The beauty and diversity of plants has interested botanists for centuries. Common features among plants (types of flowers, stem and branch patterns, leaf shapes and clustering, and shapes of plant organs) are all modified by variations between and within related families of plants. Understanding common features with significant variations is part of the study of plant morphology. The author of this book, Johan Gielis, is a horticulturist and mathematician. The book is a monograph describing his research into the mathematical principles underlying plant morphology originally prompted by his study of the cross-sectional shapes of bamboo stems. The results of his investigations were originally called superformulae but are now called Gielis transformations (not by himself, but by the mathematical biology community).

Gielis takes a strongly historical approach in describing his research, beginning with classical Greek geometrical principles taught by Pythagoras and Euclid. The entire book is based on a geometrical approach. He shows how the Pythagorean formula can be generalized to describe other shapes that are topologically equivalent to a circle. For example, the five-armed shape of a starfish is topically equivalent to a circle. Furthermore, the parameters in the formulae can be continually varied to produce a smooth transition from one shape to another, such as that which takes place as organisms mature.

The system he describes is based on the geometrical studies of Gabriel Lamé two centuries ago. Lamé described supercircles and superellipses, which Gielis used to categorize the shapes of bamboo stems. The investigation did not stop there. Gielis perceived that Lamé’s work could be developed furthered to describe mathematically the five petals on a wild rose, the shape of the calcium carbonate shells of diatoms, and other morphologically interesting and aesthetically appealing biological structures.

The book is divided into six parts, each of which is given both a Greek and a Latin name. This division follows the classical Greek and Latin way of presenting an argument. There are 12 chapters in all and an extensive bibliography. Unfortunately, there is no topical index at the end.

Part 1, “Propositio,” describes the basis of morphogenesis. The foundation consists of conic sections, well known in classical geometry, and a preference for beginning the study of naturally occurring shapes with observation, not rigorous mathematical structures. Part 2, “Expositio,” presents the theory of Lamé supercurves and surfaces using geometry and trigonometry, and shows how Gielis curves and transformations arise from Lamé’s work. Part 3, “Determinatio,” has two important chapters. Chapter 6 presents k-type curves (which can be used to describe leaves, for example) and R-functions for overlapping shapes. Chapter 7 discusses rigid and semi-rigid measures that can be applied to generate interesting and novel shapes. Part 4, “Constructio,” shows how Gielis transformations can be applied to solve boundary value problems in orthogonal functions (for example, Fourier series, Bessel functions, and Chebyshev polynomials) and how the curvature of the supershapes can be defined and used. Part 5, “Demonstratio,” applies Gielis supershapes and transformations to observable biology: bamboo leaves, tree rings, snowflakes, and Asclepiad flowering plants. Part 6, “Conclusio,” is a review of the previous five sections and the author’s personal philosophical reflections.

The first book that came to my mind when I first encountered the title of Gielis’s monograph was the classic The algorithmic beauty of plants [1]. The latter book takes an entirely different approach to plant morphology using algorithms (L-systems) and algebra in the context of generating fractals. Gielis emphasizes the continuity and differentiability of his shapes and transformations in contrast to the algorithmic and fractal approach. There is a place for both approaches. One is complementary to the other.

This book should be of interest to anyone who works in biological morphology because it describes a quantitative way of describing biological shapes. With appropriate software to modify these shapes, it would be possible to write software to fit shapes to specimens to identify and catalog them.