You are previewing Manifold Mirrors.
O'Reilly logo
Manifold Mirrors

Book Description

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.

Table of Contents

  1. Cover
  2. Half Title
  3. Title
  4. Copyright
  5. Contents
  6. Mathematics: user’s manual
  7. Appetizers
    1. A.1 Martini
    2. A.2 On their blindness
    3. A.3 The Musical Offering
    4. A.4 The garden of the crossing paths
  8. 1 Space and geometry
    1. 1.1 The nature of space
    2. 1.2 The shape of things
    3. 1.3 Euclid
    4. 1.4 Descartes
  9. 2 Motions on the plane
    1. 2.1 Translations
    2. 2.2 Rotations
    3. 2.3 Reflections
    4. 2.4 Glides
    5. 2.5 Isometries of the plane
    6. 2.6 On the possible isometries on the plane
  10. 3 Themany symmetries of planar objects
    1. 3.1 The basic symmetries
      1. 3.1.1 Bilateral symmetry: the straight-lined mirror
      2. 3.1.2 Rotational symmetry
      3. 3.1.3 Central symmetry: the one-point mirror
      4. 3.1.4 Translational symmetry: repeated mirrors
      5. 3.1.5 Glidal symmetry
    2. 3.2 The arithmetic of isometries
    3. 3.3 A representation theorem
    4. 3.4 Rosettes and whirls
    5. 3.5 Friezes
      1. 3.5.1 The seven friezes
      2. 3.5.2 A classification theorem
    6. 3.6 Wallpapers
      1. 3.6.1 The seventeen wallpapers
      2. 3.6.2 A brief sample
      3. 3.6.3 Tables and flowcharts
    7. 3.7 Symmetry and repetition
    8. 3.8 The catalogue-makers
  11. 4 Themany objects with planar symmetries
    1. 4.1 Origins
    2. 4.2 Rugs and carpets
    3. 4.3 Chinese lattices
    4. 4.4 Escher
  12. 5 Reflections on the mirror
    1. 5.1 Aesthetic order
    2. 5.2 The aesthetic measure of Birkhoff
    3. 5.3 Gombrich and the sense of order
    4. 5.4 Between boredom and confusion
  13. 6 A raw material
    1. 6.1 The veiled mirror
    2. 6.2 Between detachment and dilution
    3. 6.3 A blurred boundary: I
    4. 6.4 The amazing kaleidoscope
    5. 6.5 The strictures of verse
  14. 7 Stretching the plane
    1. 7.1 Homothecies and similarities
    2. 7.2 Similarities and symmetry
    3. 7.3 Shears, strains and affinities
    4. 7.4 Conics
    5. 7.5 The eclosion of ellipses
    6. 7.6 Klein (aber nur der Name)
  15. 8 Aural wallpaper
    1. 8.1 Elements of music
    2. 8.2 The geometry of canons
    3. 8.3 The Musical Offering (revisited)
    4. 8.4 Symmetries in music
      1. 8.4.1 The geometry of motifs
      2. 8.4.2 The ubiquitous seven
    5. 8.5 Perception, locality and scale
    6. 8.6 The bare minima (again and again)
    7. 8.7 A blurred boundary: II
  16. 9 The dawn of perspective
    1. 9.1 Alberti’s window
    2. 9.2 The dawn of projective geometry
      1. 9.2.1 Bijections and invertible funct
      2. 9.2.2 The projective plane
      3. 9.2.3 A Kleinian view of projective geometry
      4. 9.2.4 Essential features of projective geometry
    3. 9.3 A projective view of affine geometry
      1. 9.3.1 A distant vantage point
      2. 9.3.2 Conics revisited
  17. 10 A repertoire of drawing systems
    1. 10.1 Projections and drawing systems
      1. 10.1.1 Orthogonal projections
      2. 10.1.2 Oblique projections
      3. 10.1.3 On tilt and distance
      4. 10.1.4 Perspective projection
    2. 10.2 Voyeurs and demiurges
  18. 11 The vicissitudes of perspective
    1. 11.1 Deceptions
    2. 11.2 Concealments
    3. 11.3 Bends
    4. 11.4 Absurdities
    5. 11.5 Divergences
    6. 11.6 Multiplicities
    7. 11.7 Abandonment
  19. 12 The vicissitudes of geometry
    1. 12.1 Euclid revisited
    2. 12.2 Hyperbolic geometry
    3. 12.3 Laws of reasoning
      1. 12.3.1 Formal languages
      2. 12.3.2 Deduction
      3. 12.3.3 Validity
      4. 12.3.4 Two models for Euclidean geometry
      5. 12.3.5 Proof and truth
    4. 12.4 The Poincaré model of hyperbolic geometry
    5. 12.5 Projective geometry as a non-Euclidean geometry
    6. 12.6 Spherical geometry
  20. 13 Symmetries in non-Euclidean geometries
    1. 13.1 Tessellations and wallpapers
    2. 13.2 Isometries and tessellations in the sphere and the projective plane
    3. 13.3 Isometries and tessellations in the hyperbolic plane
  21. 14 The shape of the universe
  22. Appendix: Rule-driven creation
    1. Compliers/benders/transgressors
    2. Constrained writing
  23. References
  24. Acknowledgements
  25. Index of symbols
  26. Index of names
  27. Index of concepts