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Manifold Mirrors
The Crossing Paths of the Arts and Mathematics

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  • Date Published: June 2013
  • availability: Available
  • format: Paperback
  • isbn: 9780521728768

$ 24.99 (P)

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About the Authors
  • 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.

    • Uses geometry as a unifying theme to explore many different arts rather than focusing on just one
    • Assumes only a basic high-school grasp of mathematics
    • Mathematical proofs are self-contained so individual readers can pay them as much or as little attention as they wish
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    • Honourable Mention, 2013 PROSE Award for Mathematics

    Reviews & endorsements

    "Cucker [has] produced a pot au feu, an eclectic catch-all. There is much that can be learned from [his] presentation of the marriage of mathematics and art. I consider Manifold Mirrors Arcimboldesque in that it is an assemblage of many basic mathematical ideas and constructs, [adding] up to … well, to a unique work."
    Philip J. Davis, SIAM News

    "… there is certainly something new to be discovered for every reader. The book grew out of a course, and so it is obviously possible to extract some interesting lectures from the material that is presented."
    The European Mathematical Society (

    "The merits of this big, ambitious book greatly exceed its deficiencies. Felipe Cucker’s immense learning, and his often densely technical presentation of mathematical complexities, are balanced by a pervasive lightness of tone and by a flair for offbeat allusions that range from Euripides to Busby Berkeley. His book is a joy for the eye and a feast for the mind."
    Hardy Grant, MAA Reviews

    "Mathematical material is complete and rigorous, at an upper-division undergraduate level, with statements of relevant theorems and their proofs … Highly recommended. Upper-division undergraduates and faculty."
    C. A. Gorini, Choice

    'This is an impressive and ambitious book and is one well-worth taking time to work through.' Richard Talbot, Nexus Network Journal

    See more reviews

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    Product details

    • Date Published: June 2013
    • format: Paperback
    • isbn: 9780521728768
    • length: 424 pages
    • dimensions: 247 x 174 x 20 mm
    • weight: 0.86kg
    • contains: 55 b/w illus. 100 colour illus. 30 music examples
    • availability: Available
  • Table of Contents

    Mathematics: user's manual
    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
    Index of symbols
    Index of names
    Index of concepts.

  • Author

    Felipe Cucker, City University of Hong Kong
    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.


    • Honourable Mention, 2013 PROSE Award for Mathematics

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