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.
Felipe Cucker 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 decentralized systems (in particular, emergence of languages and flocking), learning theory, and foundational aspects of numerical analysis. He serves on the editorial board of several journals and is Managing Editor of the journal Foundations of Computational Mathematics, published by the society of the same name.
Mathematics: user's manual; Appetizers; 1. Space and geometry; 2. Motions on the plane; 3. The many symmetries of planar objects; 4. The many objects with planar symmetries; 5. Reflections on the mirror; 6. A raw material; 7. Stretching the plane; 8. Aural wallpaper; 9. The dawn of perspective; 10. A repertoire of drawing systems; 11. The vicissitudes of perspective; 12. The vicissitudes of geometry; 13. Symmetries in non-Euclidean geometries; 14. The shape of the universe; Appendix: rule-driven creation; References; Acknowledgements; Index of symbols; Index of names; Index of concepts.