A marvelous book, abundantly illustrated. It contains many applications in art science and technology, all with an underlying geometric flavour. There are some theorems and proofs, but these can be just skipped without harming the overall reading in case the reader is not interested in the technicalities.
What is immediately striking at first glance is the luxury of this publication: thick paper with hundreds of colorful glossy pictures and graphs. If you love books and geometry, this is one to fall in love with. Heavy stuff though: about 1.4 kg, it is a treasure that you would not like to take in your hand lugage on a plane.
In fact, the book has been available in German as Geometrie und ihre Anwendungen in Kunst, Natur und Technik (Elsevier, Spektrum Akademischer Verlag, 2005/2007). This is the English translation, extended with 60 pages and extra illustrations (there are about 900 of them). The author is professor at the Universität für angewandte Kunst in Vienna, and this might explain that this book, with geometry as the binding factor, has so many and very diverse applications. Moreover, this is not his first book on this kind of topic and he has also books on software for computer geometry (OpenGL®). Additional information about this book and links to other publications can be found at the book's website www.uni-ak.ac.at/geometrie.
This picture book is more than just a coffee-table book unless it is a table in the coffee room of a math department, because it contains not only many pictures, but also gives theorems and proofs(!). Although the proofs are not so very technical and are more descriptive geometrical than analytical with a minimum of formulas. If the reader is not interested in these proofs, there is no harm done or discontinuity in the appreciation of the global story told when they are just skipped.
The emphasis is clearly on the applications of geometry. In 13 chapters of increasing complexity, the reader is confronted with many expected but also with many unexpected applications. Sometimes, the application is more physics than geometry, but if it has an important geometrical component, it is reason enough to include it.
The author starts with points, lines, and elementary curves in the plane, to move on to projections. Already there the reader finds applications such as what can be learned from the shadows of objects or about the retro-reflector in a bicycle wheel. Entering the 3D world starts with polyhedra, then moves on to curves in 2D and 3D, to arrive at cones and cylinders as the simplest examples of what is further elaborated: developable surfaces, conic sections and surfaces of revolution. On a more advances level we find helical, spiral and minimal surfaces and an introduction to splines and NURBS for modeling general curved surfaces. All of this is amply illustrated with many applications from industrial design, architecture, cartography, connecting pipes, gear wheels, animal horns, DNA, and many more.
After that, the chapters start dealing with the more applied sciences. Chapter 9 is about optics: the human eye and photography and reflections and refraction. The next two chapters deal with the geometry of motion: curves generated by all sorts of mechanical devises, and orbits in astronomy. The last two chapters are about tilings of the plane and symmetry and other remarkable patterns appearing in nature. The latter two are are promoted from an appendix in the German edition to proper chapters in this one.
There are also two appendices left in the form of short courses. One is about free hand drawing. As the author rightfully claims, in this computer age where pictures and graphs are rendered digitally by computer software, generating a result that is unnaturally close to perfection, free hand drawing becomes a rare skill while it should be a basic one for communication. The second course is about photography: the rules of perspective brought in practice. Both of these can be read independent from the rest of the text.