In The Geometry of Evolution: Adaptive Landscapes and Theoretical Morphospaces, author George McGhee visualizes evolution by natural selection as a journey across a fitness landscape consisting of hills, mountains, and ravines, in search of some optimal solution for an organism that must adapt to certain environmental conditions. This fascinating metaphor is an interesting approach to studying evolutionary processes, one that leads naturally to the development of mathematical theories about how organisms adapt to a changing environment and how morphogenesis is linked to evolution.

In chapter 2, McGhee discusses the modeling of natural selection in adaptive landscapes, referring to the work of Kauffman (1993 Kauffman (1995). But McGhee and Kauffman take different approaches. In The Origins of Order, Kauffman (1993) presented mathematical models of a fitness landscape, then, on the basis of simulated results, visualized the landscapes in accordance with various parameters. The models discussed by McGhee in chapter 2, however, are not mathematical—they are conceptual models of how an adaptive landscape might look.

In chapter 3, on modeling evolutionary phenomena in adaptive landscapes, McGhee reproduces a fascinating illustration from Signs of Life: How Complexity Pervades Biology (Solé and Goodwin 2000) that shows how an adaptive landscape might look for extinct trilobites from the Palaeozoic era. Every peak in the landscape corresponds to some beautiful trilobite morphology. Unfortunately, the landscape is not based on measurements—it is entirely conceptual. Although it is not entirely straightforward to visualize the fitness landscape for extinct trilobites, it would have been instructive for McGhee to show not only conceptual models but also a mathematical model of an adaptive landscape, as Kauffman did in The Origins of Order.

Yet when trying to explore the adaptive landscapes of such a mathematical model, many questions need to be answered. Two key ones are, How can the fitness of an individual be determined? and How can the multiparameter spaces of an adaptive landscape be visualized? Exploring these questions is a worthy research endeavor in itself.

In chapter 4, McGhee introduces the concept of theoretical morphospaces, which he defines as n-dimensional geometric hyperspaces produced by systematically varying the parameter values of a geometric model of form. The most famous theoretical morphospace is the one constructed for mollusk shells by Raup (1966). In Raup's paper, a range of hypothetical mollusk shell morphologies was visualized as a function of the number of parameters controlling the morphology of the shell. A pioneering publication in many ways, this paper was the first to introduce the concept of morphospaces. In this early application of scientific visualization, the results of a mathematical model were visualized on a computer screen even before the field of computer graphics existed!

A general problem with morphospaces is that the underlying mathematical model is based largely on just a description of the organism's form. In the mollusk morphospace, the morphology of the shell is controlled by parameters such as the whorl expansion rate and the translation rate of the shell. The advantage of such a description is that the shell's morphology can easily be captured by a small piece of computer code; the disadvantage is that these parameters do not necessarily have a connection with the developmental process. McGhee's book would have been more interesting with examples of a connection being made between the underlying developmental gene regulatory networks and the form of an organism. There exist among the echinoderms some beautiful examples of how rewiring regulatory networks that control the body plan can result in a sea urchin or a starfish (Davidson 2006).

McGhee states on page 61 that in many cases, creating a mathematical model of growth is not very difficult—it “simply requires a little thought.” It is clear that here, McGhee is referring to simple geometrical descriptions of form. To anyone who has ever worked on modeling growth and form (e.g., gene regulation and cell movement), McGhee's statement must sound strange. An interesting topic for his next book might be the construction of morphospaces that are based on physical—or biologically relevant—parameters. In the literature on bacterial colonies, for instance, there are many examples of morphospaces (some with the form of phase diagrams). Kawasaki and colleagues (1997), for example, provided a diagram that shows the morphol ogy of a bacterial colony as a function of two biologically relevant parameters (concentration of nutrient and density of the agar medium).

Regardless of the shortcomings of descriptive and conceptual models, The Geometry of Evolution does provide an excellent overview of the role of theoretical morphospaces and adaptive landscapes in models of growth and form. McGhee has made a courageous attempt to develop a mathematical theory connecting models of morphogenesis and evolution, and his book offers an opportunity to learn about the enormous diversity of palaeontological examples of evolution. He has done a very good job in bringing all this material together in one book, and I would recommend The Geometry of Evolution to anyone interested in morpho genesis and evolution. Mathematicians and computer scientists in particular will find that the book poses many interesting questions.

Visualizing n-dimensional parameter spaces and adaptive landscapes is highly relevant to optimization problems and a good example of the challenge of information visualization. Finally, the book intrigues, enticing readers to ask new research questions: Can we develop mathematical models of growth and form that are useful for investigating the role of natural selection in evolution? What do these adaptive landscapes look like? Do many possible solutions exist in evolution, or does the evolutionary process converge on a few choice answers? The reader who is open to such questions will find much here to stimulate reflection and experimentation.