Mainly relying on two-dimensional transformations, the book is a mixture of proper mathematics explaining some aspects of Euclidean and non-Euclidean geometry, philosophical discussions on aesthetics, and all of this amply illustrated with many examples from mainly two-dimensional pieces of art, and one chapter on Bach's music. These examples illustrate many forms of symmetry, tessellations, hyperbolic and projective geometry, the use of perspective, other non Euclidean geometries, etc.
The cover states that this book grew out of a liberal arts course. Felipe Cucker is Chair Professor of Mathematics at the City University of Hong Kong. The result is a collection of chapters, some of which are just plain mathematics, others analyse the philosophical and psychological aspects of aesthetics, and many discuss a wide diversity of works of art and how the mathematics are recognized in their features (like symmetry or translation invariance for example) or how mathematics have influenced the techniques available to the artists (like perspective or hyperbolic geometry).
Almost all examples are two-dimensional, which is related to the mathematics that are covered. These are all geometric, treating transformations in the plane or different kind of projections and non-Euclidean geometry. So most art examples are graphical like paintings and drawings but also carpets, and occasionally poetry and dancing is mentioned. One exception is a complete chapter devoted to Bach's canons, but architecture and sculpture, which is clearly three-dimensional, is almost completely absent.
After an appetizing introduction showing symmetry and structure in work by Simone Martini (painting), John Milton (poetry) and Johan Sebastian Bach (music), the first chapter introduces geometry and its history from Euclid to Descartes and this is followed by a mathematical treatment of plane transforms: translation, rotation, reflection, glide, isometry, completely with definitions and proofs. Artistic examples illustrate the mathematics in another chapter where it is shown that there are exactly 7 friezes (translation invariant pattern in one direction) and 17 wallpapers (translation invariant pattern for two independent vectors). Pieces of art with planar symmetry are easily found. Tessellations and patterns in carpets from Central Asia, Chinese lattices and of course Escher's work.
Much more philosophical is an analysis of George D. Birkhoff's attempt to define a measure of aesthetics and Ernst H. Gombrich's sense of order. We often see symmetry when it is not really there. Da Vinci's Vitruvian man is not perfectly symmetric. A sense of beauty is raised by a balance disorder and boredom. This is illustrated with several examples from op-art, and for example repetitive work by Andy Warhol, but also from ballet performances and the rhyme and rhythm of poetry. Mathematics re-enter with homothecies, similarities, shears, strains and affinities and conics, which is illustrated by the use of the ellipse (a circle in perspective) in the Renaissance. More patterns are illustrated with musical canons, in particular the ones in J.S. Bach's Musical Offering.
The introduction of perspective in European paintings triggers some more mathematics introducing projective geometry and projections. This allows to produce proper representation of a reflection in a sphere, but also optical illusions based on false perspective. The rules of perspective are left with the start of cubism and modern art. The parallel in mathematics is that Euclidean geometry is left to introduce alternatives based on axiomatic systems and formal languages. In a final short chapter, Cucker ponders briefly on the geometry to describe our universe, but this requires to leave the two-dimensional world he has been discussing so far.
Rule-driven creation is moved to an appendix. Literature does not have the same geometrical basis as the other examples according to Cucker, yet he describes some patterns of constrained writing like anagrams, palindromes and other word plays.
This wonderful survey shows that, even though the author has restricted his approach mainly to two-dimensional geometry and transformations of the plane, it should be clear by now that this is still a very broad area when this is related to visual (and aural) art. From Euclidean geometry to Gödel's completeness theorem, from stone age artifacts to modern dance theater, from short biographies to quantitative aesthetics, the scope is enormous, forcing the author to be selective. The illustrations used are not always the ones that are best known though. So 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.