<i>Clifford Algebra to Geometric Calculus. A Unified Language for</i> <i>Mathematics and Physics</i>

Hestenes, David; Sobczyk, Garret; Marsh, James S.

doi:10.1119/1.14223

Cited by 316 publications

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“…He derived a rich system of formulas that have been repeatedly rediscovered by other mathematicians including myself. As I was eventually able to show [44], his progressive (exterior) and regressive (interior) products provide a foundation for invariant theory including the "theory of determinants. "…”

Section: Research Begins: Discovering a Themementioning

confidence: 99%

“…When I submitted my three papers with Sobczyk with a proposal for a book, he responded so enthusiastically that I told him I had much more along the same lines, and he gave me a strong green light to proceed. Thus began the long process of producing the book Clifford Algebra to Geometric Calculus [44]. I had no idea that it would take more than 10 years.…”

Section: Vol 27 (2017)mentioning

confidence: 99%

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The Genesis of Geometric Algebra: A Personal Retrospective

Hestenes

2016

Adv. Appl. Clifford Algebras

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Abstract. Even today mathematicians typically typecast Clifford Algebra as the "algebra of a quadratic form," with no awareness of its grander role in unifying geometry and algebra as envisaged by Clifford himself when he named it Geometric Algebra. It has been my privilege to pick up where Clifford left off-to serve, so to speak, as principal architect of Geometric Algebra and Calculus as a comprehensive mathematical language for physics, engineering and computer science. This is an account of my personal journey in discovering, revitalizing and extending Geometric Algebra, with emphasis on the origin and influence of my book Space-Time Algebra. I discuss guiding ideas, significant results and where they came from-with recollection of important events and people along the way. Lastly, I offer some lessons learned about life and science.

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Section: Research Begins: Discovering a Themementioning

confidence: 99%

Section: Vol 27 (2017)mentioning

confidence: 99%

The Genesis of Geometric Algebra: A Personal Retrospective

Hestenes

2016

Adv. Appl. Clifford Algebras

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“…Finally, we also plan to develop our formalization to capture the powerful framework of the geometric algebra [8,5] …”

Section: Resultsmentioning

confidence: 99%

A Formalization of Grassmann-Cayley Algebra in COQ and Its Application to Theorem Proving in Projective Geometry

Fuchs

Théry

2011

Automated Deduction in Geometry

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Abstract. This paper presents a formalization of Grassmann-Cayley algebra [6] that has been done in the Coq [2] proof assistant. The formalization is based on a data structure that represents elements of the algebra as complete binary trees. This allows to define the algebra products recursively. Using this formalization, published proofs of Pappus' and Desargues' theorem [7,1] are interactively derived. A method that automatically proves projective geometric theorems [11] is also translated successfully into the proposed formalization.

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“…On the other hand, the dyadic notation and the vector analysis in ordinary threedimensional space, introduced by Gibbs, suffer from a major flaw: the cross product of vectors, apart from not being associative, can only be defined in three or seven dimensions [11]. Spacetime algebra [12], the geometric algebra [13,14] of spacetime, is -from the authors' viewpoint -the most adequate formalism to achieve that union of space and time envisioned by Minkowski.…”

Section: Introductionmentioning

confidence: 99%

Doppler Shift from a Composition of Boosts with Thomas Rotation: a Spacetime Algebra Approach

Paiva¹,

Ribeiro²

2006

Journal of Electromagnetic Waves and Applications

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Abstract-As is well-known, Lorentz boosts do not form a group: the composite of two boosts is, in general, equivalent to the composite of a rotation -the so-called Thomas (or Wigner) rotation -and a single boost. Using spacetime algebra, the geometric algebra for Minkowski spacetime, we show in this paper how to relate the Thomas rotation to the Doppler shift experienced between two observers in relative motion. Both observers are moving away from a third observer, set in the laboratory, with relative non-parallel velocities.

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Clifford Algebra to Geometric Calculus. A Unified Language for Mathematics and Physics

Cited by 316 publications

References 0 publications

The Genesis of Geometric Algebra: A Personal Retrospective

The Genesis of Geometric Algebra: A Personal Retrospective

A Formalization of Grassmann-Cayley Algebra in COQ and Its Application to Theorem Proving in Projective Geometry

Doppler Shift from a Composition of Boosts with Thomas Rotation: a Spacetime Algebra Approach

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