Manifold Mirrors : The Crossing Paths of the Arts and Mathematics

Most works of art, whether illustrative, musical or literary, are created subject to a set of constraints. In many (but not all) cases, these constraints have a mathematical nature, for example, the geometric transformations governing the canons of J. S. Bach, the various projection systems used in classical painting, the catalog of symmetries found in Islamic art, or the rules concerning poetic structure. This fascinating book describes geometric frameworks underlying 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. He furthermore discusses the perceptual effects derived from the presence of particular geometric characteristics. The book began life as a liberal arts course and it is certainly suitable as a textbook. However, anyone interested in the power and ubiquity of mathematics will enjoy this revealing insight into the relationship between mathematics and the arts. Manifold Mirrors: The Crossing Paths of the Arts and Mathematics......Page 3 Contents......Page 7 Appetizers......Page 11 A.1 Martini......Page 13 A.2 On their blindness......Page 15 A.3 The Musical Offering......Page 19 A.4 The garden of the crossing paths......Page 22 1.1 The nature of space......Page 23 1.2 The shape of things......Page 24 1.3 Euclid......Page 26 1.4 Descartes......Page 30 2.1 Translations......Page 39 2.3 Reflections......Page 41 2.4 Glides......Page 42 2.5 Isometries of the plane......Page 43 2.6 On the possible isometries on the plane......Page 48 3 The many symmetries of planar objects......Page 51 3.1.1 Bilateral symmetry: the straight-lined mirror......Page 53 3.1.3 Central symmetry: the one-point mirror......Page 54 3.1.4 Translational symmetry: repeated mirrors......Page 56 3.1.5 Glidal symmetry......Page 58 3.2 The arithmetic of isometries......Page 59 3.3 A representation theorem......Page 64 3.4 Rosettes and whirls......Page 67 3.5.1 The seven friezes......Page 71 3.5.2 A classification theorem......Page 76 3.6.1 The seventeen wallpapers......Page 81 3.6.2 A brief sample......Page 88 3.6.3 Tables and flowcharts......Page 89 3.7 Symmetry and repetition......Page 92 3.8 The catalogue-makers......Page 93 4.1 Origins......Page 95 4.2 Rugs and carpets......Page 101 4.3 Chinese lattices......Page 115 4.4 Escher......Page 118 5.1 Aesthetic order......Page 123 5.2 The aesthetic measure of Birkhoff......Page 128 5.3 Gombrich and the sense of order......Page 132 5.4 Between boredom and confusion......Page 137 6.1 The veiled mirror......Page 140 6.2 Between detachment and dilution......Page 146 6.3 A blurred boundary: I......Page 150 6.4 The amazing kaleidoscope......Page 158 6.5 The strictures of verse......Page 164 7 Stretching the plane......Page 170 7.1 Homothecies and similarities......Page 171 7.2 Similarities and symmetry......Page 174 7.3 Shears, strains and affinities......Page 178 7.4 Conics......Page 186 7.5 The eclosion of ellipses......Page 189 7.6 Klein (aber nur der name)......Page 196 8 Aural wallpaper......Page 200 8.1 Elements of music......Page 201 8.2 The geometry of canons......Page 205 8.3 The Musical Offering (revisited)......Page 210 8.4 Symmetries in music......Page 218 8.4.1 The geometry of motifs......Page 220 8.4.2 The ubiquitous seven......Page 222 8.5 Perception, locality and scale......Page 225 8.6 The bare minima (again and again)......Page 228 8.7 A blurred boundary: II......Page 232 9 The dawn of perspective......Page 237 9.1 Alberti's window......Page 239 9.2 The dawn of projective geometry......Page 252 9.2.1 Bijections and invertible functions......Page 255 9.2.2 The projective plane......Page 257 9.2.3 A Kleinian view of projective geometry......Page 263 9.2.4 Essential features of projective geometry......Page 265 9.3 A projective view of affine geometry......Page 266 9.3.1 A distant vantage point......Page 267 9.3.2 Conics revisited......Page 270 10.1 Projections and drawing systems......Page 272 10.1.1 Orthogonal projections......Page 275 10.1.2 Oblique projections......Page 281 10.1.3 On tilt and distance......Page 290 10.1.4 Perspective projection......Page 294 10.2 Voyeurs and demiurges......Page 298 11.1 Deceptions......Page 305 11.2 Concealments......Page 308 11.3 Bends......Page 310 11.4 Absurdities......Page 318 11.5 Divergences......Page 323 11.6 Multiplicities......Page 327 11.7 Abandonment......Page 329 12.1 Euclid revisited......Page 333 12.2 Hyperbolic geometry......Page 337 12.3.1 Formal languages......Page 340 12.3.2 Deduction......Page 342 12.3.3 Validity......Page 345 12.3.4 Two models for Euclidean geometry......Page 347 12.3.5 Proof and truth......Page 350 12.4 The Poincaré model of hyperbolic geometry......Page 351 12.5 Projective geometry as a non-Euclidean geometry......Page 358 12.6 Spherical geometry......Page 365 13.1 Tessellations and wallpapers......Page 369 13.2 Isometries and tessellations in the sphere and the projective plane......Page 371 13.3 Isometries and tessellations in the hyperbolic plane......Page 375 14 The shape of the universe......Page 385 Compliers/benders/transgressors......Page 393 Constrained writing......Page 398 References......Page 407 Acknowledgements......Page 10 Index of symbols......Page 416 Index of names......Page 417 Index of concepts......Page 421 Manifold Mirrors: The Crossing Paths of the Arts and Mathematics 3 Contents 7 Mathematics: user's manual 11 Appetizers 11 A.1 Martini 13 A.2 On their blindness 15 A.3 The Musical Offering 19 A.4 The garden of the crossing paths 22 1 Space and geometry 23 1.1 The nature of space 23 1.2 The shape of things 24 1.3 Euclid 26 1.4 Descartes 30 2 Motions on the plane 39 2.1 Translations 39 2.2 Rotations 41 2.3 Reflections 41 2.4 Glides 42 2.5 Isometries of the plane 43 2.6 On the possible isometries on the plane 48 3 The many symmetries of planar objects 51 3.1 The basic symmetries 53 3.1.1 Bilateral symmetry: the straight-lined mirror 53 3.1.2 Rotational symmetry 54 3.1.3 Central symmetry: the one-point mirror 54 3.1.4 Translational symmetry: repeated mirrors 56 3.1.5 Glidal symmetry 58 3.2 The arithmetic of isometries 59 3.3 A representation theorem 64 3.4 Rosettes and whirls 67 3.5 Friezes 71 3.5.1 The seven friezes 71 3.5.2 A classification theorem 76 3.6 Wallpapers 81 3.6.1 The seventeen wallpapers 81 3.6.2 A brief sample 88 3.6.3 Tables and flowcharts 89 3.7 Symmetry and repetition 92 3.8 The catalogue-makers 93 4 The many objects with planar symmetries 95 4.1 Origins 95 4.2 Rugs and carpets 101 4.3 Chinese lattices 115 4.4 Escher 118 5 Reflections on the mirror 123 5.1 Aesthetic order 123 5.2 The aesthetic measure of Birkhoff 128 5.3 Gombrich and the sense of order 132 5.4 Between boredom and confusion 137 6 A raw material 140 6.1 The veiled mirror 140 6.2 Between detachment and dilution 146 6.3 A blurred boundary: I 150 6.4 The amazing kaleidoscope 158 6.5 The strictures of verse 164 7 Stretching the plane 170 7.1 Homothecies and similarities 171 7.2 Similarities and symmetry 174 7.3 Shears, strains and affinities 178 7.4 Conics 186 7.5 The eclosion of ellipses 189 7.6 Klein (aber nur der name) 196 8 Aural wallpaper 200 8.1 Elements of music 201 8.2 The geometry of canons 205 8.3 The Musical Offering (revisited) 210 8.4 Symmetries in music 218 8.4.1 The geometry of motifs 220 8.4.2 The ubiquitous seven 222 8.5 Perception, locality and scale 225 8.6 The bare minima (again and again) 228 8.7 A blurred boundary: II 232 9 The dawn of perspective 237 9.1 Alberti's window 239 9.2 The dawn of projective geometry 252 9.2.1 Bijections and invertible functions 255 9.2.2 The projective plane 257 9.2.3 A Kleinian view of projective geometry 263 9.2.4 Essential features of projective geometry 265 9.3 A projective view of affine geometry 266 9.3.1 A distant vantage point 267 9.3.2 Conics revisited 270 10 A repertoire of drawing systems 272 10.1 Projections and drawing systems 272 10.1.1 Orthogonal projections 275 10.1.2 Oblique projections 281 10.1.3 On tilt and distance 290 10.1.4 Perspective projection 294 10.2 Voyeurs and demiurges 298 11 The vicissitudes of perspective 305 11.1 Deceptions 305 11.2 Concealments 308 11.3 Bends 310 11.4 Absurdities 318 11.5 Divergences 323 11.6 Multiplicities 327 11.7 Abandonment 329 12 The vicissitudes of geometry 333 12.1 Euclid revisited 333 12.2 Hyperbolic geometry 337 12.3 Laws of reasoning 340 12.3.1 Formal languages 340 12.3.2 Deduction 342 12.3.3 Validity 345 12.3.4 Two models for Euclidean geometry 347 12.3.5 Proof and truth 350 12.4 The Poincaré model of hyperbolic geometry 351 12.5 Projective geometry as a non-Euclidean geometry 358 12.6 Spherical geometry 365 13 Symmetries in non-Euclidean geometries 369 13.1 Tessellations and wallpapers 369 13.2 Isometries and tessellations in the sphere and the projective plane 371 13.3 Isometries and tessellations in the hyperbolic plane 375 14 The shape of the universe 385 Appendix: Rule-driven creation 393 Compliers/benders/transgressors 393 Constrained writing 398 References 407 Acknowledgements 10 Index of symbols 416 Index of names 417 Index of concepts 421 Most Works Of Art, Whether Illustrative, Musical Or Literary, Are Created Subject To A Set Of Constraints. In Many (but Not All) Cases, These Constraints Have A Mathematical Nature, For Example, The Geometric Transformations Governing The Canons Of J.s. Bach, The Various Projection Systems Used In Classical Painting, The Catalog Of Symmetries Found In Islamic Art, Or The Rules Concerning Poetic Structure. This Fascinating Book Describes Geometric Frameworks Underlying This Constraint-based Creation. The Author Provides A Development In Geometry, A Description Of How These Frameworks Fit The Creative Process Within Several Art Practices, And Discusses The Perceptual Effects Derived From The Presence Of Particular Geometric Characteristics. Space And Geometry -- Motions On The Plane -- The Many Symmetries Of Planar Objects -- The Many Objects With Planar Symmetries -- Reflections On The Mirror -- A Raw Material -- Stretching The Plane -- Aural Wallpaper -- The Dawn Of Perspective -- A Repertoire Of Drawing Systems -- The Vicissitudes Of Perspective -- The Vicissitudes Of Geometry -- Symmetries In Non-euclidean Geometries -- The Shape Of The Universe -- Appendix: Rule-driven Creation. Felipe Cucker, City University Of Hong Kong. Includes Bibliographical References (pages 395-401) And Indexs. Felipe Cucker presents a unifying mathematical structure to explore the relationship between mathematics and the arts, including architecture, music, poetry and more. The book emerged from the author's undergraduate course, but requiring only basic high-school knowledge of mathematics it makes a fascinating read for anyone interested in the arts. This fascinating book will interest anyone wanting to learn more about the relationship between mathematics and the arts