"In this work we present a whole series of [problems which] remain unsolved, or have been solved only recently ... while showing the necessity of abstract concepts and how they enter progressively into the solution. These are conceptual notions, each built 'above' the preceding and permitting an increase in abstraction, represented metaphorically by Jacob's ladder with its rungs."
Geometry Revealed makes no attempt to be comprehensive, but covers a broad range of material. Its twelve chapters group ideas about points and lines in the plane, circles and spheres, conics and quadrics, plane curves, smooth surfaces, convex sets, polygons and polytopes, lattices and packings in the plane and in higher dimensions, and the dynamics of billiards and geodesic flows. The problems centred are various. Sylvester's conjecture: Given a finite set of points in the plane, not collinear, then there exists at least one line that contains only two of them. Borsuk's conjecture: In d-dimensional Euclidean space, can we decompose any bounded part E into d+1 parts of diameter strictly less than the diameter of E? Can we evenly distribute points on the surface of a sphere? The isoperimetric inequality: Among all plane curves of a given length, the circle is the one which encompasses the largest area possible. If we slice a cube by a plane, what's the largest cross-section we can get? And so forth.
To give an idea of the style and approach, a fairly random excerpt:
"A simple remark: a complete minimal surface of E³, without boundary, has negative or zero curvature and thus is never compact. In fact, we saw in Sect. VI.5 that every compact surface necessarily has points of positive curvature. Now comes a celebrated example, that of the one-parameter family of minimal surfaces that joins the two most celebrated (complete) minimal surfaces, the helicoid (which is ruled) and the catenoid (which is a surface of revolution).
This family of surfaces has the remarkable property that each of its members is minimal and isometric to all other surfaces of the family. Of course, this for the intrinsic metrics of these surfaces and not the restriction of isometries of E³. Thus in particular they all have the same principal curvatures k₁ and k₂ since k₁ = -k₂ and K = k₁k₂ is invariant under the deformation (although variable on each surface). Thus we find surfaces having the same principal curvatures, but different in E³. This doesn't contradict the fact that the two fundamental forms determine the surface within an isometry of the space; what happens here is that the principal directions 'turn' during the deformation."
Some proofs are included, but in outline: "only the crucial ideas and above all the abstract concepts introduced for attaining those ideas are elucidated".
"Numerous tilings are known for pentagons, but to this day they have not been classified. It seems to us that this genre of problem is one of curiosities, i.e. it is contrary to the philosophy of the present book, which is: citing open problems, but only when their solution would seem to require an ascent of the ladder and not just some work by hand, as difficult as that might be."
and
"So, at least to this day, the positive solution of the four color theorem has not contributed any new concept and has not renewed the mathematical landscape; the proof is computerized and formal. ... it is much the same with the proof of Kepler's conjecture. ... This opinion is shared by numerous mathematicians, but not by all."
Some more technical material, with additional definitions and formalism, is consigned to "XYZ" sections at the end of each chapter. Each chapter also has its own bibliography, though the volume has a unified index. The text is illustrated with numerous, small, hand-drawn diagrams, which often demand thought but are invariably insightful and helpful in making the ideas clearer.
The mathematics is placed in its context. Berger usually touches only briefly on the history:
"We find the notion of convexity with Archimedes (250 B.C.), with Newton (1720) with respect to the classification of singularities of algebraic curves, with Poinsot (toward 1800) regarding statics and the notion of a sustentation polygon, with Fourier where we find the seeds of linear programming. Then comes Minkowski ... apart from specific results, he had an entire program, but he died very young."
There is more detail in rapid surveys of recent and current research, looking at key figures and at other books and papers of relevance and offering pedagogical hints. The only treatment of fractals, for example, is a single page on Hausdorff dimension, but there is the useful aside: "For fractals see the classic references: among others, after the historic (Mandelbrot, 1977-1982), the three books Feder (1988), Falconer (1990) and Cromwell (1997)."
There are also occasional glances at practical applications, looking at the workings of calipers and micrometers, dimples on golf balls (a Slazenger ball has twelve with only five neighbours, the rest have six), methods for manufacturing spherical balls, and so forth. A few scattered half-tones mostly illustrate such aspects of the history of technology.
This is not then a textbook, such as Berger's own Geometry I and Geometry II, but nor is it what Berger calls a "cultural text", since it remains focused on the mathematics. It has some similarities to Aigner and Ziegler's Proofs From the Book, but is more concerned with concepts and abstractions than with proofs (and it is four times as long and focused on just one field, so it climbs rather higher up the ladder).
A solid familiarity with mathematics is assumed, at the level of at least a good undergraduate degree, and there are places where more would be helpful. I can imagine Geometry Revealed being useful for research mathematicians as a still reasonably up-to-date survey. There was much that I only half-grasped, especially towards the higher rungs, but even that was quite satisfying.
I did the equivalent of a four year degree in pure mathematics, but barely touched on geometry. Euclidean geometry as taught at school had degenerated into a method for teaching proof formalism, some after-school training for mathematics competitions emphasized spatial intuition rather than ideas and abstractions, and the one differential geometry course in my degree was pitched at too high a level. So for me Geometry Revealed offered an ascent into the wonders of a new world.
July 2015
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