A solution of tartaric acid, a compound found in the "tartar" formed during the fermentation of wine, will rotate polarized light to the *right*. There is another form of tartaric acid, called racemic acid (from the Latin word for a bunch of grapes). It has the same chemical composition, but is optically inactive. In 1848, the chemist Louis Pasteur noticed that crystals of racemic acid had two forms, which were mirror images of each other. He separated the crystals by hand. The resulting solutions were now optically *active*, the one rotating polarized light to the *right*, the other to the *left*. Thus, racemic acid was a mixture of two optically active forms of tartaric acid.

Pasteur suspected that the two optically active forms of tartaric acid must be composed of asymmetric molecules which were mirror images. However, he had no idea what the actual shapes of the molecules were. The first specific suggestion of a possible structure for an optically active molecule was made in 1874 by Jacobus Henricus van't Hoff, who later received the first Nobel prize in chemistry, and, independently, by Joseph Achille Le Bel. Van't Hoff and Le Bel considered a molecule formed of a carbon atom linked by chemical bonds to four other atoms at the four corners of a tetrahedron. If the four attached atoms are all different, then they can be arranged in two distinct ways which are mirror images. This proposal of van't Hoff and Le Bel was, as it turns out, completely correct; indeed, the molecule CHFClBr is today the simplest known optically active molecule. (It's amusing to read the reaction to the 22-year-old van't Hoff's suggestion by one of the scientific big-shots of the day: see Martin Gardner's *New Ambidextrous Universe*, pp. 112-113.)

A molecule which has two mirror-image forms is said to be *chiral* (from the ancient Greek word for 'hand'). Erica Flapan explains, in the last chapter of the book under review, why molecules of DNA tend to be chiral. This fact helps to explain why molecules produced by living things are typically chiral, while molecules produced by inanimate natural processes are not.

The chirality of organic molecules often has significant consequences. In the late 1950s, the drug thalidomide was prescribed to treat morning-sickness in pregnant women. By the early 1960s, it became clear that the drug was responsible for producing horrible birth defects. It was not until about 1980 that it was understood that thalidomide is a "racemic" mixture of two mirror-image chiral molecules. It is one of these two molecules which relieves morning-sickness; the other induces birth defects.

During the last several decades, chemists have synthesized molecules which are more and more complex, among them some which are chiral. For example, in the early 1980s, David Walba and his co-workers synthesized a "molecular Möbius ladder". This is a molecule which has the shape of a Möbius band. The edge of the band is traced out by chemical bonds, while the interior of the band is represented by a sequence of "rungs" formed by carbon double bonds.

Walba was able to show by nuclear magnetic resonance techniques that the molecules he had synthesized were chiral, but he wanted to show that they were really molecular Möbius ladders. Hoping to strengthen the evidence, he asked whether molecular Möbius ladders were chiral, something that seemed plausible, but which he was unable to prove. Then, in 1986, the topologist Jonathan Simon proved mathematically that Möbius ladders with three or more rungs are chiral. Simon's work caught the attention of many topologists, including the author of the present book, and in the ensuing decade-and-a-half many new results were obtained.

Erica Flapan's *When Topology Meets Chemistry* is an introduction to this field of investigation. It presents several techniques for showing that particular molecules are chiral. Many of the results are due to Flapan herself.

It is natural to model a chemical molecule mathematically by the mathematical structure of a *graph*; that is, a discrete collection of *vertices* joined by *edges*. In discrete mathematics, graphs of this kind are often considered simply as abstract structures. For the applications to stereochemistry, however, we need to consider graphs which are actually embedded in a particular way in three-dimensional space.

In the case of a simple molecule consisting, say, of a carbon atom attached to four other atoms, it is obvious geometrically that the molecule is chiral. The problem is more difficult, however, for the complex molecules being synthesized nowadays by chemists. These molecules are so large that their chemical bonds have a certain degree of flexibility. Consequently, it is possible for the embedded shape of the graph to become *deformed*. This raises the question: for a given embedded molecular graph, could it be possible to *deform* the graph to the shape of its mirror image? Flapan calls a graph which cannot be continuously deformed to its mirror image *topologically chiral*. Thus, the problem becomes a problem in topology.

A *tree* is a graph whose edges do not form any loops. From the topological point of view, trees are trivial. If an embedded graph contains a loop, however, that loop may be *knotted*. Furthermore, two or more separate loops in the graph may be *linked*. Thus, the mathematical theory of knots and links becomes a useful tool for studying the topology of embedded graphs. Roughly speaking, if a graph contains a knot or link which is chiral, then we may be able to show that the graph itself is chiral.

Knot theory has had an explosive growth over the past few decades, particularly since the discovery of new polynomial invariants in the 1980s. Flapan gives a quick introduction to some of these knot polynomials, and shows how they can sometimes be used to prove that molecular graphs are chiral. (Incidentally, the tag HOMFLY has six letters, not five [p. 42].)

The proof of Simon's theorem on Möbius ladders, however, requires more sophisticated ideas from the topology of three-dimensional manifolds. Specifically, ordinary three-dimensional space can be completed by adding a point at infinity to form the *three-sphere*. Given a knot embedded in the three-sphere, it is possible to construct another 3-manifold which is a double covering of the three-sphere, branched over the given knot; that is, each point of the three-sphere is covered by two points in the covering, except for the points on the given knot, which are each covered only once. The topological nature of the knot is reflected in the topology of the double covering. Flapan presents the proof of Simon's theorem in Chapter 3, together with an intuitive explanation of the necessary topological ideas. (There is a confusing error in the proof. On pp. 89-90, Flapan claims that if the loops α_{1}, α_{2}, and α_{3} are such that the pairs α_{1} and α_{2} and α_{1} and α_{3} both have linking number 1, then the pair α_{2} and α_{3} also has linking number 1. Actually, α_{2} and α_{3} will then have linking number –1. Fortunately, this error does not invalidate the proof, which depends only on the fact that these linking numbers cannot transform consistently under an orientation-reversing homeomorphism.)

One might ask — and Flapan did ask — whether the chirality of Möbius ladders was an inherent property of the abstract graph, or was determined by the particular way in which that graph was embedded in space. The answer is: both. Flapan shows that Möbius ladders with an *odd* number of rungs are *intrinsically* chiral: the corresponding graphs are topologically chiral, no matter how they are embedded. On the other hand, ladders with an *even* number of rungs have particular embeddings which are topologically *achiral* (*i.e.*, *not* chiral). (The diagram illustrating the two types of "pass moves" on p. 139 is incorrect; both "moves" illustrated are actually the same. Compare Colin Adams, *The Knot Book*, p. 223.)

Another train of ideas stems from Walba's thoughts on rubber gloves. A glove is, of course, a chiral object. (So is a shoe; recall the White Knight's lines, "*And now if e'er by chance I put/ My fingers into glue,/ Or madly squeeze a right-hand foot/ Into a left-hand shoe*".) A *rubber* glove, however, can be turned inside out. It is *topologically* achiral. On the other hand, at no stage during the deformation which turns it inside out does it cease to be (geometrically) chiral. Walba called a molecular graph with such a property a *topological rubber glove*. Flapan provides techniques for showing that a given embedded graph is a topological rubber glove, and gives examples of actual organic molecules which have this property. It turns out that an essential notion here is that of a *finite-order homeomorphism*, that is, a homeomorphism of three-dimensional space (or of the three-sphere) which has *finite order*, in the sense that if we take its composition with itself a finite number of times, we obtain the identity homeomorphism. The basic mathematical tool is a 1939 theorem of the topologist Paul Smith which classifies the possible sets of invariant points under a finite-order homeomorphism.

An actual molecule is of course not the same as an abstract graph. Each vertex corresponds to an atom of a particular kind; and a carbon atom, say, cannot be interchanged with an atom of oxygen or nitrogen. To capture some of this information in our mathematical model, we could consider *labelled* graphs; that is, graphs in which each vertex is assigned a *label*, chosen from some appropriate set. We would then restrict ourselves to considering automorphisms of the graph which preserve the given labelling. A beautiful theorem of Flapan's which is based on these notions is the following: A *nonplanar* graph (*i.e.*, a graph which cannot be embedded in a plane) which has no automorphisms of order two is necessarily intrinsically chiral.

Flapan tells us that this book originated in lectures she presented to graduate students. However, she has tried to make the material accessible to undergraduates, as well as to chemists or molecular biologists.

Although the basic ideas of knot theory and topological graph theory are very intuitive, "there are genuine practical difficulties in attempting to give a totally self-contained introduction to knot theory," as Raymond Lickorish has observed. "To avoid pathological possibilities, in which diagrams of links might have infinitely many crossings, it is necessary to impose a piecewise linear or differential restriction on links. Then all manoeuvres must preserve such structures, and the technicalities of a piecewise linear or differential theory are needed" (*An Introduction to Knot Theory*, p. vi). Flapan's approach to this problem is simply to leave the technicalities unstated. So most of her proofs are really sketches or proof outlines which give the main ideas, but do not include all details. This approach is a reasonable compromise, but some students may feel a little uneasy about what is being swept under the rug. For example, if we have an ambient isotopy of a graph which is embedded in the three-sphere, Flapan remarks without explanation (p. 37) that there is at least one point of the three-sphere which the graph does not pass through during the isotopy. A student who has heard of space-filling curves may wonder how we can be sure.

The range of mathematical ideas in the book is really quite remarkable. On the one hand, many of the proofs involve fairly straightforward, if intricate, combinatorial arguments. At the other extreme, there are a couple of results (her Theorems 6.1 and 7.1) which involve very sophisticated ideas and techniques from topology. In these cases, Flapan gives only a rough sketch of the proof, but I think even her sketches could hardly be understood except by an expert. (I have to admit that I did not understand them myself.) An instructor who would like to use this book for an undergraduate course, but is not a topologist, might find these parts of the book somewhat uncomfortable.

Students might also have some difficulty visualizing the relations among three-dimensional manifolds which arise elsewhere in the book; for example, that the three-sphere can be dissected into two linked solid tori. (The undergraduate students I know have trouble visualizing the region of integration of a triple integral.)

As to whether the book might be useful for chemists or molecular biologists, Flapan clearly displays the ingrained habits of thought of a mathematician. Though she provides some useful discussions of the chemical background, she simply takes it for granted that the main task is to give proofs of theorems. Thus, in content if not in format, the book is mostly theorem-proof, theorem-proof. I can only say that the chemists I know have never shown much interest in studying mathematical proofs.

Flapan also expects the reader to be able to disentangle the kind of language commonly used by mathematicians. Thus (p. 183), "the subscripts are considered mod *m*". The reader is supposed to know what "considered mod *m*" means. On p. 12, she defines an *automorphism* in one sentence; thereafter, the reader is expected to understand what an automorphism is. One of my favorites (p. 36): "Adding a point at infinity is somewhat disturbing to chemists, who tend to live in the real world, which they imagine to be **R**^{3} without any ∞." Of course, this is intended to be humorous, but — does she really think that chemists imagine the "real world" to be "**R**^{3}"?

In a study which deals with mathematical models which are intended to be applied to problems in chemistry or molecular biology, it is necessary to consider the relations between those models and the physical phenomena they are intended to represent. For example, as Flapan points out, *topological chirality* and *chemical chirality* need not be the same. In topology, we consider all possible continuous deformations of a given graph; but not all these deformations need be realizable for an actual molecule.

On the other hand, Flapan seems to accept the implication

(topologically chiral) implies (chemically chiral)

(for example, pp. 32, 201). But what kind of a proposition is this? It certainly is not a mathematical theorem, because the right-hand side has no mathematical definition. On the other hand, it cannot be a statement of experimental fact, since the left-hand side is not accessible to experiment. I think it is a muddle.

Each of the seven chapters in the book is furnished with a set of exercises, most of which require the student to construct a proof. No answers are provided. Some of these exercises are fairly straightforward: "Prove that any complete graph with four or more vertices is three connected" (p. 195). Others require more sophistication: "Prove that a surface embedded in an orientable 3-manifold is orientable if and only if it is two sided" (p. 107). Some are open-ended: "Explain why a mirror reverses your left and your right sides but not your head and your feet" (p. 67). (The student who solves this one might like to compare his or her answer with Martin Gardner's; see *The New Ambidextrous Universe*, p. 19-22.)

According to the blurb on the back cover of the book, "Reading this fascinating book, undergraduate mathematics students can escape the world of pure abstract theory and enter that of real molecules". I presume that Flapan herself is not responsible for this blurb. However, as the book is co-published by Cambridge University Press and the MAA, it occurs to me that those in charge of publications for the MAA might like to consider whether this is the sort of attitude we want to encourage in undergraduates.

Apparently, the blurb-writer is under the impression that American undergraduates are imprisoned in a world of "pure abstract theory", from which they must "escape". Such a view of American higher education seems to me to be highly eccentric. However that may be, the reader who wishes to escape from "pure abstract theory" will hardly do so through the pages of this book. It is, I would say, quite abstract and very theoretical. I hasten to remark that I (for one) do not consider these attributes to be bad ones.

Indeed, I would draw a quite different moral from that of the blurbist: Flapan's book is a wonderful illustration of the power and practical usefulness of "pure abstract theory"!

**References:**

Martin Gardner, *The New Ambidextrous Universe*, Third Revised Edition, Freeman, 1990, ISBN: 0-7167-2092-2. Much of this book is devoted to mirror symmetry at the sub-atomic level, but there are also chapters on molecular chirality.

Jon Applequist, "Optical Activity: Biot's Bequest", *American Scientist*, vol. 75, January-February 1987, pp. 58-68. A treatment of the history and physics of optical activity.

René Dubos, *Pasteur and Modern Science*, Doubleday Anchor, 1960. Pasteur's experiment on racemic acid is described in Chapter 2.

Thalidomide may be coming back into use; see Richard Horton, "Thalidomide Comes Back", *New York Review of Books*, May 17, 2001, pp. 12-15 (a review of the book *Dark Remedy*, by Trent Stephens and Rock Brynner, Perseus, 2001). This article does not mention the concept of chirality. For the chirality of thalidomide, see, for example, Ernest L. Eliel and Samuel H. Wilen, *Stereochemistry of Organic Compounds*, Wiley, 1994, p. 204.

A few books on Knot Theory:

- W. B. Raymond Lickorish,
*An Introduction to Knot Theory*, Springer, 1997, ISBN: 0-387-98254-X. A graduate text.
- Colin C. Adams,
*The Knot Book*, Freeman, 2001, ISBN: 0-7167-4219-5. A wide-ranging introduction at a more elementary level than Flapan's book. It has some material on molecular chirality.
- Gerhard Burde and Heiner Zieschang,
*Knots*, Walter de Gruyter, 1985, ISBN: 0-89925-014-9. Knot theory in the context of algebraic topology. This book explains the concept of a branched covering over a knot or a link. However, some of the topological results Flapan requires are apparently still available only in research papers.

The White Knight's verses are found in Lewis Carroll's *Through the Looking Glass*, Chapter VIII.

Stacy G. Langton (langton@sandiego.edu) is Professor of Mathematics and Computer Science at the University of San Diego. He is particularly interested in the works of Leonhard Euler, a few of which he has translated into English.