is then used to obtain compact expressions for the integer powers of scator elements. A scator in S1+n can be factored into a product of n scators that are geometrically represented as its projections onto n two dimensional planes. A geometric interpretation of scator multiplication in terms of rotations with respect to the scalar axis is expounded. The powers of scators, when the ratio of their director components is a rational number, lie on closed curves. For 1 + 2 dimensional scators, twisted curves in a three dimensional space are obtained. Collecting previous results, it is possible to evaluate the exponential of a scator element in 1 + 2 dimensions.

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The Genesis of Geometric Algebra: A Personal Retrospective

Cited by 16 publications

References 48 publications

Some Applications of Clifford Algebra in Geometry

Some Applications of Clifford Algebra in Geometry

Some Applications of Clifford Algebra in Geometry

Powers of Elliptic Scator Numbers

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