Introduction

D’Arcy Thompson’s most influential ideas about evolution can be found in a single chapter of a single book: Chapter 17, entitled “On the Theory of Transformations, or the Comparison of Related Forms,” in his magnum opus, On Growth and Form. The material in this chapter has been published at least four times in English alone. It was first published as a preliminary paper (Thompson 1915), then in the original edition of the book (Thompson 1917), then in the revised and much expanded second edition (Thompson 1942), and also in the abridged edition annotated by John Tyler Bonner (Thompson 1961). This material has probably been more widely read than anything else Thompson wrote.

Why has this chapter influenced so many biologists—and indeed scholars more generally? Despite the beauty of his prose, the answer lies not in the text but in the illustrations. It is difficult to find other images that have been so influential in the field that we now refer to as evo-devo. Among the closest competitors are Haeckel’s 19th century embryo drawings (see Hopwood 2015) and the much-reproduced diagrams of the colinearity (alias co-linearity, collinearity) of the chromosomal positions of Hox genes, the regions of the body in which they are expressed, and their temporal order of expression in embryogenesis (see, for example, Carroll et al. 2005). Pictures of Thompson’s morphological transformations are familiar to most readers; I reproduce a representative example here as Fig. 1. All are of broadly the same type: they show the outlines of structures belonging to typical individuals of several related species—in this case crabs, in others fish, birds, mammals, and various other animals, as well as the leaves of plants.

Fig. 1
figure 1

Thompsonian transformations applied to the carapaces of six genera of crabs. The starting point for this series of transformations is Geryon (top left), with a carapace of this genus drawn on a Cartesian grid. Five types of transformation, as shown by the five different distortions of the initial square grid, reveal the shapes of five other genera. As noted in the text, one of these, Paralomis, is not closely related to the others and is the product of convergent evolution, though it seems likely that Thompson was unaware of this fact

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In the world of art, images can be iconic for many reasons. For example, they can be intriguingly beautiful, such as Leonardo da Vinci’s Mona Lisa; strikingly original, such as Claude Monet’s Impression: Sunrise; disturbingly emotive, such as Edvard Munch’s The Scream; or puzzlingly paradoxical, such as depictions of the stairways of M. C. Escher that always go upward and yet whose bottom and top adjoin. However, the situation is very different in the world of science (Hopwood 2015, Chap. 1). Scientific images generally become iconic because, in addition to being eye-catching and aesthetically appealing, they tell us something important—or appear to tell us something important—about how nature works. Haeckel’s embryo illustrations suggest—notwithstanding the various criticisms of them—that in animal groups with simple life cycles, such as mammals, evolution works by modifying later developmental stages more readily than earlier ones. Hox gene colinearity suggests some sort of coordinated activation of the genes in this group, in many kinds of animals, even though all the details are not yet clear. But what do Thompsonian transformations suggest about nature in general, or evolution in particular?

If multiple morphological differences between animals of related species can all be captured in a single transformation, as Fig. 1 and many of Thompson’s other drawings indicate, this suggests that evolution somehow acts in a holistic way, treating the organism as an integrated entity rather than as a collection of individual traits or characters that are modified independently. Of course, evolution is a messy probabilistic process, not a clean, deterministic one. Even if it acts in the way that Thompsonian transformations suggest in some lineages some of the time, it acts in a more piecemeal way in other lineages and at other times. There are many cases where individual characters are much modified, with relatively minor effects on the rest of the organism’s structure. For example, the third digit of the hand of the aye-aye (a species of lemur) has become greatly modified in evolution—it is unusually thin and is used to extract food such as insect larvae from narrow holes in wood. Other species of lemur (there are about 100 of them) do not possess this adaptation, so it must have arisen in the aye-aye stem lineage. However, the other digits of the aye-aye’s hand are “normal,” as are all the digits of the foot.

If Thompson had only shown a single series of transformations—such as those of crab carapaces in Fig. 1—his illustration wouldn’t have had so much impact. It would only have provided a single example of the evolution of multiple coordinated characters, just as the aye-aye provides us with a single example of the opposite, namely evolution of one seemingly independent character. It would thus have said nothing of the relative commonness of the two. Since all biologists accept that there is much variation in the ways that evolution changes animals, a single example of any particular form of evolutionary change is limited in the information it conveys.

However, the fact that Thompson’s many illustrated series of transformations ranged across the animal kingdom—and beyond—suggests that coordinated evolution is the norm and independent evolution the exception. In other words, it suggests a potentially important conclusion about how evolution works in general. In doing so, it also suggests something generally important about development, and in particular about the developmental process of morphogenesis, in other words, shape formation. In terms of the connection between the two—the focus of evo-devo—it would suggest that the coordinated nature of evolution somehow derives from the coordinated nature of morphogenesis. This is an exciting idea. But excitement about the general nature of evolution, development, and the link between the two, must be tempered with careful reasoning, and with a commitment to understanding causality. Such reasoning and commitment are at the heart of the present article.

Before proceeding further, it is worth considering what might be described as the dual usefulness of pictorial representations in biology—especially in evolution, development, and evo-devo. Images can be interpreted broadly, as abstracting certain features upon which an author wants to concentrate, and ignoring other features that are taken to be of lesser interest. In such a general usage, the images do not necessarily connect with specific hypotheses of causality, but they may be helpful in a more general sense in terms of guiding thinking about the process concerned. Alternatively, images may be interpreted in a more concrete way, with clear connections to causality. This point about two different scientific usages of pictures has been made by Fusco et al. (2014) in relation to the landscape visualizations of Wright (1932) in evolution and Waddington (1940, 1957) in development. It has also been made by Nuño de la Rosa (2018) in relation to pictures of particular embryonic stages, and the advances made in contemplating transitions between these stages that have derived from time-lapse and other modern imaging techniques.

I should make it clear that in this article I am considering Thompson’s transformations from what I have just called a “concrete” perspective, with explicit connections to hypotheses about causality. I show that from this perspective morphological transformations as drawn and discussed by Thompson are fundamentally inadequate because of their limited dimensionality. Some biologists and philosophers may claim that these pictures nevertheless retain a usefulness of the more general sort. However, I am inclined to believe that they do not, and that they may even hinder, as opposed to help, clear thinking about the evolution of development.

Thompson’s Views on Causality

D’Arcy Thompson was a scholar of considerable breadth. He is often deservedly described as a polymath. His knowledge spanned many areas of biology, natural history, mathematics, and classics. However, he was less strong on the causality of the transformational patterns to which he drew attention than he was at describing the patterns themselves; indeed he often shied away from issues of causality. Nevertheless, he did make some comments on the causality of transformations, and it’s worth looking at these in order to see what they do—and do not—say. As a prelude to this, I should note that “causality” can mean different things in different academic disciplines, for example biology and philosophy. As a biologist, I use “causality” as referring to “the relationship between cause and effect” in a physical sense; this expression being given under both causality and causation in the Concise Oxford English Dictionary. With one exception, the page numbers of the quotations from Thompson that follow refer to the Project Gutenberg online version of the first edition of On Growth and Form (1917), which is available here: https://www.gutenberg.org/files/55264/55264-h/55264-h.htm. The exception is a quote from the second edition, where the page number refers to the abridged version of this edition (Thompson 1961).

At the start of Chapter 17, Thompson discusses the advantages of taking a mathematical approach to morphology instead of a verbal one. He argues that a benefit of making the transition to a mathematical approach is that it enables a dynamic rather than static concept of form. In other words, as he puts it on p. 720: “we rise from the conception of form to an understanding of the forces that gave rise to it.” Importantly, “gave rise to” is synonymous with “caused.” Regarding these causal forces, he goes on to say (p. 720) that in a morphological transformation “we discern the magnitude and the direction of the forces that have sufficed to convert the one form into the other.”

A few pages later, Thompson continues his account of causal “forces,” reiterating what he said in the previous section and expanding it somewhat. He describes the general approach of morphologists whose aim is to understand the relationship between two related forms as follows (p. 724): “The morphologist…will enquire whether two different but more or less obviously related forms can be so analysed and interpreted that each may be shown to be a transformed representation of the other.”

He continues: “This once demonstrated, it will be a comparatively easy task (in all probability) to postulate the direction and magnitude of the force capable of effecting the required transformation.” Here, “capable of effecting” is synonymous with “capable of causing.” Indeed, the latter is simply a more modern version of the former. Today, “effect” is usually a noun, but the old verb “to effect” and the modern one “to cause” both mean “to make (something) happen.”

Thompson’s optimism that transformations point the way to causal explanations is linked to his enthusiasm for Occam’s razor. He says that if a single transformation can be shown to describe multiple morphological changes

we may find ourselves able to dispense with many widely current and more complicated hypotheses of biological causation. For it is a maxim in physics that an effect ought not to be ascribed to the joint operation of many causes if few are adequate to the production of it. (p. 725, italics added)

However, there is a useful piece of counter-advice that proponents of Occam’s razor should heed: “seek simplicity but distrust it.” This is a shortened and slightly altered version of a quote from the British mathematician and philosopher Alfred North Whitehead (1920, p. 163), as follows:

The aim of science is to seek the simplest explanations of complex facts. We are apt to fall into the error of thinking that the facts are simple because simplicity is the goal of our quest. The guiding motto in the life of every natural philosopher should be, Seek simplicity and distrust it.

In his enthusiasm for simple explanations, Thompson was too ready to ascribe an explanatory power to his transformations that they didn’t possess. There are two ways of looking at this issue. First, we can ask whether particular examples of Thompson’s hand-drawn transformations do indeed adequately capture the multiple morphological differences among a group of related species. Scholtz et al. (2020) did exactly that with the crab example, and discovered that most of the transformations were oversimplifications, and some of them were seriously misleading. Second, we can ask whether Thompson’s general notion of causal “forces” responsible for transformations holds up to scrutiny. This is my main purpose here. In particular, when proposing causal forces it is necessary to consider not just their magnitude and direction, which is what Thompson focused on, but also their nature. Indeed, it can be argued that the last of these three attributes is logically prior to the other two.

There is a widespread perception that when Thompson referred to “more complicated hypotheses” that could be dispensed with, he was thinking in particular of hypotheses involving the “force” of natural selection. This may be true, because he had a view of selection as a process that modified different morphological characters independently of each other. Also, there is a widespread perception that the “force” he had in mind when thinking of the potentially simpler hypotheses that might explain transformations was some sort of physical force related to the ways in which cellular structures can be modified—perhaps in broadly developmental terms. But Thompson’s silence on the exact nature of either his favored or his disfavored forces is problematic, as is a resort on the part of readers to guessing what he had in mind based on his other writings.

Thompsonian Transformations Never Happen in Nature

Here I will approach the issue of the causality of morphological differences between related species from what might be called first principles, with a focus on the need to have adequate dimensionality in any causal hypotheses of this kind. This approach leads to the conclusions (1) that none of the transformations Thompson illustrated ever took place as physical processes, and therefore (2) that a search for the “forces” that caused the transformations, and for an elegantly simple hypothesis involving one or a few such forces, is futile.

A good starting point for my “dimensional sufficiency” approach is a point made by Gilbert (2007, p. 371), as follows: “For evolutionary developmental biology, the current challenge is producing a five-dimensional representation: the four standard dimensions of space and time placed into the context of the paleontological temporal dimension.”

In my view, Gilbert is correct, but two caveats to his point are necessary. First, strictly speaking, developmental and evolutionary time are not two distinct dimensions, but rather two different timescales. However, labelling them as dimensions adds impact to Gilbert’s point and is thus very helpful in combatting hypotheses that fail to distinguish the two. Second, the way Gilbert makes his point, and in particular his perception of developmental time as “standard,” is influenced by his training as a developmental biologist. As someone who came to evo-devo from a starting point in evolutionary rather than developmental biology, I might consider evolutionary time as “standard” and developmental time as an add-on. However, the best approach is to abandon the use of any “standard” in this respect, and simply think of five “dimensions”—three spatial and two temporal.

Now we will consider Thompson’s transformations from a dimensional perspective. The illustrations of these in Chapter 17, whether of crabs, fish, or other creatures, are all two-dimensional. However, at the end of the chapter, Thompson briefly considers the possible extension of transformations from 2 to 3D. He prefaces his thoughts on this issue by saying that 3D transformations “must be left to other times, and to other hands” (p. 774). But then he does allow himself to consider a single example—that of the transformation of a “normal” fish into a flatfish. He concludes his consideration as follows (p. 776):

In short, if we might conceive of a haddock being transformed into a plaice, a very large part of the change would be accounted for by supposing the round fish to be “rolled out” into the flat one, as a baker rolls a piece of dough.

Let’s now reflect on Thompson’s example of haddock and plaice, or, perhaps better, a simpler example of “flattening,” such as shark and ray, where there is no body rotation involved. The main problem with Thompson’s view about the differences in form between the two types of fish is that, although we now have a 3D rather than 2D spatial view of the transformation, we still have only a 1D rather than 2D temporal view. In other words, there is an issue with the identity of the “baker”: is it development, evolution, or both? In this context, it is important to note that all Thompson’s illustrations of claimed transformations depict adults—either whole adults, such as his fish transformations, or parts of them, such as his drawings of the skulls of humans and chimpanzees. This is the most serious flaw in Thompson’s approach. Evolution cannot change the shape of an adult of one species directly into the shape of an adult of another. Such changes, whether thought of as coordinated (transformations) or piecemeal (independent characters), never happen in nature. Thus we should not search for “forces” that cause them. Instead, we should concentrate on the changes that take place in reality, and only focus on their causality when we are certain what they are.

As an animal develops from a fertilized egg (or other starting point) into an adult, it undergoes various types of change. The changes to which transformations can be most readily applied are those of late rather than early development. Specifically, they are the stages that follow the laying down of the broad body plan of the animal concerned, and primarily involve growth (as indicated in Thompson’s book title). Occasionally, shape stays the same with growth, as in the case of gastropod shells that approximate to equiangular (or logarithmic) spirals; Thompson devoted some space to such cases of isometric growth in Chapter 11. However, more often growth is allometric (Huxley 1932; Young et al. 2010), with shape changing as the animal develops. An often-cited example is the reduction in the ratio of head size to overall body size during the development of a human. Allometric growth and transformations are linked. To be more specific, allometric growth produces transformations in shape, while isometric growth does not. However, this use of “transformation” is different from Thompson’s: these are developmental transformations; no evolutionary process is involved. The changes all occur within the lifespan of a single individual.

In evo-devo, we focus on evolutionary change in something that is itself a process of change—development. This is a complex matter, and it leads to both conceptual and verbal difficulties. It is something different from either development itself or evolution itself. But is “evolution itself,” in other words evolution without a developmental angle, a useful concept? I think it is; and here is a hypothetical example. Suppose that in species A there is one version of an enzyme involved in a routine metabolic process that occurs in all cells at all stages of development, and that in a closely-related species B there is an alternative version. The difference is a single amino acid at a particular position along the chain of them that constitutes a molecule of the enzyme involved. The simplest explanation of this interspecific difference is that the switch from one amino acid to another was initially caused by a gene mutation in the germline of a single individual of species B, and that this change spread through populations of species B by natural selection. In such a situation, we have a genetic process (mutation) followed by a population-level process (selection), but no developmental process per se because the gene concerned is a classic “housekeeping gene” and has no developmental effects.

Allometric growth is a process that takes place in one of Gilbert’s time dimensions, while the evolution of the structure of an enzyme is a process that occurs in the other. So in both of these cases—development alone and evolution alone—the two time dimensions, or scales, can be thought of individually, with a focus on one of them. But, when we consider the evolution of development, we must think of them together. To do this, it helps to have a phrase for the long-term process that modifies the short-term one—something more specific than the catch-all “evolution,” which applies to all long-term changes, including those in which the developmental process is unaltered (e.g., the case mentioned above of the evolution of a housekeeping gene that’s expressed at all stages of development).

When I first became aware of the difficulty caused by the lack of such a phrase, I began to use “developmental reprogramming” (Arthur 2000). However, some readers pointed out that this phrase came with too much philosophical baggage, and smacked of what has sometimes been called “genetic imperialism.” Development is not completely pre-programmed, as we see in cases of phenotypic or developmental plasticity, which are common (West-Eberhard 2003). Around the same time as I realized the need to take on board this criticism, I discovered the earlier use of “ontogenetic repatterning,” for a closely-related but nonidentical purpose, by Roth and Wake (1985). So, ever since then, I have used “developmental repatterning” as a fusion of the earlier two phrases (see, for example, Arthur 2011). With this usage, we have one process—development—which causes a series of changes during the lifetime of a single individual, and another process—developmental repatterning—which causes changes in development itself over multiple generations.

This approach is in one sense neutral with respect to causality but in another sense not so. It is neutral in that it does not claim one particular dominant cause for either process. For example, it does not claim that the main cause of development is gene switching or that the main cause of developmental repatterning is natural selection. Either or both of these may be the case—but so far that’s left open. However, what is not left open is the fact—and I believe this word to be applicable here—that the causes of the processes occurring on the two different timescales are themselves different. Developmental repatterning may be caused by a combination of germline mutation, natural selection, and genetic drift, but none of these causal agencies is involved in development itself. Likewise, development may be caused primarily by a mixture of gene switching and cell–cell signaling via morphogens, but neither of these is the cause of developmental repatterning.

We are now in a position to think more clearly about the realms in which “transformations” in a general sense might apply, having already concluded that they do not apply to the conversion of the form of an adult of one species into that of another species, since such conversions are nonexistent in nature. There are two potential realms of application: that of development itself, and that of developmental repatterning. It seems appropriate to take them in that order, so we’ll look at development now, repatterning in the following section.

Since allometric growth is common in development, and its isometric counterpart rare, the shapes of structures are usually altered as an animal develops from a late embryonic stage, through a series of juvenile stages, to the “final” adult form (with apologies to Minelli (2003, 2009); I don’t mean to be unduly adultocentric here). However, such changes during the later stages of development need not be “transformations” in the sense that they are coordinated with each other across different characters. Character coordination happens in some cases, but not in others. There are several factors involved in this variation, including (a) the degree of modularity of developmental systems and (b) the extent of pleiotropic effects of developmental genes (see Young et al. 2010).

Another factor that is important is the “directness” of development. There are many different types of development (Minelli 2021), including both those where the process of getting from egg to adult occurs directly and those where it occurs indirectly (via larvae and metamorphosis). Coordinated quantitative shape change can occur in both types of developmental system. However, in an indirect system such as the development of a butterfly from a caterpillar, transformations have restricted applicability. They can be applied to the quantitative growth of the caterpillar that occurs before metamorphosis, but not to metamorphosis itself, since the changes in shape during this radical developmental process are typically qualitative. Thompson himself was clear that his transformations were not applicable in cases of qualitative shape change (Arthur 2006). He said this in the context of comparing two qualitatively different shapes that are distantly related in evolutionary terms; he pointed out that you could transform one fish into another or one crab into another, but not a fish into a crab or vice versa. It is important to note that his point also holds true of qualitatively-different forms that are developmentally rather than evolutionarily related, such as caterpillar and butterfly.

We have now reached a point where we can see that although the idea of transformations cannot be applied to the conversion of one adult form into another, it can be applied to the conversion of one stage to another in the development of a single individual, with its applicability in this realm depending on both developmental time (more appropriate to late development than early) and the type of developmental system involved (not applicable to the metamorphosis phase of indirect development). Now the remaining question about realms of applicability is whether, or to what extent, transformations can be applied to developmental repatterning.

Transformations in the Context of Developmental Repatterning

The best way to deal with this issue, so that it does not become too rarefied, is to consider it in the context of a particular group of animals. I will use crabs, since this allows us to make a contrast with the morphological transformations shown in Fig. 1. Our first task, then, is to be clear about (a) the phylogenetic context of crabs and (b) the ways in which they develop.

The clade of “true crabs” is Brachyura, which is usually thought of taxonomically as an infraorder. Its stem species probably lived in the Jurassic period. It now includes over 7250 species (Wang et al. 2021) belonging to about 100 families. Five of the six genera of crabs illustrated by Thompson belong to this clade. The exception is Paralomis, which belongs to the clade (infraorder) Anomura, most of whose members are hermit crabs (Richter and Scholtz 1994). It seems that Paralomis stemmed from a hermit crab ancestor and evolutionarily converged with the general body form of Brachyura. This is interesting, because Thompsonian transformations are normally thought of as being related to divergent rather than convergent evolutionary processes. As Scholtz et al. (2020, p. 305) say: “There is no indication that Thompson was aware of the fact that one of the carapaces does not belong to a brachyuran.”

Now we shift our attention from crab phylogeny to crab development. Naturally, development varies in detail across a clade of more than 7000 species. Direct development without a larval stage typically occurs in freshwater species (Wu et al. 2010). However, in most marine species there is indirect development as follows. Fertilized eggs are brooded by the adult female until they hatch into mobile zoea larvae, which are planktonic. These undergo several molts, the last producing a different larval form called a megalopa, which eventually metamorphoses into a juvenile. Like adult crabs, the juveniles are benthic. They molt several times before becoming reproductively mature. The part of this overall indirect developmental trajectory to which transformations can be most easily applied is the growth of juveniles into adults. In some species, the juvenile shape change that occurs with growth may be coordinated (transformative), while in others it may be piecemeal with multiple quasi-independent changes. Note that to establish which of these models is a better fit in the case of any particular species, it would be necessary to undertake the digitization of multiple juvenile shapes in the same way as Scholtz et al. (2020) digitized the shapes of adult crabs.

The question of whether coordinated transformations are applicable to the developmental repatterning that over evolutionary time produces one pattern of crab development from another is even more difficult to answer. For this we would need to have the above digitized data on carapace development not just from one species of crab but from at least two. Furthermore, this is a good point to remind ourselves (following Jenner 2022) that what we need is not data on two present-day species (“cousins”), but rather one extant species and one earlier one (ancestor and descendant). This is another aspect of the lack of applicability of Thompsonian transformations, as shown in Fig. 1, to evolutionary change in the real world.

The fossil record of crabs, like that of animals in general, is biased towards adults. The idea that we could find enough intact fossil juveniles of various stages to perform reliable digitizations is optimistic to say the least, as is the idea that we could identify a particular fossil species as being definitively ancestral. Given this unfortunate situation, we are left with having to conduct a sort of thought experiment, which I will now describe, with the help of Fig. 2. Although this is a poor substitute for real data, there is much we can learn from it.

Fig. 2
figure 2

The difference between developmental transformations on the one hand and evolutionary transformations on the other. The juvenile growth of the ancestral species (left) is isometric, so there is no developmental transformation. In contrast, the juvenile growth of the descendant species (right) is allometric, and so a developmental transformation takes place (bold vertical arrow), involving a coordinated change of shape with increasing size. The transition from ancestral to descendant developmental trajectories is brought about by a coordinated form of developmental repatterning, which can be referred to as an evolutionary transformation (bold horizontal arrow), though it must be stressed that this is not the same as a Thompsonian transformation of the sort shown in Fig. 1. Note that a developmental transformation takes place in the short term (within a single lifespan), while an evolutionary transformation is a long-term process, often requiring millions of years

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Figure 2 shows a hypothetical situation in which an ancestral species is characterized by isometric growth of the juvenile carapace—in other words the shape remains constant from the earliest juvenile stage to the adult, despite the large increase in size. The descendant species, in contrast, exhibits allometric growth, with the shape changing in a coordinated (transformative) manner as the animal gets bigger. Thus in this case the developmental repatterning that has occurred over evolutionary time has altered isometric growth into its allometric counterpart, and in particular a form of allometry in which growth of the carapace from side to side exceeds growth from front to back. In Fig. 2, there are four “dimensions.” The carapace is shown in two spatial dimensions (its depth is ignored, as in Thompson’s illustrations); the developmental time dimension is vertical, its evolutionary counterpart—repatterning—is horizontal.

Now we ask the question: what might have happened in the stem lineage of the descendant species to cause this kind of repatterning of development? Morphogens involved in late development, such as juvenile to adult, often permeate the growing body via the circulatory system. A good example of this is the circulation of the molting hormone ecdysone in flies, which is produced by a gland at the anterior end of the larval body and leads to pupation. Experiments have shown that if circulation to the posterior part of the body is restricted at a particular developmental stage, only the anterior part pupates (Fraenkel 1934). Crabs have a circulatory system consisting of a heart, blood vessels, and a hemocoel. A morphogen being transported by this system may reach all parts of the developing body. It is likely to achieve its effects via certain receptor molecules of the cells it reaches. How might the evolution of such a system result in enhanced lateral growth?

Suppose that an evolutionary change takes place in a developmental gene, such that a morphogen causing cell proliferation tends to accumulate in lateral areas of the developing carapace. As a result, there is an increased rate of proliferation in those areas, and thus enhanced lateral carapace formation. So, a system of isometric growth in the development of an ancestor could be evolutionarily converted into a system of allometric growth in a descendant. In terms of “transformations,” an ancestor in which the growth of juveniles wasn’t characterized by a developmental transformation of shape has evolved into one in which such a transformation does take place—presumably for some selective reason. We could call the kind of coordinated developmental repatterning that caused this change an evolutionary transformation, but this is of course very different from the morphological transformations that Thompson depicted. Developmental and evolutionary transformations may be linked; but such a link is far from guaranteed. This is where we need to build into our mental picture something that Thompson had a tendency to ignore: variation among individuals within species.

Darwin and Thompson

The Darwin-Wallace approach to evolution is centered on the existence—and the importance—of individual variation. Without this, natural selection would be impossible. It’s no accident that Darwin (1859) began The Origin of Species with two chapters on variation. Only in the light of widespread variation, plus the struggle for existence (Chap. 3), did natural selection (Chap. 4) make sense. Darwin built a very logical foundation for his theory of the mechanism driving evolutionary change. Interestingly, though, he went back to variation for Chapter 5, “Laws of Variation,” in which he devoted considerable attention to what he called “correlation of growth” (see also Abzhanov 2017). One way to interpret this split treatment of variation is that the first two chapters were focused on establishing the ubiquity of variation, while the later chapter was focused on considering its nature. Now we need to connect both of these things with the views of Thompson.

The first problem we encounter in trying to connect transformations with a Darwinian world-view is Thompson’s tendency to downplay intraspecific variation—to effectively remove it from sight. This is not to say, of course, that he denied its existence. But, like many other biologists interested primarily in macroevolution, he seemed to see it as a distraction from greater things. Early in his chapter on transformations, he says (1961, p. 271) that

we must learn from the mathematician to eliminate and to discard; to keep the type in mind and leave the single case, with all its accidents, alone; and to find in this sacrifice of what matters little and conservation of what matters most one of the peculiar excellences of the method of mathematics.

From the perspective of making morphological transformations tractable, it makes sense to imagine each species as if it were characterized by a single invariant adult form. But for thinking about evolution more broadly it makes no sense at all. Not only that, but it isn’t really a question of the distinction between mathematical and biological approaches, rather the distinction between one branch of mathematics and another: geometry and statistics. To understand the evolution of form, we need to incorporate both. We need to combine the geometry of individual form with its statistical distribution—normal or otherwise—in the population. Such a combined approach is now taken by many authors in the field of geometric morphometrics, for example Bookstein (1977, 1996), Mitteroecker and Gunz (2009), and Klingenberg (2022).

The second problem we encounter in relating transformations to Darwinism is Thompson’s insistence that independent variation of different characters is fundamental to natural selection in particular, and to a Darwinian worldview in general. He stresses that for transformations to occur (p. 726): “it is essential that our structure vary in its entirety, or at least that ‘independent variants’ should be relatively few.”

This stance was echoed later by Gould and Lewontin (1979) in their famous “spandrels” paper. These authors criticize what they call the “adaptationist programme” of neo-Darwinism (1979, p. 581) because it “proceeds by breaking an organism into unitary ‘traits’ and proposing an adaptive story for each considered separately.”

Gould and Lewontin’s (1979) criticism of some strands of neo-Darwinism seems very pertinent. However, as neo-Darwinism evolved from its Darwinian origins, there were many shifts of emphasis, which varied from one author to another. Criticism of the ways in which neo-Darwinism evolved are fine in themselves, but they do not necessarily apply to Darwin’s own thinking or writing.

In contrast to Gould and Lewontin (1979), Thompson was clearly criticizing Darwin himself when he said that the Darwinian view of evolution requires the “independent variants” model to be the rule rather than the exception (p. 727):

unless I am greatly mistaken, it is precisely on such a conception of the easy, frequent, and normally independent variability of parts that our conception of the process of natural selection is fundamentally based.

I fear that he was indeed mistaken on this point, as two quotes from The Origin of Species (below) will show. In footnote 646 (p. 727) Thompson goes even further: “Since the Origin of Species appeared, the terms variation and variability have always [italics added] referred to single characters.”

Here’s a final quote from Thompson on this subject (p. 727):

the morphologist, when comparing one organism with another, describes the differences between them point by point, and “character” by “character.” If he is from time to time constrained to admit the existence of ‘correlation’ between characters… yet all the while he recognises the fact of correlation somewhat vaguely… and he falls readily into the habit of thinking and talking of evolution as though it had proceeded on the lines of his own descriptions, point by point, and character by character.

Is this description true of Darwin, as Thompson apparently thought? In my view, it is not. Chapter 4 in The Origin of Species includes a section entitled “Correlation of Growth.” This begins with Darwin (1859, p. 182) explaining what he means by that phrase:

I mean by this expression that the whole organization is so tied together during its growth and development, that when slight variations in any one part occur, and are accumulated through natural selection, other parts become modified. This is a very important subject, most imperfectly understood.

He goes on to list some obvious examples, such as the tendency for left and right sides, or front and rear limbs, to vary in a correlated manner. Then he makes this comment (1859, p. 183): “These tendencies, I do not doubt, may be mastered more or less completely by natural selection.”

This is a very different perspective from that which Thompson criticizes—that of a morphologist who recognizes correlation of growth but then “falls into the habit” of thinking in terms of the evolution of independent characters. Darwin was not guilty of lazy, “habitual” thinking. Rather, he was making an important claim here—that the tendency to covary will itself vary in magnitude, and that natural selection can act on this latter type of variation to diminish a particular correlation of growth, or perhaps even eliminate it altogether.

This is a bold claim, and the extent to which it is true is not yet known even today, more than 150 years after Darwin made it. Experimental studies of the way in which natural selection and correlation of growth interact with each other are relatively recent (see below for an example using butterflies). It can be difficult to relate them to Darwin’s and Thompson’s accounts, because they often use different language. In particular, the phrase “correlation of growth” now seems somewhat archaic. More recent approaches to this subject have used a variety of different terms—not just different from the early accounts but different from each other too.

The disciplines that have contributed most to this issue are quantitative genetics and evo-devo. In the former, a phrase that is often encountered is the “covariance” of two characters (a technical measure of the degree of covariation). Quantitative geneticists attempt to estimate covariance, but are careful to distinguish two types: that of the phenotypic values of the characters concerned (e.g., limb lengths) and that of the genetic bases for these values. This distinction is important because, in a case where the covariance between two characters is entirely due to direct environmental effects on development, natural selection would not be able to modify it.

In evo-devo, different terms are used, most commonly developmental constraint (e.g., Gould 1989) and developmental bias (e.g., Arthur 2004). In a two-character context, these terms imply that the character values are linked. For example, in tetrapods high values of the character “forelimb length” usually go along with high values of the character “hindlimb length” (with some exceptions, such as tyrannosaurs). In other words, the developmental processes generating these values are causally linked, and the system is “constrained” or “biased” accordingly. Despite the different terminology from that of quantitative genetics, these two approaches are intrinsically connected; they are different ways of looking at the same thing. However, we must be careful about the exact way in which they are connected. For example, Cheverud (1984) says that “the genetic variance/covariance matrix of quantitative genetics measures developmental constraints.” Perhaps this is true in terms of the strengths and effects of the constraints, but the matrix says nothing of their causality. Ideally, we would also like to know in mechanistic terms how constraints or biases come about.

The crucial point, in terms of how evolution works, is whether selection can “break” developmental bias, or, to use Darwin’s words, whether selection can “master” correlation of growth. Perhaps the best set of experiments to date on this issue is that conducted by the research groups of Paul Brakefield and Patricia Beldade on covariation of eyespots in the African butterfly Bicyclus anynana (Beldade and Brakefield 2002; Allen et al. 2008). In particular, they examined the sizes and pigmentation patterns of two prominent eyespots on the dorsal forewings of these butterflies. In artificial selection experiments carried out in the laboratory, it was found that selection could break the covariance in eyespot sizes, but could not break it in the case of pigmentation patterns. This is an interesting result because it tells us that there is no simple general answer to the question of whether selection can or cannot break bias; rather, the answer depends on the specifics of the case. Of course, artificial and natural selection are different in various ways. Whether this case-by-case answer applies to natural selection just as much as it does to its artificial counterpart remains open, but I see no obvious reason why it should not.

Conclusions

D’Arcy Thompson’s idea of morphological transformations has been inspirational to many biologists, and to scholars more generally. However, while inspirational ideas have an important place in science, they do not suffice on their own to make scientific progress. Careful analyses, critiques, and testing of such ideas are equally important. Having subjected Thompsonian transformations to critical analysis here, what are my main conclusions? There are six of them:

  1. 1.

    The transformations that Thompson depicted, which I call morphological transformations, never occur in the real world, because evolution cannot directly alter the form of an adult of one species into the form of an adult of another.

  2. 2.

    In contrast, developmental transformations do occur. These involve the shape of an animal at one developmental stage being altered in a coordinated way into its shape at a later stage. Developmental transformations necessarily involve allometric growth, because, where growth is isometric, shape remains unchanged with increasing size.

  3. 3.

    Evolution of body form involves the repatterning of development. This can be piecemeal, involving changes in one character but not others, or it can be coordinated, involving related changes in many characters. In the latter case, we could use the term evolutionary transformation or an alternative one such as coordinated developmental repatterning.

  4. 4.

    Neither in development nor in evolution is there a clear cutoff between piecemeal (character-by-character) and coordinated (transformational) changes. Indeed, changes on both timescales often take the form of what Thompson called “partial transformations,” with some characters changing in a coordinated way, others not.

  5. 5.

    The causal agents of developmental and evolutionary change are different. A morphogen gradient is a common cause of developmental change. But such a gradient per se cannot cause evolutionary change. The latter requires a mutational alteration in the gradient, which results in altered development of an individual, followed by that individual’s progeny coming to dominate the population and species, for example via natural selection.

  6. 6.

    The above phrase “mutational alteration” needs to be interpreted broadly. There is not a single ancestral pattern of development that, following mutation of one of the genes that influence it, is altered into a single derived pattern. Rather, developmental variation is the norm in most populations most of the time. However, only heritable aspects of it can be the basis for evolutionary change. These include both genetic and epigenetic inheritance.

I would like to make one final point. D'Arcy Thompson is sometimes described as a structuralist (Gould 2002; Iurato and Igamberdiev 2020); and biological structuralism is sometimes perceived to be at odds with Darwinism (or selectionism or functionalism). I don’t feel that a perceived clash between such “isms” helps us much in our quest to better understand how evolution, development, and the interaction between the two work in causal terms. As I said earlier, it is not a question of a mathematical approach versus a biological one, nor a question of geometry versus statistics. Both the geometric form of individual animals (at all stages of their development) and the statistics of natural populations (at all stages of their history) are necessary components of an overall understanding of the field that has come to be known as evo-devo.