Geometry Revealed. A Jacob's Ladder to Modern Higher Geometry
Marcel Berger is a well known and influential French geometer. His Géométrie I and Géométrie II appeared in 1977. English translations appeared in 1987. In the nineteenth century the millennium-long supremacy of Euclidean geometry was replaced by an outbreak of non-Euclidean geometries but in the course of the twentieth century, geometry had somewhat disappeared from the mainstream of mathematics, but near the end of that century, geometry got married to several other aspects of mathematics and a new impetus was given to the field. Now in the 21st century, it seems that geometry, or at least in its current form, is back at the forefront in answering fundamental questions as well as helping to solve practical problems.
Berger's Geometry I,II books were already stressing the artistic aspect of geometry and showed that even with simple geometry, easily understood problems can be formulated with non-obvious answers or for which the proofs are all but simple if known at all. However, although brought in a relatively colloquial style, those books are precise, with formulas and fully worked out proofs. The current book is again a translation of the French original Géométrie vivante ou l'échelle de Jacob. It first appeared in 2009, 21 years after the Geometry books. It is essentially a continuation and a supplement of those with of course recent developments included, but still pursuing the same objective. A major difference is that now proofs are not always fully included. The discussion is essentially at a conceptual level. For the elementary details, one is referred to his Geometry books, and for more advanced problems and recent solutions to the literature. In the introduction Berger states that it is his aspiration to write a modern version of Geometry and the Imagination by Hilbert and Cohn-Vossen (German original 1932, English translation 1952), bringing `cultural geometry' for a broad audience. The idea is similar. It is not a book that one reads from cover to cover. There are 12 chapters, and although there are cross references, they can be read practically independently. Each chapter groups somehow related problems mostly starting with a relatively simple one but which is easily, and quickly, lifted to a quite complicated one for which a solution is far from obvious. This requires the reader to go one or more `steps up the ladder', i.e., the Jacob's ladder, a stairway to heaven, which is a recurrent theme in the book. Many times a relatively simple solution comes from a transformation of the problem, or using a solution method that comes from a different discipline. Berger is a purist in the sense that he is always searching for the minimal possible conditions, and if the problem formulation does not need a metric then a solution or proof should be found without a metric.
Each chapter has its own bibliography, but there is a global name, subject, and notation index. There are not perfect, but nevertheless very useful instruments since there is not really a linear structure throughout the book and even within a chapter there are many digressions. A somewhat more systematic approach with definitions and further discussions are collected at the end of each chapter as a section XYZ. There are too many subjects covered to give a survey in this review. As Berger suggests in the introduction, one can pick up the book and open it at random and see if there is something that interests you. Once you are hooked, you will read on. There is not a strict logical order. The best way to describe it is that it is a kind of exposition with beautiful, curious, strange, or challenging problems, theorems and proofs, and Berger is the guide leading us through the different exposition rooms with his expert comments and clarifications, drawing our attention to the magnificent details here and there, telling about the history, the background and the utility of the results.
As for the particular contents, let me just list the titles of the chapters, although these give only a superfluous idea of what they contain.
1. Points and lines in the plane,
2. Circles and spheres,
3. The sphere by itself: can we distribute points on it evenly?
4. Conics and quadratics,
5. Plane curves
6. Smooth surfaces,
7. Convexity and convex sets,
8. Polygons, polyhedra, polytopes,
9. Lattices, packings and tilings in the plane,
10. Lattices and packings in higher dimensions,
11. Geometry and dynamics I: billiards,
12. Geometry and dynamics II: geodesic flow on a surface.
So there is a lot of material, but even with about 850 pages, not every aspect of geometry is covered and not all aspects covered are covered exhaustively. To give an idea of what these chapters look like, let me go through one example: Chapter 3. It starts with a metric on the sphere and spherical trigonometry and the Möbius group, to come to the Tammes problem: uniform distribution of a set of points on a sphere. For 2,3,4, or 6 points, the optimal solution is a symmetric ones. First surprise: with 5 points, the solution is the same as for 6 points, by removing one of them. Known solutions are vertices of regular polyhedron. For example with 6 points, the vertices of a octahedron. But that is not always the case, a phenomenon known in physics as symmetry breaking. Next surprise: optimal solutions are known up to 12 points and also for 24 points. Nature has found solutions in some alloys with 16 points. Asymptotics estimates can be given for the distance between the points as their number approaches infinity. The problem is of practical interest to distribute the dimples on golf balls, forming a pattern that has been patented, but alternatives are proposed. Potential theory gives another solution for the distribution that can be computed numerically. It relates to computing the average of a function over the sphere. A different topic related to spheres is how many identical spheres can be arranged, around one of them if they are just touching: the kissing problem. Proving the kissing number for a sphere in the familiar 3D space is an old problem that has been settled only last century, but for higher dimensions it is only known for some and only upper and lower bounds are known in other cases.
At first sight, the book is addressing a general (mathematical) public, although not every reader will enjoy all the chapters with the same enthusiasm. Moreover, if one is prepared to really think about some of the problems or follow Berger's suggestions and work out some of the proofs, some level of geometric training, for example being familiar with his Geometry books, is necessary or at least advisable. You need to be familiar with the lower rungs of the ladder to be able to climb some steps higher.
The quality of the many figures is not uniform. Some are just hand-drawn with hand-written text, others are reprinted from other publications and some are computer generated without reference. Of course this does not affect their main purpose which is to clarify the text, but it is rather unusual in a book published in an era with all that wonderful graphical software available. I was not particularly looking for typos, but nevertheless I could spot some. References to figures were sometimes pointing to the wrong section. But these are just minor flaws that are largely overcompensated by the contents and the entertaining qualities of the guide and curator Marcel Berger who brought all these mathematical gems together in this geometric exposition for the reader to enjoy.