Manifold mirrors : the crossing paths of the arts and mathematics / Felipe Cucker, City University of Hong Kong.

Format:

Book

Author/Creator:

Cucker, Felipe, 1958- author.

Language:

English

Subjects (All):

Arts--Mathematics.

Arts.

Physical Description:

1 online resource (x, 415 pages) : digital, PDF file(s).

Edition:

1st ed.

Place of Publication:

Cambridge : Cambridge University Press, 2013.

Language Note:

English

Summary:

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.

Contents:

Intro

Manifold Mirrors: The Crossing Paths of the Arts and Mathematics

Contents

Mathematics: user's manual

Appetizers

A.1 Martini

A.2 On their blindness

A.3 The Musical Offering

A.4 The garden of the crossing paths

1 Space and geometry

1.1 The nature of space

1.2 The shape of things

1.3 Euclid

1.4 Descartes

2 Motions on the plane

2.1 Translations

2.2 Rotations

2.3 Reflections

2.4 Glides

2.5 Isometries of the plane

2.6 On the possible isometries on the plane

3 The many symmetries of planar objects

3.1 The basic symmetries

3.1.1 Bilateral symmetry: the straight-lined mirror

3.1.2 Rotational symmetry

3.1.3 Central symmetry: the one-point mirror

3.1.4 Translational symmetry: repeated mirrors

3.1.5 Glidal symmetry

3.2 The arithmetic of isometries

3.3 A representation theorem

3.4 Rosettes and whirls

3.5 Friezes

3.5.1 The seven friezes

3.5.2 A classification theorem

3.6 Wallpapers

3.6.1 The seventeen wallpapers

3.6.2 A brief sample

3.6.3 Tables and flowcharts

3.7 Symmetry and repetition

3.8 The catalogue-makers

4 The many objects with planar symmetries

4.1 Origins

4.2 Rugs and carpets

4.3 Chinese lattices

4.4 Escher

5 Reflections on the mirror

5.1 Aesthetic order

5.2 The aesthetic measure of Birkhoff

5.3 Gombrich and the sense of order

5.4 Between boredom and confusion

6 A raw material

6.1 The veiled mirror

6.2 Between detachment and dilution

6.3 A blurred boundary: I

6.4 The amazing kaleidoscope

6.5 The strictures of verse

7 Stretching the plane

7.1 Homothecies and similarities

7.2 Similarities and symmetry

7.3 Shears, strains and affinities

7.4 Conics

7.5 The eclosion of ellipses

7.6 Klein (aber nur der name)

8 Aural wallpaper

8.1 Elements of music.

8.2 The geometry of canons

8.3 The Musical Offering (revisited)

8.4 Symmetries in music

8.4.1 The geometry of motifs

8.4.2 The ubiquitous seven

8.5 Perception, locality and scale

8.6 The bare minima (again and again)

8.7 A blurred boundary: II

9 The dawn of perspective

9.1 Alberti's window

9.2 The dawn of projective geometry

9.2.1 Bijections and invertible functions

9.2.2 The projective plane

9.2.3 A Kleinian view of projective geometry

9.2.4 Essential features of projective geometry

9.3 A projective view of affine geometry

9.3.1 A distant vantage point

9.3.2 Conics revisited

10 A repertoire of drawing systems

10.1 Projections and drawing systems

10.1.1 Orthogonal projections

10.1.2 Oblique projections

10.1.3 On tilt and distance

10.1.4 Perspective projection

10.2 Voyeurs and demiurges

11 The vicissitudes of perspective

11.1 Deceptions

11.2 Concealments

11.3 Bends

11.4 Absurdities

11.5 Divergences

11.6 Multiplicities

11.7 Abandonment

12 The vicissitudes of geometry

12.1 Euclid revisited

12.2 Hyperbolic geometry

12.3 Laws of reasoning

12.3.1 Formal languages

12.3.2 Deduction

12.3.3 Validity

12.3.4 Two models for Euclidean geometry

12.3.5 Proof and truth

12.4 The Poincaré model of hyperbolic geometry

12.5 Projective geometry as a non-Euclidean geometry

12.6 Spherical geometry

13 Symmetries in non-Euclidean geometries

13.1 Tessellations and wallpapers

13.2 Isometries and tessellations in the sphere and the projective plane

13.3 Isometries and tessellations in the hyperbolic plane

14 The shape of the universe

Appendix: Rule-driven creation

Compliers/benders/transgressors

Constrained writing

References

Acknowledgements

Index of symbols

Index of names

Index of concepts.

Notes:

Title from publisher's bibliographic system (viewed on 05 Oct 2015).

Includes bibliographical references and indexes.

ISBN:

1-107-23303-8

1-107-34737-8

1-139-01463-3

1-107-34860-9

1-107-34112-4

1-107-34487-5

0-521-72876-2

1-107-34362-3

OCLC:

842929972