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.
Author(s): Felipe Cucker
Publisher: Cambridge University Press
Year: 2013
Language: English
Pages: 424
Tags: Математика;Популярная математика;
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
