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GEOMETRIC ALGEBRA:Parallel Processing for the Mind

Timothy F. Havel (Nuclear Engineering)

AN IAP-2001 PRODUCTION

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THE TIME IS 1844

The place, the rural town of Stettin, inwhat is now a part of Poland

A second generationgymnasium Lehrer bythe name of HermannGunther Grassmannhas just completed hisMagnum Opus, theAusdehnungslehre, orCalculus of Extension.In it, he lays forth ageneral theory of formswhich today would becalled abstract algebra,and applies it to geometry,mechanics, electro-dynamics and crystal-lography. He sends copies to all the leading mathematicians ofhis time, including Gauss, Möbius, and Cauchy, none of whomwere willing or able to read it.

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The following excerpt gives some idea of why:

This provoked the following comment from his contemporaryHeinrich Baltzer:

Grassmann tried repeatedly but never received a university post(he was rather more successful as a scholar of Sanskrit), and in1860 the remaining copies of his book were shredded by thepublisher. A second edition in 1862 did no better.

The primary division in all the sciencesis into the real and the

formal. The former represents in thought the existent as

existing independentlyof thought, and their truth consistsin

their correspondencewith the existent.The formal scienceson

the other hand have astheir objectwhat hasbeenproducedby

thought alone, and their truth consistsin the correspondence

betweenthe thoughtprocessesthemselves.Puremathematicsis

thus thescienceof theparticular existentwhich hascometo be

through thought. The particular existent,viewed in this sense,

we name a thought form or simply a form. Thus pure

mathematics is the theory of forms.

It is not possiblefor meto enter into thosethoughts; I become

dizzy & see sky-blue before my eyes when I read them.

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THE TIME IS 1843

October 16, to be precise. The place isBrougham Bridge in Dublin, Ireland

The gentleman shown hereis William Rowan Hamilton,a poet, philologist parexcellence, the RoyalAstronomer of Ireland sincethe age of 21, knighted at28 and now President ofthe Royal Society ofIreland. Walking to work,he suddenly stops and ingreat agitation scratcheswith his pocket knife thefollowing equations on thestonework of Brougham bridge:

Quaternions, which Hamilton regards as the algebra of puretime, have been born.

i2 j2 k2 ijk 1–= = = =

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Hamilton spent the last 22 years of his life developingQuaternions, publishing his results in 109 papers and a 735page book Lectures on Quaternions. In the process he intro-duced such now common terms as scalar, vector, real andimaginary. Despite his fame (Hamilton’s equations!), his newideas were only slowly assimilated, as indicated by the followingcomment from his contemporary J. T. Graves:

This is also indicated by Hamilton’s own remarks:

This eventually got to him, for he died of alcoholism at the age of60, still working on Elements of Quaternions.

Thereis still somethingin this systemwhich gravelsme.I have

not yetanyclearviewson theextentto which weareat liberty to

arbitrarily create new imaginaries, and to endow them with

supernatural properties(such as noncommutativity).

... it requireda certain capital of scientificreputation,amassed

in formeryears,to makeit other than dangerouslyimprudentto

hazardthepublicationof a work which has,althoughat bottom

quiteconservative,ahighly revolutionaryair. It wasapartof the

ordeal through which I had to pass,an episodein the battleof

lif e,to know that evencandidandfriendly peoplesecretly, or, as

it might happen, openly censuredor ridiculed me for what

appeared to them my monstrous invention.

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WE ARRIVE IN THE LATE19TH CENTURY

A Few People Begin to Take Interest

William Kingston Cliffordrealizes the connectionsbetween Grassmann andHamilton’s works, developsbiquaternions, applies themto non-Euclidean geometry,and anticipates Einstein’stheory of relativity in 1870.But he dies at the age of 35.

Arthur Cayley, a lawyer beforeaccepting a chair at Cambridgeand the supreme master ofdeterminants, discovers in1885 something that escapedHamilton and Clifford, namelyhow to express general 3-Drotations using quaternions.But he finds them unintuitive!

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ANOTHER CENTURY PASSES

In it, the Only Good Mathematician is anAbstract Mathematician

Oliver Heaviside & JosiahGibbs succeed in getting avery small part of Grass-mann & Hamilton’s ideasincorporated into physicsand engineering under thename of “vector algebra”.

In pure mathematics, piecesof “geometric algebra” arerediscovered by MarcelRiesz & Elie Cartan, butcloaked in a new notationand vocabulary so as to becompletely unrecognizable.

These new names include “tensor algebra”, “differential forms”and “spinor calculus”.

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A NEW DAWN COMES

In a Way as Unlikely as Anything Else

David Hestenes, the son of awell-known mathematician, astudent of the physicist JohnArchibald Wheeler, and adedicated physics teacher in atArizona State University, Tempe,recognizes that all theseseemingly different formalismsare, like life itself, pieces of thesame thing united by theircommon lineage. Moreover, hesees that geometric algebraprovides a language which, dueit is geometric content, capturesmuch of the logic that makesphysics what it is. His first book,Space-Time Algebra, appears in1966, and applies geometric algebra to relativity and quantummechanics.

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Together with his first student, Garret Sobczyk, he writes acomprehensive treatise on the underlying mathematics, FromClifford Algebra to Geometric Calculus. An editor at his firstpublisher sits on it for several years, until they give up and tryanother publisher. Unbeknownst to them, the editor takes a newjob at that same publisher and continues to delay its publicationseveral more years.

Finally, they send a copy to GianCarlo Rota in the MIT math-ematics department. Rota, whouses higher algebra to studyprobability and combinatorics, butalso has an appreciation forphysics through long associationwith Los Alamos, recognizes thevalue of the work and gets itpublished soon thereafter by D.Reidel. Rota’s recent & untimelydeath, however, prevented themfrom ever meeting each other.

We have arrived in the present. This woefully incomplete historywill close with Rota’s remarks on Grassmann in his collection ofautobiographical sketches, Indiscrete Thoughts:

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Mathematicianscanbedividedinto two types:problemsolversandtheorizers.Most mathematiciansare a mixture of the two althoughit is easy to find extreme examples of both types.To the problemsolver, the supremeachievementin mathematicsisthe solution of a problem that had beengiven up as hopeless.Itmatterslittle that thesolutionmaybeclumsy;all that countsis thatit is correct. Once the problem solver finds the solution, he willpermanentlylose interest in it ... The problem solver is the rolemodelfor buddingyoungmathematicians.Whenwedescribeto thepublic the conquestsof mathematics,our shining heroes areproblem solvers.To the theorizer, the supremeachievement of mathematicsis atheory that shedssuddenlight on someincomprehensiblephen-omenon.Successin mathematicsdoesnot lie in solvingproblems,but in their trivialization. The moment of glory comeswith thediscovery of a newtheorythat doesnot solve any old problems,butrendersthemirrelevant.To thetheorizer, theonly mathematicsthatwill survive are the definitions. Great definitions are whatmathematicscontributesto the world. Theoremsare toleratedasanecessaryevil since they play an essentialrole in understandingthe definitions.Grassmannwasa theorizerall the way. His greatcontribution wasthe definition of geometricalgebra. Evil tongueswhisperedthattherewas really nothing new in Grassmann’s algebra: “What canyou prove with it that you can’t prove without it?” they asked.Whenever you hear this question,beassuredthat you are likely tobe in the presence of something important.