During the last quarter century, the ancient notion of a material as a graph-like structure with atoms or molecular building blocks as vertices and bonds or ligands as edges has been explored using tools developed by topologists. Crystals are among the most readily described materials, at least at the atomic or molecular level, and this book is a focused introduction to the problem of generating geometric realizations of graphs representing crystals.

This is a departure from the classical approach to mathematical or geometric crystallography, in which a crystal was represented as an infinite array of points (each representing an atom or molecule) in d-dimensional Euclidean space. In the classical view, the occupied points would make up finitely many orbits of a group of isometries on that space. The only restrictions were that for that any orbit of points of that group, the orbit had no accumulation points,and that for any hyperplane in that space, there would be points on both sides.

Such a group of isometries was called crystallographic, and the “Fundamental Theorem of Mathematical Crystallography” of Arthur Schönflies, Evgraf Federov, Ludwig Bieberbach, and Hans Zassenhaus states that a group is isomorphic to a crystallographic group if and only if it has a maximal abelian subgroup which is free and of finite index (see "Crystallography and Cohomology of Groups", The American Mathematical Monthly 93 (1986) pp. 765-779.) Further, for each \( d\), there are finitely many isomorphism classes of crystallographic groups in \(d\)-dimensional space. This is the approach taken in Schwarzenberger's *N-Dimensional Crystallography* and Engel's *Geometric Crystallography: An Axiomatic Introduction to Crystallography.*

The development of X-ray diffraction about a century ago led ultimately to another notion of what a “crystal” might be. For most materials, an X-ray of that material produces a blur, but for crystals, the result is a symmetric array of dots. This array can be predicted using Fourier analysis, and it turned out that there were materials that were not crystals - in the sense that they did not have crystallographic symmetry groups - and yet their X-ray diffraction patterns were also symmetric arrays of dots. These “quasicrystals” have been explored in Senechal's *Quasicrystals and Geometry* and *Aperiodic Order: Volume I* by Baake and Grimm.

Returning to graph-like representations of crystals, while some crystals (e.g. salt) consist of atoms held together by electrostatic charges, others (e.g. sugar) consist of atoms or molecules linked together by bonds or ligands, respectively. Such covalent crystals may be represented by graphs, with atoms or molecular building blocks represented by vertices and bonds or ligands represented by edges. The automorphism group of such a graph \( \Gamma \) might not be crystallographic (!), but it will have a crystallographic subgroup \( G \), which in turn will have a maximal abelian subgroup \( L \) that is free and of finite index in the crystallographic subgroup. Then the quotient graph \( \Gamma / L \) will be finite. A graph \( \Gamma \) with a free abelian group \( L \) of automorphisms such that \( \Gamma / L \) is finite is called periodic. Sunada calls periodic graphs *topological crystals*, although crystallographers call these *crystal nets* (See O’Keeffe & Hyde's *Crystal Structures I: Patterns and Symmetry*.)

But one can go the reverse direction: given a finite graph \( X \), there exist graphs \( \Gamma \) that can be mapped onto \( X \) via graph homomorphisms \( \omega \) which, for each vertex of \( \Gamma \), maps incident edges of each vertex \( x \) of \( \Gamma \) bijectively onto incident edges of \( \omega(x) \). We can say that \( \Gamma \) covers \( X \), and the set of automorphisms \( \sigma \) such that \( \omega \sigma = \omega \) form a cover transformation group \( G \). It follows that \( X \) is isomorphic to \( \Gamma/G \); for more and similar adaptations of such topological machinery; Gross and Tucker's *Topological Graph Theory* addresses the issue via voltage graphs, which are not addressed in this text. Sunada is primarily concerned with free abelian cover transformation groups as he is concerned primarily with periodic graphs.

A crystal is a physical object, so a geometric representation is desirable. Given a periodic graph \( \Gamma \), a *realization* of \( \Gamma \) is the image of a map \( ρ \) from \( \Gamma \) into \( d \)-dimensional Euclidean space \( R^{d} \). Certain realizations seem more likely than others, and Sunada focuses on three criteria for realizations. First of all, a realization is *periodic* if the realization sends the free abelian group \( L \) to a translation group \( \rho(L) \) so that the translations are all symmetries of the realized graph. Second, a periodic realization is *harmonic* if each the image \( \rho(x) \) of each vertex \( x \) is the center of mass of its neighboring points: formally, if \( N_{x} \) is the neighborhood of \( x \), then \( \rho(x)=\frac{1}{|N_{x}|} \Sigma_{y \in N_{x}} \rho(y) \). Third, picking up a notion from frame theory, if \( E \) is the multiset of vectors corresponding to realizations of the edges of the graph \( \Gamma/G \), a harmonic realization is *normalized standard* if, for every \( x \in R^{d} \), \( \Sigma_{v \in E} \langle v,x \rangle v= 2x \). It turns out that any two normalized standard realizations of the same periodic graph are congruent.

Naturally, the next step is to determine the normalized standard realizations as geometric objects, modulo isometry. Treating a basis of the first homology group \( H_{1} \) of the periodic graph as if they were vectors in \( R^{d} \), Sunada derives the edge vectors of the realization, from which the entire structure may be derived.

This construction is the grand finale of the book, but there are two “advanced topics” at the end of the book. One of these is a chapter on random walks, which are used to relate the automorphism groups of the periodic graphs to the symmetry groups of their realizations. The other concerns “discrete Abel-Jacobi maps”, which provide a representation of periodic realizations.

This is a very useful book for a mathematician interested in the mathematical theory of the more abstract components of contemporary crystal design. While it is somewhat accessible, it does presume more background than the appendices on set theory, group theory, linear algebra, free groups, and crystallographic groups would suggest. In addition, the exposition is a bit thin and has leaps; a dedicated reader could treat these bare spots as exercises. Both the index and the references are also a bit thin.

As of this writing, this book is unique in its coverage, and is therefore a necessary addition to the library of any mathematician working on mathematical or geometric crystallography, particularly with respect to crystal design and engineering.

Greg McColm is an associate professor of mathematics & statistics at the University of South Florida–Tampa. He worked in mathematical logic but is now active in mathematical crystallography.