This is a unique, unusual, and important book on the conceptual ideas underlying modern geometry, written by Marcel Berger, one of the most influential French geometers in the second half of the twentieth century. He passed away less than a year ago, on October 15, 2016, at the age of 89.

In *Geometry Revealed* Berger celebrates “the unceasingly renewed vigor of the spirit of geometry” that underlies much of contemporary mathematical research. More broadly, he presents readers with the elements of a “modern geometric culture.” This culture includes the ubiquitous use of geometrical metaphors within many seemingly non-geometric subjects, allowing “pictures” to be formed and geometric intuition to be applied to non-geometric questions. In this way geometric thinking, through its flexibility and adaptability, becomes a universal tool in all fields of mathematics.

Moreover, Berger wishes to illustrate how mathematical topics can gain new life and new direction via contact with geometric thinking and techniques. Examples of this influence are found in the recent advances in geometric probability theory, Thurston’s geometrization conjecture in topology, and those aspects of functional analysis which have morphed into noncommutative geometry.

The results analyzed in *Geometry Revealed* are not usually given complete proofs in the book. “Only the crucial ideas and above all the abstract concepts introduced for attaining these results are elucidated.” The author’s more standard two-volume *Geometry* is a helpful reference for many of these omitted proofs as well as for more formal presentations of topics discussed in *Geometry Revealed*. Conversely, *Geometry Revealed* provides extensions of much of the material of *Geometry*, in certain cases “illuminating it with some very recent results.”

Each chapter of the book examines a series of problems, all quite visual and presentable in a simple manner, and all sharing at least one of the following properties:

- each problem has been solved only recently, or for some problems remains unsolved;
- each problem, to be well understood and/or eventually solved, requires the creation of new concepts and tools whose degree of abstraction exceeds what is required for just stating the problem; or
- the tools used to solve each problem were originally created for very different purposes.

To understand and solve the problems requires introducing further abstract concepts, each built upon previous concepts and increasing levels of abstraction. Each new concept is the rung of a ladder ascending higher and higher in abstraction, the *Jacob’s ladder* of the book’s subtitle.

Chapters can be read independently. More to the point, the book is not intended to be read from cover to cover: readers are encouraged to jump into random chapters depending on their “intuition and inclination.” This is not to say that the chapters ignore each other. To the contrary, there are numerous references, both back to earlier chapters and forward to later chapters,. Berger hopes his book will be seen as a modern sequel to the classic *Geometry and the Imagination* by Hilbert and Cohn-Vossen, serving as a contemporary and accessible introduction to “cultural geometry.”

*Geometry Revealed* is divided into twelve chapters, the titles of which show Berger’s intent to start each discussion with problems that are visual and simple to present.

- Points and lines in the plane.
- Circles and spheres.
- The sphere by itself: can we distribute points on it evenly?
- Conics and quadrics
- Plane curves
- Smooth surfaces
- Convexity and convex sets
- Polygons, polyhedra, polytopes
- Lattices, packings and tilings in the plane
- Lattices and packings in higher dimensions
- Geometry and dynamics I: billiards
- Geometry and dynamics II: geodesic flow on a surface

Consider how these chapters relate to Felix Klein’s 1872 *Erlanger Programm*, a philosophy summarized compactly in § IV.5 as “a geometry is characterized by its group of automorphisms.” The first four chapters consider topics that, acknowledged or not, essentially fall within the borders of the *Erlanger Programm*. However, at the end of § IV.5 Berger observes that Klein’s theory doesn’t include all the interesting geometric systems (e.g., systems where the group of automorphisms is not *transitive* on the space), and that the major part of *Geometry Revealed* studies topics which fall outside the *Erlanger Programm*. Most of Chapters V through XII are in this category.

Each chapter introduces a set of basic problems followed by conceptual/intuitive discussions which *climb the ladder* of increasing abstraction in reformulating, proving, and generalizing the initial problems or considering related topics. The scope and detail of the exposition (not in proof details but in conceptual analysis and description of intuitive mathematical thinking) is impressive. We now provide a specific and illustrative example of “climbing Jacob’s ladder” from the beginning of the first chapter.

Chapter I concerns points and lines in the plane, both affine and projective. The discussion begins by considering “an utterly simple problem of Sylvester about the collinearity of points”:

- [(I.1.2)] If \(E\) is a finite set of points in the plane not composed of points belonging to a single line, then there exists at least one line that contains only two points of \(E\).

Sylvester posed this conjecture in 1893 and it is not hard to convince oneself of its truth by trying to construct a finite set \(E\) of non-collinear points in the plane such that no line contains only two points of \(E\). Any such effort appears to require an unending addition of points to \(E\), strongly suggesting that the set must be infinite.

However, it took almost forty years for a proof to be devised. As related by Berger, in 1932 Tibor Gallai proved the conjecture for the *real affine plane*. His proof depended heavily on what is sometimes called the *Plane Separation Axiom*: any line in the real affine plane divides the plane into two distinct convex (hence connected) regions such that the line segment joining a point in the first region to a point in the second must intersect the line. Another proof was given in 1948 by Leroy Kelly, this one short enough to be summarized in Berger’s Section I.1. However, this beautiful proof uses Euclidean distance to prove a purely affine result, and hence is not fully satisfying if you desire to employ only affine concepts in your verification.

Neither of the proofs just mentioned required new concepts and hence did not rise higher on Jacob’s ladder. However, Berger describes several ways in which it is natural to climb the ladder when starting from (I.1.2). It should not be surprising that one method involves moving from the real affine plane to the real *projective* plane. Such an ascent from (I.1.2) was published by Eberhard Melchior in 1940. The first step is to use projective duality: in the real projective plane the truth of (I.1.2) is equivalent to the truth of the following dual statement:

- [I.1.2*] If \(D\) is a finite set of lines in the real projective plane not all coincident at one point, then there exists at least one point that is incident to only two lines of \(D\).

Berger summarizes Melchior’s proof of this (a proof which incidentally uses the fact that the Euler-Poincaré characteristic of the real projective plane equals 1). Climbing Jacob’s ladder from affine geometry to projective geometry is of course a standard technique, but this is a lovely example of its usefulness.

A different ascent up Jacob’s ladder from (I.1.2) is to consider the *complex* affine plane. Using complex plane *cubics*, i.e., algebraic curves defined by a polynomial of degree 3, we can show (I.1.2) to be *false* in the complex plane. A generic cubic in the complex plane has nine distinct inflection points, and the line determined by any two of these points must contain yet a third inflection point. Hence the set \(E\) consisting of the nine points of inflection proves (I.1.2) to be false in the complex plane. In Section I.8 Berger gives a specific set \(E\) of nine points in \(\mathbb{CP}^2\) with the desired properties.

However, (I.1.2) can be generalized in such a way that the altered claim is indeed true in complex affine geometry, but this requires moving to complex dimension two. In 1966 Jean-Pierre Serre conjectured the following generalization of (I.1.2) for complex affine spaces of dimension at least 2:

- [I.1.2**] Suppose \(E\) is a finite set of points in a complex affine space of arbitrary dimension. If \(E\) is not composed of points belonging to a single complex plane, then there exists at least one complex line that contains only two points of \(E\).

Serre’s conjecture is stated in Berger’s § I.8 along with a long discussion of the first proof of the conjecture given by Leroy Kelly in 1986. The discussion is long because the proof is extremely difficult, depending on several deep results in algebraic geometry. Berger’s path from (I.1.2) had to ascended many rungs of Jacob’s ladder to arrive at this proof of (I.1.2**). (In 2006 Noam Elkies, Lou Pretorius, and Konrad Swanepoel published a new proof that is more elementary than Kelly’s, but it is still considerably higher up the ladder than the starting point of (I.1.2).)

*Geometry Revealed* is not structured like a standard textbook in that it does not intend to methodically cover the most conventional aspects of the topics under study. Selections are made instead to faithfully illustrate the Jacob’s ladder metaphor of rising to increasing heights of abstraction. This flexible structure is well-suited to an informal tone, resulting in a book that seems like the transcription of a (very long) conversation with a brilliant geometer who is eager to tell you about the mathematical delights and insights he’s collected over his life.

Adding to the conversational tone, Berger is not shy about stating his opinions, often with biting candor and sharp wit. In a passage from § II.2 concerning the number of spheres tangent to all four faces of a tetrahedron in space (the answer, 5 to 8 depending on the tetrahedron, is nicely derived by the use of barycentric coordinates), Berger introduces a historical reference as follows: “For a study, repulsive because of its quixotic refusal to use barycentric coordinates, see…” Berger often mentions what he believes is an unhealthy “attachment to the romance of ‘pure’, i.e., coordinate-free, geometry”. One example in § IV.5 refers to the reluctance of some to use the unifying power of Felix Klein’s group based *Erlanger Programm* formulation of geometry: “The refusal to see that we have an explanation that is systematic and quick and that it is absolutely necessary to make use of it, and to want at all costs to do things by pure geometry, leads to the sort of aberrations already mentioned….” As a final example, when discussing Gauss’s *theorema egregium* Berger points out that to define the total curvature of a surface given as the graph of a function \(f\) it suffices that \(f\) be of class \(C^2\), whereas for computing the Gaussian curvature it appears as though a third derivative of \(f\) is required. As Berger comments, “[t]here is a mystery here, invisible somehow to less reflective people who work in \(C^\infty\) without moral concern.” (The requirement of a third derivative for \(f\) was in fact removed by van Kampen in 1938.)

Most of the material described above in the example of “climbing Jacob’s ladder” is covered in the first five pages of *Geometry Revealed*, with only a few of the extensions located further along in Chapter I. Even though most proofs are omitted or only partially sketched, the level of sophistication of the discussion is quite high. In fact, it might be argued that the absence of most proofs makes the discussions more abstract and in instances more difficult to understand in a detailed manner unless the reader already has prior knowledge of the material. This is an unavoidable feature of the intended structure of the book.

There are also unintended reasons, however, that make *Geometry Revealed* a challenge. The informal, intuitive descriptions given in place of most proofs are usually excellent, but occasionally I found some mystifying, especially when they concerned topics with which I was less familiar. Some references, both those to external sources and those internal to the book itself, are not precisely specified or are specified by confusing instructions. This also applies to the illustrations in the book: they are not always well integrated with the written text. Some are reprints from other books, occasionally with irrelevant labels that don’t match items in *Geometry Revealed*. Some are not referenced directly in the book and are placed in locations that don’t make their purpose clear. However, these small complaints pale into insignificance against the mathematical power of this book.

Could *Geometry Revealed* be used as a course textbook? It would be a challenge to use *Geometry Revealed* as the primary text for a first undergraduate geometry course. The book makes no attempt to systematically cover the conventional results of the subjects it considers, which is fully consistent with its stated purpose. This limits its effectiveness as a course text, at least if the intent is to cover a conventional curriculum. The lack of formal exercises could also be a stumbling block.

The primary obstacle to *Geometry Revealed* as a course textbook, however, is that the discussions in the book would be difficult for a reader who does not already possess knowledge beyond Euclidean geometry, in particular familiarity with other geometric systems such as real and complex projective geometries. Such prerequisites do not fit the students in a first undergraduate geometry course. More plausible would be an advanced undergraduate course or a graduate course. An independent reading course for talented and advanced students would also be a plausible way to utilize *Geometry Revealed* as a textbook. Given the massive amount of information to choose from, using this book for any type of a course would require careful preparation on the part of the instructor.

Berger claims in his introduction that a course in geometry “as seen from the cultural aspect” could be constructed using *Geometry Revealed* as the primary text, with the author’s earlier two-volume text *Geometry* as a natural reference for the details. In theory this is possible, but for the reasons given above I believe it would require unusually gifted and advanced students.

While *Geometry Revealed* might be a nonconventional choice for a course text, it is a *must own* book for anyone serious about developing a conceptual understanding of the interconnected web of modern geometry and the ever-growing intertwining of geometry with practically all other branches of mathematics. I was overwhelmed by *Geometry Revealed*. Berger’s deep understanding of geometry in all its forms is evident in every page of this 831 page tome. I gained many insights from reading this book.

Of greater importance, *Geometry Revealed* is an immersion in current topics of mathematical research, taking the reader out to the edge of current knowledge. It is remarkable for a book to provide such a detailed glimpse of contemporary geometry via well developed discussions of so many questions of current interest. It provides the most extensive exposition of geometric thinking I’ve ever seen in a book at this level.

**Chapter by Chapter**

I can only give an incomplete sampling of the contents of *Geometry Revealed*, both because of the over 800 page length of the book and the amount of information given in each chapter. What follows are topics that I found particularly interesting or noteworthy; other readers will no doubt have their own list of preferences. My coverage is more cursory for the first five chapters since I felt this was somewhat more conventional material. The notes are more detailed for the later chapters.

**Chapter I**, as indicated in earlier in this review, is concerned with aspects of *affine* and *projective* spaces especially in two dimensions. The discussion in § 5 of the “irresistible necessity of projective geometry” provides a well-developed argument for the importance of a projective extension even when considering essentially affine questions. The discussion — and accompanying set of illustrations — on the topology of the real projective plane \(\mathbb{RP}^2\) in §7 is particularly informative.

**Chapter II**, set primarily in the Euclidean plane, considers topics involving circles. § 2 provides an interesting discussion of several circle configurations: the *5 circles*, *6 circles*, and *7 circles* theorems, *Feuerbach’s theorem*, *Steiner’s alternative*, and the *Poncelet theorem for circles*. The chapter also devotes significant time to *inversion in a circle*, which leads into the various models for the *hyperbolic plane*.

**Chapter III** considers the ordinary sphere \(S^2\). Berger emphasizes that the geometry of the sphere “harbors many pitfalls” and is more difficult that would be expected. Important aspects of *spherical geometry* are developed in § 1, followed in § 2 by the *Möbius group* consisting of the *conformal transformations* of \(S^2\). The centerpiece of the chapter is § 3, a 21 page discussion on attempting to uniformly distribute \(N\) points on the sphere, both why this is an important question in applications and the current state of our knowledge on solutions. (At the time *Geometry Revealed* was published optimal configurations were known only for \(2\leq N\leq 12\) and \(N=24\).) To emphasize the non-triviality of spherical geometry §5 consists of the presentation of four open problems for the sphere.

**Chapter IV** presents a powerful overview of *conics*, starting in the Euclidean plane and ending in the complex projective plane. There is also a relatively brief but still informative discussion of *proper quadrics*, the natural generalizations of proper conics to three dimensions. A major goal for the chapter is to explain why, in order to obtain “the good definition of a conic,” we are naturally and necessarily led to the use of the real projective plane (with its points at infinity) and the use of complex numbers (with their “imaginary aspect”). The development starts with classical material on conics, known (or obtainable by techniques known) prior to Descartes’s identification of the Euclidean plane with \(\mathbb{R}^2\). Then the sections climb the ladder: the introduction of coordinates to the Euclidean plane, the move into the real projective plane \(\mathbb{RP}^2\), and finally the ultimate generalization to conics in the complex projective plane \(\mathbb{CP}^2\). Along the way there are many lovely results including *Pascal’s theorem*, the *Poncelet theorem*, and “the 3264 conics of Chasles.” The chapter concludes with a discussion of proper quadrics in space. Chapters 13–17 in Berger’s two-volume *Geometry* is an excellent reference and companion for all this material.

**Chapter V** examines *plane curves*. It starts with the statements of three famous theorems: the *Jordan curve theorem*, the *turning tangent theorem*, and the *isoperimetric inequality*. These results often seem obvious to the “person in the street,” but a mathematician needs proofs and these are not trivial results. In fact the *turning tangent theorem* was not proven until 1935. Berger weaves an engrossing story using these three theorems as the scaffolding for his chapter. The discussion does not shy away from technical distinctions: *algebraic curves* versus *smooth curves*, or *geometric curves* (locally graphs of a \(C^1\) numerical function) versus *kinematic curves* (curves parametrized by a non-zero \(C^1\) function into \(\mathbb{R}^2\)). The chapter includes proofs or proof sketches for the original three theorems — in fact, there are four different approaches discussed in §12 for the *isoperimetric inequality*.

**Chapter VI** considers *smooth surfaces*, both those in Euclidean space \(\mathbb{E}^3\) (as differentiable submanifolds of dimension 2) and those which are *abstract surfaces* (i.e., differentiable manifolds of dimension 2). The stated goal for the chapter is to “concentrate on what isn’t known about surfaces.” For a surface \(S\) in \(\mathbb{E}^3\) the *intrinsic* metric is defined for all \(p,q\in S\) by \[ d_S(p,q)=\inf\{\text{length}(c)\mid c \text{ a curve traced on \(S\) joining \(p\) to \(q\)}\}.\] For any surface \(S\) in \(\mathbb{E}^3\) the question considered in §2 is *what is the shortest path from one point to another* when distance is measured with the intrinsic metric? Hence, given \(p,q\in S\) we would hope to determine — or at least prove the existence of — a curve \(c\) joining \(p\) to \(q\) such that \(\text{length}(c)=d_S(p,q)\). If such a *shortest path* does exist, then there are properties that \(c\) must satisfy: these become the defining properties of the curves defined in §3 as *geodesics* of the surface \(S\). The concepts of distance, geodesics and shortest paths can be extended to those abstract surfaces which have the structure of connected *Riemannian surfaces*. As discussed in §5, any connected smooth surface \(S\) in \(\mathbb{E}^3\) automatically inherits the structure of an abstract Riemannian surface.

Gauss’s *theorema egregium* is discussed in §6. Suppose \(S\) is a surface in \(\mathbb{E}^3\). Then it is automatically a Riemannian surface and hence at each point \(p\) has a *Gaussian curvature* \(K=K(p)\), defined from the intrinsic structure of \(S\), independent of the placement of \(S\) in \(\mathbb{E}^3\). However, there is another concept of curvature at \(p\) defined via how \(S\) actually sits in \(\mathbb{E}^3\): the *total curvature* \(K(ext)\) of \(S\) at \(p\) given by \(K(ext)=k_1k_2\) where \(k_1, k_2\) are the *principal curvatures* of \(S\) at \(p\). The *theorema egregium* states that \(K(ext)=K\) which, in particular, shows that \(K(ext)\) is actually an *intrinsic* measure in spite of its original definition. Further properties of total curvature such as the *Gauss-Bonnet theorem* are discussed in §7.

Another portion of Chapter VI that should be highlighted is §9: *What we don’t entirely know how to do for surfaces.* This provides a collection of open problems for surfaces in \(\mathbb{E}^3\).

**Chapter VII** is concerned with the concepts of *convex sets* and *convex functions*, primarily in the context of real affine geometry. This is a subject which was, with a few notable exceptions, essentially ignored until the 1960s. As Berger observes, the subject contains many “results whose statements are elementary but whose proofs [are] very difficult, and problems whose statements are equally elementary, but which are still unresolved.”

In §§ 2 and 3 Berger introduces *convex sets* in a real affine space and numerical *convex functions* defined over a convex set, then discusses the basic elementary properties of these objects. Many examples of convex sets are given in §4, the primary ones being *solid ellipsoids* and *solid parallelepipeds*. (During the chapter Berger illustrates why the ellipsoids are “the most beautiful of convex sets” while the parallelepipeds are “the ‘worst’ among the convex sets.”) *Symmetrization* operations are defined on convex sets in §5 that take a convex set \(P\) and produces a new, *more symmetric* but still convex set \(P^*\). The *Minkowski sum* of two convex sets in the same affine space is also defined; this operation results in a third convex set.

An *isoperimetric inequality* is discussed in §7 for the Euclidean case: *Among all the convex sets of a given volume there is exactly one of least possible diameter, the ball.* Measures of the “degree of badness” for a convex set are considered in §10. One, introduced in § 10.E, is the *volumic quotient* of a center-symmetric convex set \(C\). This affine invariant is the ratio of volumes \(qv(C)=\text{Vol}(C)/\text{Vol}(E)\) where \(E\) is the ellipsoid of maximum volume contained in \(C\). There is an upper bound for \(qv(C)\), with equality occurring only for \(C\) a parallelepiped.

**Chapter VIII** considers *polytopes* of dimension \(d\geq2\) in an affine (or Euclidean) space. A *polytope* is the convex hull of a finite number of points with \(d\) defined as the dimension of this convex hull. (Polytopes are *polygons* when \(d=2\) and *polyhedra* when \(d=3\)). Like convexity, this subject was not of major interest until the 1960s. Polytopes appear simple but this is deceptively misleading — there are many results whose proofs are surprisingly difficult and open conjectures that still defy solution. Foundational definitions (\(k\)-*faces* and *flags*) and results (e.g., characterizing polytopes as intersections of a finite number of closed half-spaces) are developed in §§ 1 and 2. Then a list is given in §3 of problem types that consume the rest of the chapter. The results for \(d=2\) are given in §3, those for \(d=3\) in §§ 4 through 9, and those for \(d\geq4\) (such as they are) in §§6, 7, 10, 11, and 12.

*Combinatorics*: Relations between the numbers \(f_0, \ldots, f_{d-1}\) for a polytope \(P\) where each \(f_k\) equals the number of \(k\)-faces of \(P\) for \(k=0,\ldots,d-1\). The basic result is *Euler’s formula* \(f_0-f_1+f_2=2\) for \(d=3\), which can be generalized to cover all \(d\geq2\).
*Regularity*: Classification of the polytopes \(P\) whose isometry groups are transitive on the flags of \(P\). When \(d=3\) there are five regular Euclidean polyhedra, commonly known as the *Platonic solids*. When \(d=4\) there are six regular polytopes, and for \(d\geq5\) there are three regular polytopes. If the requirement of convexity is dropped there are other polytopes that exhibit regularity. One collection consists of the *regular star polytopes*. There are four when \(d=3\), ten when \(d=10\), but none in higher dimensions.
*Rigidity*: The *face lattice* \(\mathcal{L}(P)\) of a polytope \(P\) of dimension \(d\) is the set of all \(k\)-faces (\(0\leq k\leq d-1\)) of \(P\) partially ordered by set inclusion. Suppose polytopes \(P\) and \(Q\) have isomorphic face lattices such that any two corresponding faces in \(\mathcal{L}(P)\) and \(\mathcal{L}(Q)\) are congruent. If \(d\geq3\), then *Cauchy’s Rigidity Theorem* claims \(P\) and \(Q\) are congruent. Thus polytopes for dimensions \(d\geq3\) are *rigid*.
*Isoperimetry*: Two polytopes \(P\) and \(Q\) are of the same *combinatorial type* if they have isomorphic face lattices. In 1842 Steiner posed the question *Is it the case that, in their combinatorial type, the regular polyhedra are those having the best isoperimetric ratio* \(A^3/V^2\)? Here \(A\) is the area of the boundary and \(V\) is the enclosed volume. This question is still not entirely resolved, and generalizations to \(d\geq4\) are extremely sparse.
*Inscribability*: Given a combinatorial type for polyhedra, can it be realized by a polyhedron inscribed in a sphere? Alternately, can it be realized by a polyhedron circumscribed around a sphere? The answers to both questions are “usually not.” Necessary and sufficient conditions are given in §8 for circumscribability. Conditions for inscribability can be obtained by passing to polar (dual) polyhedra.
*Rationality*: In §9 it is observed that every combinatorial type for polyhedra can be realized by a polyhedron whose vertices belong to \(\mathbb{Z}^3\). While this result is trivial for any combinatorial type that has only triangular faces, proving the result when the faces have four or more vertices is far more difficult. However, the real surprise comes in §12 with the observation that a combinatorial type for 4-dimensional polytopes exists which cannot be realized by a polytope with vertices in \(\mathbb{Z}^4\).

**Chapter IX** considers geometric questions concerning lattices in the plane. The basic lattice \(\mathbb{Z}^2\) is considered in §1. In particular, a geometric approach to *continued fractions* is developed using \(\mathbb{Z}^2\) to obtain rational approximations to any real number. Continuing with \(\mathbb{Z}^2\), §2 considers results on counting points in \(C\cap\mathbb{Z}^2\) for various regions \(C\subset\mathbb{R}^2\): the *Pick formula*, *Ehrhart formula*, and *Gauss circle problem*.

In §3 more general lattices \(\Lambda\) in the Euclidean plane are discussed, the concept of \(\text{Area}(\Lambda)\) is defined, and the *Minkowski theorem* for *center-symmetric* regions \(C\subset\mathbb{R}^2\) is obtained: if \(\text{Area}(C)\geq 4\,\text{Area}(\Lambda)\), then \(C\cap(\Lambda\!\setminus\!0)\) is non-empty. A more detailed analysis of lattices in the Euclidean plane is initiated in §4. In particular, the set of lattice equivalence classes — up to direct similitudes — is classified and the geometry of the resulting collection is developed. This yields the *modular space*, *SpMod*.

Berger goes on to show that *SpMod* is the homogeneous space \(\text{SL}(2;\mathbb{Z})\backslash\mathcal{H}\) where \(\mathcal{H}\) is the Poincaré upper half-plane. Berger also initiates a study of the geometry of a single lattice \(\Lambda\) in \(\mathbb{R}^2\) starting with the *tiling* of the plane by the *Voronoi domains* associated to \(\Lambda\). An initial discussion of the *density* \(\text{Dens}(\Lambda)\) of the lattice is also begun. The density is defined from \(\text{NormMin}(\Lambda)\), the *minimal norm* of the lattice, the lower bound of the norm of non-zero vectors in \(\Lambda\).

In §5 the optimal packing of a region in \(\mathbb{R}^2\) by congruent circles is considered, but this time not restricted to packings governed by a lattice. This requires extending the definition of *density* to a packing of the entire plane, which is done with reasonable precision. Consideration of optimal packing continues in §6, this time with congruent shapes more general than circles, and perhaps even using several shapes for the packing. There are some interesting open questions mentioned, even concerning the optimum packing of a finite number of congruent squares.

A particular type of tiling of the plane is considered in §7, governed by a group of isometries of the Euclidean plane that contain two linearly independent translations. These result in the famous *wallpaper patterns*. There are only seventeen such patterns given the identification of natural equivalents. Other tilings related to the wallpaper patterns are also considered in this section. Tilings in higher dimensions are considered in the short §8; as Berger observes, “we don’t understand much in dimension 3, and … higher dimensions are still richer in unexpected phenomena.”

*Penrose tilings* are the topic of §9. Roger Penrose in 1974 produced two quadrilaterals (with carefully placed indentations along the boundaries) — the *kite* and the *dart*. Congruent copies of these shapes can tile the plane, but any such tiling has no translational symmetry (the boundary indentations prevent constructing such a tiling). Because of these properties the kite and the dart are said to form an *aperiodic* set. Berger describes many of the truly amazing properties of Penrose tilings.

**Chapter X** is concerned with lattices and packings in dimensions 3 and higher, and is therefore a natural sequel to Chapter IX. Given a lattice \(\Lambda\) in \(\mathbb{R}^3\), §1 introduces the definitions of the *minimal norm* and the *density* of the lattice: \(\text{NormMin}(\Lambda)\) and \(\text{Dens}(\Lambda)\), generalizing the two-dimensional concepts of § IX.4. Normalizing so that \(\text{NorMin}=1\), the main problem of §1 is to find the lattice which yields the largest density and hence provides the optimal lattice packing of \(\mathbb{R}^3\) with congruent spheres. As proven by Gauss in 1831, the densest such packing is achieved with \(\text{A}_3\), the *cubic face centered lattice*, giving a density of \(\pi/(3\sqrt{2})\). In §2 Berger considers if this result remains true if expanded to all not-necessarily-lattice-based packings. It was conjectured by Kepler in 1611 that the \(\text{A}_3\)-based packing does indeed produce the maximum density for a packing of congruent spheres, but this conjecture remained unproven for centuries. It was actually part of the eighteenth problem in David Hilbert’s famous 1900 list. Thomas Hales established its truth in 1999. (Berger also explains that an infinite number of related but non-lattice-based packings yield the minimum density).

In §4 Berger introduces examples and concepts for lattices \(\Lambda\) in \(\mathbb{R}^d\) with \(d\geq4\). As in dimensions 2 and 3, the *minimal norm* and the *density* of the lattice are defined and denoted as \(\text{NormMin}(\Lambda)\) and \(\Delta(\Lambda)\). There is a natural lattice \(\text{A}_d\) in \(\mathbb{R}^d\) based on the regular simplex, generalizing \(\text{A}_3\) from §1. However, there is now another related lattice, \(\text{D}_d\), such that the density of \(\text{D}_d\) is better than the density of \(\text{A}_d\) for all \(d\geq4\). Other lattices are also introduced: \(\text{E}_6\), \(\text{E}_7\), \(\text{E}_7\), and \(\Lambda_{24}\) (the *Leech lattice*, which has rather extraordinary properties).

The sphere packing problem for lattices in \(\mathbb{R}^d\) (\(d\geq4\)) is discussed in §5. The goal is to find the densest possible lattices. We begin with some important definitions:

- the
*central density*, \(\text{DensCentr}(\Lambda)=\Delta(\Lambda)/\beta(d)\) where \(\beta(d)\) is the volume of the unit ball in \(\mathbb{R}^d\),
- \(c(d)=\) the lower bound of \(\text{DensCentr}(\Lambda)\) for all lattices \(\Lambda\) in \(\mathbb{R}^d\), and
- the
*Hermite constant*, \(\gamma_d=4\,c(d)^{2/d}\).

Here is the basic question: *What are the values of \(\gamma_d\) for different dimensions \(d\), and above all what are the lattices that realize them if they exist?* The Hermite constants are known only for \(d\leq4\). Moreover, in these cases the densest lattices have been determined and shown to be unique. Considering all values \(d\geq2\) the following asymptotic bounds are known for the Hermite constants: \[ \frac{1}{2\pi e}\leq \liminf_{d\to\infty}\frac{\gamma_d}d\ \text{ and }\ \limsup_{d\to\infty}\frac{\gamma_d}d \leq \frac 1{\pi e}. \]

In §6 we consider packings of congruent \(d-1\) dimensional spheres in \(\mathbb{R}^d\) not necessarily associated to lattices. The *density* of such a packing in \(\mathbb{R}^d\) is the natural generalization of that given in § IX.5 for \(\mathbb{R}^2\). There is an upper bound \(\sigma_d\) known for the packing density for any \(d\geq2\) — the *Rogers’ bound* — though its sharpness deteriorates as \(d\) increases. Moreover, as of the publication of *Geometry Revealed*, there exist dimensions (\(d=10, 11, 13\)) where the densest **currently known** packings are *not* lattice packings. This provides some support for the following surprising conjecture: *Starting with a certain dimension there exist packings not of lattice type which are denser than any lattice packing.* The section ends with a discussion of various theories for analyzing finite packings in \(\mathbb{R}^d\).

§7 centers on a discussion of *error correcting codes* and their important connections to lattices in \(\mathbb{R}^d\) for large values of \(d\). In §8, given a lattice \(\Lambda\) in \(\mathbb{R}^d\) Berger defines the *dual lattice* \(\Lambda^*\) and then uses it to define particular types of lattices in \(\mathbb{R}^d\): \(\Lambda\) is *integral* if \(\Lambda\subseteq\Lambda^*\) and, in particular, *unimodular* if \(\Lambda=\Lambda^*\). Examples of unimodular matrices are \(\mathbb{Z}^d\), \(E_8\), and \(\Lambda_{24}\) (the Leech lattice). Then for any lattice \(\Lambda\) in \(\mathbb{R}^d\) Berger defines the *theta function* \(\Theta_\Lambda\colon\mathbb{C}\to\mathbb{C}\). Computations of the theta functions for several lattices are carried out, in some cases requiring results about *modular forms*.

**Chapter XI** considers billiards as a particular case of *dynamical systems* with applications to more general questions in mechanics. In §1 the problem of two particles of the same mass oscillating and colliding along the interval \([0,1]\) is reduced to studying the trajectories of one billiard on an isosceles right triangle, which is then reduced to a square billiard table. The solution of this problem is carried out in §2: for the square billiard table we see that the trajectory is *periodic* or *never closes* depending on if the slope of the trajectory is *rational* or *irrational* respectively. However, there are two additional questions that are natural and essential in dynamics that Berger answers for the square billiard table. The first is determining the nature of the *counting function* \(FC(L)\) via an asymptotic formula: if \(FC(L)\) equals the number of periodic trajectories of length \(\leq L\), then \(FC(L)\sim \frac 3{4\pi}L^2\) as \(L\to\infty\). In the non-periodic case Berger shows each such trajectory is *uniformly everywhere dense* in the square.

In §3 the problem of two particles oscillating and colliding along the interval \([0,1]\) is considered where the particles have different masses \(m\) and \(m'\). This is reduced to studying the trajectories of one billiard on a right triangle with side lengths \(\sqrt{m}\) and \(\sqrt{m'}\). However, it is not possible (except in very special cases) to further reduce this right triangle into a square as was done in §1 for an isosceles right triangle. Instead Berger considers the more general situation of a (convex) polygonal billiard table with \(n\geq3\) sides. A trajectory in a polygon \(P\) is composed of a series of (oriented) line segments, each specified by a direction, i.e., a non-zero vector. Sketching trajectories in a square quickly shows that each trajectory has a maximum of four (oriented) directions. Similarly a trajectory in an isosceles right triangle has at most eight directions. The general result is the following. Define a polygon \(P\) to be *rational* if all its \(n\) vertex angles are rational multiples of \(\pi\). Then *the billiard trajectories of a polygon \(P\) all have a finite number of directions if and only if the polygon is rational.* This is an elementary but important result.

Analysis of the possible trajectories in a polygon \(P\) are carried out in §4. In the 1970s, using *interval exchanges* (which are among the simplest dynamical systems considered in ergodic theory), each billiard trajectory of a rational polygon was shown to have three possible behaviors: either (1) it *darkens everywhere*, (2) it is *periodic*, or (3) it *partially darkens* the polygon. Cases (2) and (3) are countable, and hence occur with probability zero, so case (1) occurs with probability one. (A trajectory *darkens* a region essentially means that points on the trajectory are *everywhere dense* in the region.) The use of dynamical systems raised the analysis up Jacob’s ladder. On the other hand, “[i]nterval exchanges do not permit us to answer several questions about rational polygon billiards that have been left hanging: the exact density of the trajectories, [and] the counting function of periodic trajectories.”

In §5 Berger describes the method for addressing these gaps. The approach, developed in the 1980s, is a major jump up the ladder. It starts with a rational polygon \(P\), constructs an associated *abstract surface* \(\mathcal{S}(P)\), and then associates with \(\mathcal{S}(P)\) a *Riemann surface* \(\mathcal{R}(P)\). Using this structure allows a refinement of the trichotomy of §4: each billiard trajectory of a rational polygon can be shown to have one of the following behaviors: either (1a) it *darkens everywhere* **with equal density**, (1b) it *darkens everywhere* but with **nonuniform density**, (2) it is *periodic*, or (3) it *partially darkens* the polygon. Moreover, in 1990 Howard Masur determined the nature of growth for the counting function \(FC(L)\) of periodic trajectories of each rational polygon: there exist constants \(a\) and \(b\) such that \(aL^2 < FC(L) < bL^2\) for all \(L\).

The case of irrational polygons is considered in §6. We have one major positive result: *“Almost all” polygons are phase-ergodic for the dynamical system that it defines in its phase space.* (This statement uses the language of dynamical systems. Moreover, “almost all” is in a topological sense since there is no good measure on the space of all polygons.) However, “at present it is not known how to exhibit explicitly even a single polygon that can be proved to be phase-ergodic.” Considering periodic trajectories for an irrational polygon, “it is not known how to show (even though everyone believes it) that in each polygon there exists at least one periodic trajectory.” As to the counting function \(FC(L)\), “such a function presently makes no sense.”

§7 returns to the case of two masses \(m\) and \(m'\) on the interior of an interval first raised in §3, applying the results obtained for a polygonal billiards table to the specific case of an isosceles right triangle and translating back to the particles in an interval.

§8 considers results for a concave billiards table and §9 considers results for circular or elliptical tables. Various types of convex billiards tables are considered in §10. In particular, given a smooth boundary, it is very easy to show that such a table has plenty of periodic trajectories. Finally, billiards in higher dimensions are considered in §11. As Berger states at the start: “Billiards in Euclidean spaces of dimension three or more are essentially in an infantile state. In brief: practically nothing is known in general. … The only well-understood case is that of hyperbolic billiards. … The polyhedral case (polytopes more generally) is a complete mystery. … [T]oday no one has the slightest idea of what happens in a regular tetrahedron, the fundamentally simplest polyhedron that there is.”

**Chapter XII** considers dynamical questions about geodesics on surfaces, primarily oriented compact Riemannian manifolds of dimension two. If there exists a shortest path between two points \(p\) and \(q\) on such a surface \(S\), we know from § VI.3 that this curve is a geodesic. However, a geodesic joining \(p\) and \(q\) in \(S\) need not be a shortest path. In this chapter geodesics are studied simply as trajectories in \(S\), and the overarching question is: what do they become when we follow them indefinitely? In §2 the basic dichotomy is established: a geodesic is either *periodic* — at the end of a certain time it returns to the same point with the same velocity vector and then continues to repeat along the same curve — or *non-periodic*. The questions Berger asks about periodic geodesics are: do any/many exist? Can we count those with a given length? Is the set of them *evenly distributed* on the surface? Do the set of them exhibit *ergodicity in phase*, i.e., *darken* the space of the set of tangent directions at different points of the surface?

To illustrate the difficulties associated with these questions about periodic geodesics Berger introduces several example surfaces in §3:

*Surfaces of revolution*: As Berger observes, “they are practically the only ones where calculation is easy.” On a compact surface of revolution there are an infinite number of periodic geodesics since at least every meridian is such a curve. Existence of many others is indicated for the “general” case.
*Spheres and Zoll surfaces*: The usual sphere \(S^2\) is a compact surface of revolution whose set of geodesics equals the set of *great circles*. In particular, *all* the geodesics of \(S^2\) are periodic. Berger describes other compact surfaces of revolution with the same property: the *Zoll surfaces*. In addition to having only periodic geodesics, the geodesics are all of the same length and the surface is topologically homeomorphic to the usual sphere \(S^2\). All Zoll surfaces have been classified. However, if we drop the requirement of being a surface *of revolution*, then not all members of this larger collection of *completely periodic surfaces* are known.
*Weinstein surfaces of revolution*: The unit sphere \(S^2\) is a surface of revolution with constant curvature 1 and an equator of length \(2\pi\). In fact, we can find, for any \(0 < D < 1\), compact surfaces of revolution \(S\), roughly shaped like circular ellipsoids, with constant curvature 1 everywhere except at small “smoothing caps” at the top and bottom, and with an equator of length \(2\pi D\). When \(D\) is chosen to be irrational, it can then be shown that any geodesic leaving from the central portion of \(S\) (i.e., not at either “cap”) with a tangent vector sufficiently close to horizontal cannot be periodic. The conclusion: a Weinstein surface does not exhibit *ergodicity in phase*.
*Ellipsoids with three unequal axes.* Such an ellipsoid \(S\) is not a surface of revolution, so we should expect a more complicated geodesic structure. There are three *basic* geodesics, those which are the intersections of the ellipsoid with the three planes of reflective symmetry for \(S\). However, these ellipsoids have the following curious property, established by Morse in 1934: *For any number \(L\) (think very large) there exists an \(\varepsilon>0\) such that any ellipsoid with unequal axes whose axis lengths are strictly between 1 and \(1+\varepsilon\) has all of its periodic geodesics of length larger than \(L\) except for the three basic geodesics.* (This is an alternate statement of (XII.3.D.1). The original statement is missing the necessary normalizing bounds on the axis lengths. The substituted result is from another Berger book, *A Panoramic View of Riemannian Geometry*, Theorem 201.)
*The flat torus*: This abstract surface is obtained by starting with a square and (i) identifying corresponding points on the left and right sides and (ii) identifying corresponding points on the top and bottom sides. It is made “flat” by having the same local geometry as Euclidean space, i.e., without curvature. There are precisely two types of geodesics on the flat torus: *periodic geodesics* from lines with rational slopes, and *everywhere dense geodesics* from lines with irrational slopes. Moreover, there isn’t any ergodicity in phase space.

Questions concerning the existence of a periodic geodesic in various general types of oriented compact surfaces are considered in §4. Such surfaces are topologically classified by *genus* \(g\geq0\), intuitively the number of “handles” or “holes”. For \(g\geq1\) the essential idea is to surround any hole with a closed curve, then use continuous deformation to proceed to the shortest such curve. This must be a periodic geodesic, as desired. The more difficult case is when \(g=0\), i.e., when there is no hole to encircle with a curve. Birkhoff established the existence of a periodic geodesic for this case in 1917; Berger carefully outlines Birkhoff’s procedure. It is of interest to note that Birkhoff’s method does not always yield a *simple* periodic geodesic, i.e., a geodesic without self-intersections. The existence of a simple periodic geodesic for any smooth oriented compact surface was not fully established until 1978. Differentiability is an essential assumption: as an example, an argument is sketched indicating that a *generic tetrahedron* will rarely have a simple periodic geodesic. (On the other hand, a *regular tetrahedron* has infinitely many simple periodic geodesics.}

In §5 Berger considers questions on the number of periodic geodesics on compact, oriented surfaces. Two geodesics are *geometrically equivalent* if their supports (or *traces*) are identical subsets in the surface. Berger wishes to consider geodesics that are *geometrically distinct*, i.e., effectively considering equivalence classes of geodesics under geometric equivalence. Berger’s discussion is organized by the genus \(g\) of the surface.

*Tori, \(g=1\)*: Berger is considering a torus to be any compact, oriented surface with genus \(g=1\). He argues that for every geometric structure on a torus (flat or not), there exist “at least as many distinct periodic geodesics as there are relatively prime pairs of positive integers.” He then defines the *counting function* for the periodic geodesics of a surface: \(FC(L)=\) number of geometrically distinct periodic geodesics for which the length is less than \(L\). (Berger notes that in the special cases of surfaces of revolution, flat tori, and ellipsoids, “the periodic geodesics come in continuous bands that are infinite in number and thus the counting function has no sense for us.”) Berger claims that the counting function of a non-flat torus grows in at least quadratic fashion: \(FC(L)\geq a\,L^2\) where \(a\) is a constant. “It is a sad fact that at the present time the order of magnitude of the counting function isn’t known for *any* surface of the type of the torus or the sphere.”
*Surfaces of higher genus, \(g\geq2\)*: Berger’s discussion is high up Jacob’s ladder. He argues that on any surface of genus \(g\geq2\) we can place a Riemannian structure with constant Gaussian curvature equal to \(-1\). For such surfaces it has been shown that \[ \liminf_{L\to\infty} (\log(FC(L))/L)=1.\] In particular, the counting function is of exponential growth. However, any other Riemannian metric on the surface is “proportional” to a metric with curvature \(-1\) (from the *fundamental theorem of conformal representation*), which leads to the “magnificent result” by Katok in 1988: *For any metric of the same volume as for those of curvature \(-1\), the counting function satisfies* \[ \liminf_{L\to\infty} (\log(FC(L))/L)\geq1.\] Moreover, equality occurs only if the metric has constant curvature. Hence the slightest modification of a metric of curvature \(-1\) results in more periodic geodesics than in the case of constant curvature! Berger also argues that in general the periodic geodesics are not dense in phase. He also states a very simple open question (which applies to surfaces of all genus values): is the union of the supports of the periodic geodesics an everywhere dense subset of the surface? “To our knowledge and for all surface types no expert has the least idea whether the answer is yes or no.”
*Spheres, \(g=0\)*: Berger discusses (via a disguised version of *Morse theory* worded in terms of mountain passes) the *theorem of the three geodesics* (a result of Lusternik and Schnirelman announced in 1927): *Every oriented, compact Riemannian manifold of genus 0 has at least three simple periodic geodesics.* However, over the next several decades this result was extended with great difficulty by the work of several people, culminating in the following results by the early 1990s: *Every oriented, compact surface of genus 0 admits an infinite number of periodic geodesics. Moreover,* \[\text{FC(L)}\geq a\frac L{\log L} \text{ where \(a\) is a constant.}\]

Hence, combining the three cases discussed above, “we know that essentially *every* surface — in space or abstract — admits an infinite number of periodic geodesics.” However, as Berger emphasizes at the end of §5, “the exact order of magnitude of \(FC(L)\) is not known for any surface of toroidal or spherical type. It seems that it isn’t even possible at present to conjecture whether, for example, the growth is always at least quadratic or greater; or again to relate this growth to the geometry of the surface.”

The remaining ten pages of the chapter considers questions related to ergodicity and entropy as well as closing comments on open questions.

William Barker is the Isaac Henry Wing Professor of Mathematics at Bowdoin College. He received his Ph.D. at M.I.T. in 1973, writing a thesis under the guidance of Sigurdur Helgason in analysis on Lie groups. His most recent work has been an undergraduate geometry textbook, Continuous Symmetry: From Euclid to Klein, co-authored with Roger Howe of Yale University. A second volume is currently under development. He can be reached at barker@bowdoin.edu.