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Geometry and the Imagination

American Mathematical Society/Chelsea
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This is the 1952 translation by Paul F. Nemenyi of the German book Anschauliche Geometrie of 1932. A more literal translation of the German title would be “Descriptive Geometry,” a term that, in English, carries too many overtones of technical drawing and the work of Monge. The Italian translation is called Geometria intuitiva and includes P. S. Alexandroff’s Einfachste Grundbegriffe der Topologie (originally commissioned as an appendix to Anschauliche Geometrie) as an appendix with the title “I primi fondamenti della topologia,” whereas the English translation (by Alan F. Parley) of Alexandroff’s work Elementary Concepts of Topology is available as a separate book through Dover. An alternative translation of the title, then, would be “Intuitive geometry.”

David Hilbert is, of course, known to all mathematicians. Stephan Cohn-Vossen died tragically from pneumonia as a Jewish refugee in Moscow, driven from Germany by the Nazi racial legislation forbidding him to teach there, a mere four years after the original publication of this book at the age of 34. This book is his memorial. (See Sanford Segal Mathematicians under the Nazis, footnote on page 383, and Reinhard Siegmund-Schultze, Mathematicians Fleeing from Nazi Germany. and).

Geometry and the Imagination is a mathematics book, written at the popular level by a great mathematician and a younger colleague, but without condescension to the intended audience. Although richly illustrated with over 300 pictures, it is not an easy read — the pages are densely packed with information, presented with a minimum of technicalities. It is beautifully written and carefully translated,

This is a book that is dear to the hearts of many geometers and topologists. It is impossible to do justice to its astonishing breadth in a review, but perhaps mentioning a few highlights to my eyes will suffice. It would be a rare reader today who likes each chapter equally. So, in order to differentiate this review from numerous other, much earlier reviews (such as Turnbull’s 1933 review in The Mathematical Gazette of the German original, MacDuffee’s 1952 review in Science and Coxeter’s 1953 review for the American Association for the Advancement of Science, both of the English translation), and to accommodate those readers with an appetite for more, suggestions will be made for follow-up or companion books to each of the chapters, many of them of much more recent vintage, and so unable to be mentioned by Turnbull, MacDuffee or Coxeter. This also compensates for the more than eighty years that have passed since the book was written.

In the second chapter, there is a delightful proof of a famous series for \(\pi/4\). The series is usually attributed to either Gregory or Leibniz, but was known to Kerala mathematicians of the 14th century, such as Madhava of Sangamagrama. (See George Gherveghese Joseph’s books, The Crest of the Peacock and A Passage to Infinity: Medieval Indian Mathematics from Kerala and its Impact.) The proof involves counting lattice points in circles, and some elementary number theory on Pythagorean triples. Minkowski’s Geometrie der Zahlen is a possible sequel to this chapter, as is Cassels’ An Introduction to the Geometry of Numbers.

The third chapter emphasizes configurations, a thriving area for the 50 years prior to the book being written, ever since Theodor Reye’s Die Geometrie der Lage was published in 1876 — with a (now obscure) book devoted to the subject published (in German) by Friedrich W. Levi in 1929. With the prominent exception of the papers of Branko Grunbaum, and of the little booklet from 1957 of Argunov and Skornyakov, this subject has been somewhat quiescent until the very recent publication of two books devoted to the subject: Grunbaum’s 2009 Configurations of Points and Lines, and Tomaz Pisanski and Brigitte Servatius’ 2013 Configurations from a Graphical Viewpoint, both of which take inspiration from the third chapter of Hilbert and Cohn-Vossen, and would make good sequels to this chapter. In the chapter, there are lovely treatments of Schläafli’s double-six (beautifully drawn), the 27 lines on a cubic surface, and Reye’s configuration of 12 points, 16 lines and 12 planes (with a wonderful drawing of the related four-dimensional regular polytope). The point of view is that of configurations that express theorems, so that if all but one of the incidences hold, the remaining incidence must follow. (The more recent books take a more combinatorial perspective, only occasionally reverting to this older perspective.) The Desargues and Pappus configurations are also prominent, with the latter named after Pascal, as is usual in Hilbert’s geometric work.

The fourth chapter is a discussion of curvature and its ramifications, including non-Euclidean geometry. The recent Differential Geometry of Curves and Surfaces, by Banchoff and Lovett, might be one place an interested reader might go from here. Much of this cannot be explained at an elementary level, especially without coordinates, and so the authors often resort to “it can be shown that” in this chapter. A highlight of this chapter is the famous list of eleven properties of the sphere. Another possible sequel is John Stillwell’s Geometry of Surfaces.

The fifth chapter, which might be entitled the geometry of the engineer, contains a discussion of linkages and associated instruments. An example is Yom Tov Lippman Lipkin’s and Charles-Nicolas Peaucellier’s linkage of seven bars and six joints for translating circular motion into linear from the 1860s, which unfortunately turned out to be mechanically impractical in industrial settings, owing to a tendency to wobble too much because of the large number of moving components. There is an animation of this linkage on Wikipedia. The mathematics of this linkage is based on inversion in a circle. Bill Thurston’s Three-Dimensional Geometry and Topology is a book that a reader stimulated by this chapter to seek out more may choose to read after this, perhaps with Reid and Szendroi’s Geometry and Topology in between.

The sixth chapter contains a classification of surfaces, essentially in terms of the Euler-Poincaré characteristic, and a discussion of the problem of colouring a map with four colours, so that no two adjacent countries have the same colour. Two good companions for this chapter are Alexandroff’s work mentioned above, and Lakatos’ incomparable Proofs and Refutations. For the four colour problem, the interested reader should seek out Robin Wilson’s Four Colo(u)rs Suffice.

In summary, this book is a masterpiece — a delightful classic that should never go out of print.

Tim Penttila is Professor of Mathematics at Colorado State University.

Date Received: 
Wednesday, September 6, 2006
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David Hilbert and S. Cohn-Vossen
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Tim Penttila
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  • The simplest curves and surfaces
  • Regular systems of points
  • Projective configurations
  • Differential geometry
  • Kinematics
  • Topology
  • Index
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Monday, January 19, 2009