This book consists of research papers and articles on phyllotaxis, all of which were written by Irving Adler (1913–2012). Four of the papers appeared in the *Journal of Theoretical Biology* between 1974 and 1977, another was published in the *Journal of Algebra* in 1998, and two ‘articles’ are really excerpts from the book *Symmetry in Plants* (World Scientific, 1998).

Phyllotaxis analyses the positioning of botanical units such as leaves on a stem or the scales on a pineapple. For example, trees like basswood and elm have leaves occurring alternately on opposite sides of a stem (½ phyllotaxis). For beech and hazel, the upward journey from one leaf to the next is obtained by a screw displacement along the stem via a rotation of ⅓ turn (⅓ phyllotaxis). On other trees, such as poplar and pear, one observes ⅜ phyllotaxis which, when the sense of rotation is reversed, can also be described as ⅝ phyllotaxis. In all such cases, the fractions are quotients of alternate Fibonacci numbers.

Patterns, known as ‘parastichies’, can be seen on florets on the heads of sunflowers. These take the form of oppositely directed spirals, of which 55 are clockwise and 34 anticlockwise. On pineapples, the hexagonal cells occur in spirally ascending rows, and one may see 8 sloping spirals, 13 steeper spirals, and 21 very steep spirals. The numbers 8, 13, 21 are consecutive members of a Fibonacci sequence, yielding the parastichy pairs (8,13) and (13,21). If there are m upward spirals to the left and n spirals upwards to the right, then the plant has (m,n) phyllotaxis.

In spiral phyllotaxis, botanical elements grow one by one, each at a constant *divergence angle* *d *from the previous one. The fundamental theorem of phyllotaxis relates d to a parastichy pair (m,n), and the precise connection was first established by Adler in the first of the papers in this book. The style of that paper, and the remaining contents of this book, confirms Irving Adler’s position as an original researcher in this field and also a first class expositor. But non-specialists in this field would find it easier to grasp the essential principles of this topic if they were to begin by reading it from back to front. This is because the later articles are of an introductory nature for readers with little prior knowledge of this botanical discipline.

Adding to the readability of this book is the historical commentary that pervades it from start to finish. We read that the earliest recorded thoughts on phyllotaxis are due to Theophrastus (370–285 B.C.), and that Leonardo da Vinci was the first known person to subject it to mathematical analysis. Subsequently, Johannes Kepler recognised that Fibonacci numbers are of particular relevance to this subject. Many naturalists, such as Linnaeus, Charles Bonnet and the Bravais brothers made great strides in systematising this botanical discipline — although various mathematicians have contributed to its theoretical basis. Among major contributors were Alan Turing and HSM Coxeter and the author of these collected papers — Irving Adler who consolidated the modern mathematical theory of phyllotaxis.

At first sight, phyllotaxis would seem to be a topic pertaining exclusively to the realm of theoretical biology, but patterns similar to those of phyllotaxis arise in a range of surprising contexts. For example, Frey-Wyssling (1954) showed that a helical polypeptide chain displays characteristics similar to those of a genetic spiral on a plant stem. And Levitov (1991) observed phyllotaxis patterns in a flux lattice in a layered superconductor. Moreover, Irving Adler himself has shown that the central concept of cylindrical lattices pertains to Minkowski’s geometry of numbers.

Despite having no prior knowledge of this subject, this collection of articles and papers gripped my attention from start to finish. They are the hallmark of a fine expositional writer in the field of pure and applied mathematics. What’s more, the original research that forms the basis of this book, was carried out by Irving Adler in his retirement. In fact, the book itself was published only a few months before he died at the age of 99.

Peter Ruane’s** **nearest prior insight into this field was investigational work on symmetry of plants and flowers with schoolchildren in the 1960s and1970s.