Table Of ContentNew Foundations in Mathematics
Garret Sobczyk
New Foundations
in Mathematics
The Geometric Concept of Number
GarretSobczyk
DepartamentodeF´ısicayMatema´ticas
UniversidaddeLasAme´ricas
Puebla,Mexico
ISBN978-0-8176-8384-9 ISBN978-0-8176-8385-6(eBook)
DOI10.1007/978-0-8176-8385-6
SpringerNewYorkHeidelbergDordrechtLondon
LibraryofCongressControlNumber:2012948769
Mathematics Subject Classification (2010): 11A05, 15A16, 15A18, 15A21, 15A63, 15A63, 15A66,
15A69,15A75,17B45,22E60,51N10,51N15,53A04,53A05,53A30,58A05,65D05,83A05,20B30
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Preface
This book provides an introduction to geometric algebra and its application to
diverse areas of mathematics. It maintains the spirit of its predecessor, Clifford
AlgebratoGeometricCalculus:AUnifiedLanguageforMathematicsandPhysics,
and as such it has many unique features not seen in any other undergraduate
textbook. It provides many innovative ways of looking at geometrical ideas and
topicsforstudentresearchandthesisprojects.
Thematerialhasbeendevelopedoverthemanyyearsthattheauthorhastaught
undergraduatecoursesattheUniversidaddeLasAme´ricas-Puebla,Mexico,inlinear
algebra,vectorcalculus,differentialgeometry,numericalanalysis,modernalgebra,
and number theory. Whereas this book cannot be considered a textbook for all
of these different subjects, there is a common theme they all share: they can all
be efficiently formulated using the unified geometric number system advocated
here.Geometricalgebra,whichhasundergoneextensivedevelopmentinthesecond
half of the twentieth Century, has its origins in the seminal works of Grassmann,
Hamilton,andCliffordinthenineteenthcentury.
The bookbeginswith the introductionof the spectralbasis in modularnumber
systems and in modular polynomials. This often overlooked concept provides
insight into and greatly simplifies the proofs of many basic theorems in number
theoryandthecorrespondingcloselyrelatedstructuretheoremsofalinearoperator.
Sincegeometricnumbersobeyexactlythesamealgebraicrulesassquarematrices
of real numbers, the languages are completely compatible and structure theorems
thatarevalidforoneareequallyvalidfortheother.
Theconceptofamatrixasanarrayofnumberswithanunintuitivemultiplication
rule hardly providesa geometric way of lookingat things. Nevertheless, matrices
haveproventobeanextremelyeffectivecomputationaltoolandhaveplayedamajor
roleinthedevelopmentofdiverseareasofmathematics.Geometricalgebrarectifies
this defect by providing a geometric perspective, and many new algebraic tools.
Combiningbothofthese powerfulsystemsbysimplyconsideringmatriceswhose
elementsaregeometricnumbersaddsmuchneededgeometriccontentandflexibility
to both languages.The authorhopesthat this book capturesboth the idea and the
vii
viii Preface
spirit of the powerful geometric number system that has kept him going since he
learned about the subject as a graduate student at Arizona State University many
yearsago.
Weassumereaderstohavehadundergraduatedifferentialandintegralcalculus,
afirstcourseinmodernalgebra,andthemathematicalmaturitythatanupper-level
mathematicsorphysicsundergraduatestudentmightbeexpectedtohave.Themany
topics covered in the book should also appeal to first-year graduate students in
mathematics,physics,engineeringand computerscience.Anyunfamiliaritythata
readermighthaveregardingmathematicalterminologycanbequicklyovercomeby
aquickreferencetotheunlimitedresourcesontheinternet.Wealsorecommendthat
thereaderhasknowledgeofandaccesstosymbolicmathematicalsoftwaresuchas
MathematicaorMaple.Suchsoftwareconsiderablylightensthecomputationalwork
requiredandmakesforeasyverificationofresults.AsimpleMathematicapackage
isprovidedforcalculatingthespectralbasisforamodularpolynomial.
Therearethreemaingroupingsofinterrelatedcorechapters:
(cid:129) Chapters 1–5introducethefundamentalconceptsofaspectralbasisofmodular
numbersandmodularpolynomialswithapplicationsinnumbertheory,numerical
analysis, and linear algebra. The hyperbolicnumbers,introducedalongside the
well-knowncomplexnumbers,areusedtosolvethecubicequationandprovide
a mathematical foundation for the theory of special relativity. The geometric
extension of the real numbers is achieved by introducing new anticommuting
square roots of plus or minus one which represent orthogonal directions in
successivelyhigherdimensions.
(cid:129) Chapters 7–10 lay down the ideas of linear and multilinear algebra. Ma-
trices of geometric numbers are considered throughout. New proofs of the
Cayley–Hamilton Theorem,Gram–Schmidtorthogonalization,and the spectral
decompositionof a linear operatorare givenin geometric algebra,as well as a
comprehensivegeometricinterpretationofcomplexeigenvaluesandeigenvectors
inanHermitian(definiteorindefinite)innerproductspace.
(cid:129) Chapters 13–16 develop the basic ideas of vector calculus and differential
geometryinthecontextofgeometricalgebra.Theclassicalintegrationtheorems
are derived from a single fundamental theorem of calculus. Manifolds are
embedded in Euclidean or pseudo-Euclidean spaces and consequently have
bothintrinsicandextrinsiccurvature,characterizedbytheprojectionandshape
operators. Highlighted is a special treatment of conformal mappings and the
conformalWeyltensor,whichhaveapplicationsinphysicsandengineering.
Chapter6coverssomeofthemoretraditionaltopicsinlinearalgebrawhicharenot
otherwiseusedinthisbook.Chapters11,12,17,and18provideadditionalbreadth
and scope by treating the symmetric group,by givinga novellook at the concept
of space-time in special relativity, by laying down the basic ideas of projective
geometry,andbygivinganintroductiontoLiealgebrasandLiegroups,topicswhich
arenotusuallycoveredinanundergraduatecourse.IntheTableofContents,a“*”
isusedtoindicatethosesectionswhichareconsiderablymoretechnicalandmaybe
omittedonfirstreading.
Preface ix
The author is indebted to many students and collegues from around the world
who have contributed much during the many stages of the development of these
ideas. Foremost he is indebted to David Hestenes, who first introduced him to
geometricalgebramanyyearsagoasagraduatestudentatArizonaStateUniversity
during the years 1965–1971. The author is indebted to Roman Duda of the
Polish Academy of Sciences, Bernard Jancewicz, Jan Łopuszan´ski, and Zbigniew
Oziewicz of the Institute of Theoretical Physics in Wrocław, Poland, and Stony
BrookUniversity,forencouragementandsupportduringthesedifficultearlyyears.
He wants to thank Rafal Ablamowicz (USA), Timothy Havel (USA), William
Baylis(Canada),andPerttiLounesto(Finland),whocontributedin differentways
tothewritingofthisbook.Inaddition,hewantstothankJaimeKellerforinviting
himto Mexico,LuisVerde-Star(Mexico),WaldyrRodrigues(Brazil),JosepParra
(Spain), and Eduardo Bayro-Corrochano (Mexico). Among former students and
now sometimes collaborators, he wants to thank Jose´ Mar´ıa Pozo (Spain), Marco
Antonio Rodr´ıguez (Mexico), Omar Leon Sanchez (Mexico), and Alejandra C.
Vicente(Mexico).GraphicsdesignartistAnaSa´nchezStonewasagreathelpwith
all of the figures in Chap.11 and in particular Fig.11.6. The author is greatly
indebted to Universidad de Las Ame´ricas-Puebla and Sistemas Nacionales de
Investigadores de Me´xico, for many years of support. This book could not have
beenwrittenwithouttheconstantsupportandencouragementovertheyearsbymy
wife,Wanda,mymotherandfatherandisapresentformymother’s100thbirthday.
Cholula,Me´xico GarretSobczyk
