The Crossing Paths of the Arts and Mathematics: Cucker

428 Pages·46.602 MB·English

by  Mirrors, Manifold

Manifold Mirrors
TheCrossingPathsoftheArtsandMathematics
Most works of art, whether illustrative, musical or literary, are created
subject to a set of constraints. In many (but not all) cases, these con-
straints have a mathematical nature; for example, the geometric trans-
formations governing the canons of J. S. Bach, the various projection
systems used in classical painting, the catalogue of symmetries found
inIslamicartortherulesconcerningpoetic structure.Thisfascinating
book describes geometric frameworksunderlying this constraint-based
creation. The author provides both a development in geometry and
a description of how these frameworks fit the creative process within
several art practices. Furthermore, he discusses the perceptual effects
derivedfromthepresenceofparticulargeometriccharacteristics.
The book began life as a liberal arts course and is certainly suitable
as a textbook. However, anyone interested in the power and ubiquity
of mathematics will enjoy this revealing insight into the relationship
betweenmathematicsandthearts.
FelipeCucker is Chair Professor of Mathematics at the City University
of Hong Kong. His research covers a variety of subjects, including
semi-algebraic geometry, computer algebra, complexity, emergence in
decentralizedsystems(inparticular,emergenceoflanguagesandflock-
ing),learningtheoryandfoundationalaspectsofnumericalanalysis.He
servesontheeditorialboardofseveraljournalsandisManagingEditor
ofthejournalFoundationsofComputationalMathematics,publishedby
theSocietyofthesamename.

Manifold Mirrors
The Crossing Paths of the Arts and Mathematics
FELIPE CUCKER
CityUniversityofHongKong
cambridge university press
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Firstpublished2013
PrintedandboundintheUnitedKingdombytheMPGBooksGroup
AcataloguerecordforthispublicationisavailablefromtheBritishLibrary
LibraryofCongressCataloging-in-Publicationdata
Cucker,Felipe,1958–
Manifoldmirrors:thecrossingpathsoftheartsandmathematics/
FelipeCucker,CityUniversityofHongKong.
pagescm
Includesbibliographicalreferences.
ISBN978-0-521-42963-4(Hardback)–ISBN978-0-521-72876-8(pbk.)
1. Arts–Mathematics. I. Title.
NX180.M33C832013
(cid:3)
700.105–dc23
2012046405
ISBN978-0-521-42963-4Hardback
ISBN978-0-521-72876-8Paperback
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CONTENTS
Mathematics:user’smanual pageix
Appetizers 1
A.1 Martini 1
A.2 Ontheirblindness 3
A.3 TheMusicalOffering 7
A.4 Thegardenofthecrossingpaths 10
1 Spaceandgeometry 11
1.1 Thenatureofspace 11
1.2 Theshapeofthings 12
1.3 Euclid 14
1.4 Descartes 18
2 Motionsontheplane 27
2.1 Translations 27
2.2 Rotations 29
2.3 Reflections 29
2.4 Glides 30
2.5 Isometriesoftheplane 31
2.6 Onthepossibleisometriesontheplane 36
3 Themanysymmetriesofplanarobjects 39
3.1 Thebasicsymmetries 41
3.1.1 Bilateralsymmetry:thestraight-linedmirror 41
3.1.2 Rotationalsymmetry 42
3.1.3 Centralsymmetry:theone-pointmirror 42
3.1.4 Translationalsymmetry:repeatedmirrors 44
3.1.5 Glidalsymmetry 46
3.2 Thearithmeticofisometries 47
3.3 Arepresentationtheorem 52
3.4 Rosettesandwhirls 55
3.5 Friezes 59
3.5.1 Thesevenfriezes 59
3.5.2 Aclassificationtheorem 64
3.6 Wallpapers 69
3.6.1 Theseventeenwallpapers 69
v
ListofContents
3.6.2 Abriefsample 76
3.6.3 Tablesandflowcharts 77
3.7 Symmetryandrepetition 80
3.8 Thecatalogue-makers 81
4 Themanyobjectswithplanarsymmetries 83
4.1 Origins 83
4.2 Rugsandcarpets 89
4.3 Chineselattices 103
4.4 Escher 106
5 Reflectionsonthemirror 111
5.1 Aestheticorder 111
5.2 TheaestheticmeasureofBirkhoff 116
5.3 Gombrichandthesenseoforder 120
5.4 Betweenboredomandconfusion 125
6 Arawmaterial 128
6.1 Theveiledmirror 128
6.2 Betweendetachmentanddilution 134
6.3 Ablurredboundary:I 138
6.4 Theamazingkaleidoscope 146
6.5 Thestricturesofverse 152
7 Stretchingtheplane 158
7.1 Homotheciesandsimilarities 158
7.2 Similaritiesandsymmetry 162
7.3 Shears,strainsandaffinities 166
7.4 Conics 174
7.5 Theeclosionofellipses 177
7.6 Klein(abernurderName) 184
8 Auralwallpaper 188
8.1 Elementsofmusic 189
8.2 Thegeometryofcanons 193
8.3 TheMusicalOffering(revisited) 198
8.4 Symmetriesinmusic 206
8.4.1 Thegeometryofmotifs 208
8.4.2 Theubiquitousseven 210
8.5 Perception,localityandscale 213
vi
ListofContents
8.6 Thebareminima(againandagain) 216
8.7 Ablurredboundary:II 220
9 Thedawnofperspective 225
9.1 Alberti’swindow 227
9.2 Thedawnofprojectivegeometry 240
9.2.1 Bijectionsandinvertiblefunctions 243
9.2.2 Theprojectiveplane 245
9.2.3 AKleinianviewofprojectivegeometry 251
9.2.4 Essentialfeaturesofprojectivegeometry 253
9.3 Aprojectiveviewofaffinegeometry 254
9.3.1 Adistantvantagepoint 255
9.3.2 Conicsrevisited 258
10 Arepertoireofdrawingsystems 260
10.1 Projectionsanddrawingsystems 260
10.1.1 Orthogonalprojections 263
10.1.2 Obliqueprojections 269
10.1.3 Ontiltanddistance 277
10.1.4 Perspectiveprojection 282
10.2 Voyeursanddemiurges 286
11 Thevicissitudesofperspective 293
11.1 Deceptions 293
11.2 Concealments 295
11.3 Bends 298
11.4 Absurdities 306
11.5 Divergences 311
11.6 Multiplicities 315
11.7 Abandonment 317
12 Thevicissitudesofgeometry 321
12.1 Euclidrevisited 321
12.2 Hyperbolicgeometry 325
12.3 Lawsofreasoning 328
12.3.1 Formallanguages 328
12.3.2 Deduction 330
12.3.3 Validity 333
12.3.4 TwomodelsforEuclideangeometry 335
12.3.5 Proofandtruth 338
vii
ListofContents
12.4 ThePoincarémodelofhyperbolicgeometry 339
12.5 Projectivegeometryasanon-Euclideangeometry 346
12.6 Sphericalgeometry 353
13 Symmetriesinnon-Euclideangeometries 357
13.1 Tessellationsandwallpapers 357
13.2 Isometries and tessellations in the sphere and the
projectiveplane 359
13.3 Isometriesandtessellationsinthehyperbolicplane 363
14 Theshapeoftheuniverse 373
Appendix:Rule-drivencreation 381
Compliers/benders/transgressors 381
Constrainedwriting 386
References 395
Acknowledgements 402
Indexofsymbols 404
Indexofnames 405
Indexofconcepts 409
viii