The old and new faces of morphology: the legacy of D'Arcy Thompson's ‘theory of transformations' and ‘laws of growth'


REVIEW

The old and new faces of morphology: the legacy of D’Arcy
Thompson’s ‘theory of transformations’ and ‘laws of growth’
Arhat Abzhanov1,2,*

ABSTRACT
In 1917, the publication of On Growth and Form by D’Arcy Wentworth
Thompson challenged both mathematicians and naturalists to think
about biological shapes and diversity as more than a confusion of
chaotic forms generated at random, but rather as geometric shapes
that could be described by principles of physics and mathematics.
Thompson’s work was based on the ideas of Galileo and Goethe on
morphology and of Russell on functionalism, but he was first to
postulate that physical forces and internal growth parameters regulate
biological forms and could be revealed via geometric transformations
in morphological space. Such precise mathematical structure
suggested a unifying generative process, as reflected in the title of
the book. To Thompson it was growth that could explain the
generation of any particular biological form, and changes in
ontogeny, rather than natural selection, could then explain the
diversity of biological shapes. Whereas adaptationism, widely
accepted in evolutionary biology, gives primacy to extrinsic factors
in producing morphological variation, Thompson’s ‘laws of growth’
provide intrinsic directives and constraints for the generation of
individual shapes, helping to explain the ‘profusion of forms, colours,
and other modifications’ observed in the living world.

KEY WORDS: D’Arcy Thompson, Growth and form, Theory of
transformations, Laws of growth, Evolution, Morphology,
Morphometrics

Before Thompson: Goethe’s science on natural forms
‘complete but never finished’

‘The beautiful is a manifestation of secret laws of nature, which but for
this phenomenon would have remained hidden from us for ever.’

Johann von Goethe (Bielschowsky, 1905)

There is a bewildering diversity of life on our planet, as it comes in a
huge variety of shapes and sizes ranging from bacteria to mollusks,
sea urchins, insects and birds. Multicellular organisms, both
animals and plants, have generated a remarkable panoply of
biological forms featuring bodies with segments, legs, tentacles,
wings and heads. Cataloguing, describing and explaining this vast
multiplicity of species, both living and extinct, have been some of
the main challenges for biological sciences in the past and remain so
today. In particular, we still have an incomplete understanding of
how particular biological forms are generated during development,
and how they change over evolutionary time. These questions
were a particular preoccupation of D’Arcy Wentworth Thompson
(Fig. 1A) in his book On Growth and Form (Thompson, 1917a),

which celebrates its 100th anniversary this year. In this Review, I
discuss some of the book’s key ideas in a historical perspective, in
particular geometric transformations of biological shapes and the
‘laws of growth’ underpinning biological diversity, and consider the
significance of these concepts to the modern developmental
genetics and evolutionary biology fields.

The first attempts to describe the appearance of animals and
plants, mostly for taxonomic reasons, started thousands of years ago.
Works by Aristotle and colleagues in classical times laid the
foundations of the science of morphology (morphé, meaning ‘form’,
and lógos, meaning ‘study’). This field of biology is interested in
overall appearance, both external (size, shape, colour and pattern)
and internal (anatomical features of inner organs). The main formal
principles for studying morphology were first formulated by Johann
Wolfgang von Goethe (1749-1832), a German naturalist and prolific
writer, renowned for both his scientific and literary works. The main
themes for Goethe and his followers have been to qualitatively and
quantitatively describe external and internal features of complex
organisms in an attempt to exploit their taxonomic significance
and to find their deeper biological meaning. Today, studies of
morphology include, but are not limited to, exploring and
understanding normal and abnormal variation, evolutionary
origins, developmental transitions or biomechanical and other
functions. As discussed below, a combination of Goethe’s
structuralism and the functionalism of Bertrand Russell provided
the true ideological and spiritual foundation for morphometric
mathematical work by D’Arcy Thompson, infused by an interest in
physical mechanics, as described in On Growth and Form.

Goethe’s study of morphology was both a benefactor and a
beneficiary of the growing interest in taxonomy. However, he also
realized the limitations of the Linnaean classification system based
on stereotypical assumptions about nature. Goethe wrote that:
‘Nature has no system; she has – she is – life and development from
an unknown centre toward an unknowable periphery’ (Goethe,
1823). This is important because his observations led him to believe
that nature’s patterns are not fixed – he was detecting and trying to
explain all kinds of transitions both between and within organisms.
‘From first to last, the plant is nothing but leaf’, Goethe writes in his
Italian Journey, implying that by transformations of the leaf shape
(expanding or contracting its parts during development), one might
expect to understand how unique shapes of other structures of the
same plant are generated (Goethe, 1817). He could see such
transitions apply across the different plant species as well, and from
such transitions Goethe conceptualizes an archetypal plant, or
‘Urpflanze’, an abstract common morphological denominator of all
plants seen and possible, either in the past or in the future. This is
still many years before the idea of an archetype takes on an
evolutionary meaning, specifically referring to ancestral forms. A
century before evolutionary theory, Goethe believed that all living
organisms changed under the inner physiognomic ‘drive to
formation’ or ‘Bildungstrieb’. In his publication titled First Sketch

1Department of Life Sciences, Imperial College London, Ascot SL5 7PY, UK.
2Natural History Museum, Cromwell Road, London SW7 5BD, UK.

*Author for correspondence (a.abzhanov@imperial.ac.uk)

A.A., 0000-0002-8577-8318

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© 2017. Published by The Company of Biologists Ltd | Development (2017) 144, 4284-4297 doi:10.1242/dev.137505

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mailto:a.abzhanov@imperial.ac.uk
http://orcid.org/0000-0002-8577-8318


of a General Introduction into Comparative Anatomy, Starting from
Osteology, Goethe deliberated an intrinsic law explaining the
balancing action of the Bildungstrieb in that: ‘nothing can be added
to one part without subtracting from another’ (Goethe, 1795).
Goethe’s Urpflanze, with its capacity for metamorphosis of

(plant) organs, along with the laterconcepts of Richard Owen (1804-
1892) on the animal ‘Bauplan’ (body plan) and its archetypal and
serial ‘homologies’ became crucial to our ability to relate different
species and their individual traits to each other (Owen, 1847, 1848).
From a static phenomenon worthy of a simple description,
morphology became a dynamic process that needs to be
considered both in the ontogenetic (developmental) and
phylogenetic (evolutionary) temporal dimensions. Based on
morphological observations on the anatomy of modern organisms
and their embryos, and fossils of related extinct species, the
relationship between ontogeny and phylogeny was very evident to
the early evolutionary and developmental biologists alike. First
proposed as a simple recapitulation, the ontogeny-phylogeny
connection turned out to be a much more complex, multilayered

and mutually interdependent phenomenon. Already back in the late
19th century, it was clear that understanding this relationship held
great promise for both nascent fields of embryology and Darwinian
evolutionary biology, and deserved the most rigorous investigation.
Many researchers of morphology were wondering whether a deeper
understanding of biological forms could provide important insight
into the developmental and phylogenetic principles and processes
that generated the morphologyof individuals and whole populations.
Such was the intellectual environment in the late 19th and early 20th
centuries in which D’Arcy Thompson was developing his ideas.

On ‘transformation of related forms’ and ‘laws of growth’

‘Philosophy [nature] is written in that great book which ever is before our
eyes…The book is written in mathematical language, and the symbols are
triangles, circles and other geometrical figures, without whose help it is
impossible to comprehend a single word of it; without which one wanders
in vain through a dark labyrinth.’

Galileo Galilei (The Assayer, 1623)

A B

C

D

E

F

Fig. 1. Connecting the growth and
form in morphospace. (A) D’Arcy
Wentworth Thompson circa 1906
Courtesy of University of Dundee
Archive Services, UK. (B) Geometric
transformations of human heads
drawn by artist Albrecht Dürer (Dürer,
1528; Fig. 366 from Thompson,
1917a). (C) Thompson’s comparative
illustrations of chimpanzee (left) and
human (right) cranial ontogenetic
shape changes (Figs 405 and 406
from Thompson, 1917a).
(D,E) Evolution of the horse skull from
Hyracotherium (Eocene) to the
modern horse, represented as a
coordinate transformation and to the
same scale of magnitude. (F) Diagram
of the Cartesian coordinates
projecting shape outlines of skulls in
the lineage from Hyracotherium to the
modern horse. A to H indicate
progressive changes in morphology
through evolution (Figs 401 and 402
from Thompson, 1961).

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The problem of shape (in the language of mathematics, shape is
form without scale) is as profound in biology today as in the time of
Goethe but it often receives less attention than it deserves. In the age
of advanced genetics, molecular biology and biochemistry, it is
even more important that researchers direct their main efforts to
explain biological form rather than focus exclusively on the
molecules and genes within it. The properties of biological shapes
are still far from completely appreciated at any organizational level,
and understanding them remains crucial to studies of the underlying
biomechanical and structural properties.
One very important aspect of morphology that Thompson liked to

ponder was that studying the natural laws and patterns found among
living forms could greatly benefit from the aesthetic awareness and
appreciation of nature. Thus, scientific observation of nature is
intertwined with the aesthetic experience of things orderly and
structured. This is important because in the age of Newtonian
physics, Galileo’s mathematics and Mendeleev’s chemistry,
governing natural laws and the resulting order were expected,
sought and indeed often found in nature. Thus, morphological
studies into natural patterns were a way to ‘the discovery of an
aesthetic truth’ provided for by the underlying natural laws (Brady,
1987). Such an ideology was a major inspiration for D’Arcy
Thompson’s search for the deeper meaning of complexity and
diversity of biological forms. Regarding his aspirations, Thompson
wrote: ‘We want to see how, in some cases at least, the forms of
living things, and of the parts of living things, can be explained by
physical considerations, and to realize that in general no organic
forms exist save for such as are in conformity with physical and
mathematical laws’ (p. 10, Thompson, 1917a). On Growth and
Form is a multifarious writing with parts on the structure and form
of cells and tissues, chapters on phyllotaxis and shapes of bird eggs
and shells of foraminifera. Thompson searched for and found
plentiful examples of correlations between biological forms and
mechanical phenomena and used them to emphasize the roles of
physical laws and mechanics. This is where Thompson’s thinking
about morphology echoes the ‘functionalism’ of Bertrand Russell
(1872-1970), a British philosopher and mathematician who
searched for physical causes of multiple physiological and
behavioural phenomena.
One of the main themes of the book is to argue that the form of

any object, including that of the living organism, is a ‘diagram of
forces’ that created and maintained it. Thus, a shape (of an
organism, organ or cell) can be seen and studied as an impression of
the generative forces that have acted upon it. Such thinking is at the
foundation of chemistry, physics, astronomy and other sciences.
Modern biology is now moving beyond the notion that observed
biological shapes are largely driven by unsystematic processes, such
as random mutations and genomic rearrangements, followed by
adaptation to rather unpredictable and changing environmental
factors. Thompson argued, for example, that shapes of individual
cells are produced by surface tension on their membranous walls as
if they were bubbles of soap [see Graner and Riveline (2017) for
further discussion of this aspect of Thompson’s work]. He
discussed the shape of bird eggs in different species (from the
spherical owl egg to the pointed guillemot egg) in relation to the
process that produces them as they rotate and slowly pass through
the oviduct – as described in Thompson’s earlier publication in
Nature titled On the Shapes of Eggs and the Causes Which
Determine Them (Thompson, 1908). In a series of chapters,
Thompson reflected on the role of spiralling forms (in particular the
presence of particular types of spirals, such as the equiangular or
logarithmic spirals), both in terms of mechanics and construction

under the influence of forces in structures and taxa as varied as the
horns of ruminants, claws of birds, and the shells of living and
extinct mollusks.

Probably the most famous set of observations in On Growth and
Form comes from Thompson’s attempts to relate biological shapes
to each other via geometric transformations. In so doing, he was
seeking to find mathematical logic in biological shapes by applying
methods developed 400 years earlier by Albrecht Dürer, a German
renaissance artist, and described in De Symetria Partium in Rectis
Formis Humanorum Corporum Libri (Dürer, 1528). To study
human proportions, Dürer skilfully used transformations of human
heads drawn against a coordinate grid to understand facial variation
(Fig. 1B). This was a very powerful way to demonstrate how
otherwise disparate shapes can be meaningfully compared. What
makes such a method particularly useful for analysing biological
shapes is that it allows for continuity, for gradual ontogenetic or
phylogenetic transformations of whole organisms and their body
parts.

Thompson was not so much interested in capturing a single
biological shape as in its relationship with other such shapes. He
wrote that: ‘Our essential task lies in the comparison of related
forms rather than in the precise definition of each; and the
deformation of a complicated figure may be a phenomenon easy of
comprehension, though the figure itself have to be left unanalysed
and undefined’ (p. 723, Thompson, 1917a). This statement is
important for two reasons. First, the shapes are compared by the
method of ‘deformation’, the point-by-point transformation of one
shape into another, which allows for a visual understanding of both
objects. When placed on a square grid, such transformation
becomes a geometric process that can be described
mathematically, both qualitatively and quantitatively (Fig. 1C-F).
Second, Thompson compared ‘related forms’ and insisted on the
futility of directly relating shapes of distantly related taxa. Most or
all comparisons that Thompson himself made are between
individuals of the same species or class of animals or plants. He
described his analytic approach: ‘This process of comparison, of
recognizing in one form the definitive permutation of deformation
of another, apart altogether from a precise and adequate
understanding of the original type…finds its solution in the
elementary use of a certain method of the mathematician. This is
the Method of Coordinates, on which is based the Theory of
Transformations.’ (p. 723, Thompson, 1917a). Comparisons between
truly different forms, even if they have similarly named parts,
become meaningless as these can no longer be derived from each
other either developmentally or evolutionarily. Some species could
have features that are missing or have no identifiable counterparts in
other species but, in general, related species can be compared.
Thompson believed that the very fact that two distinct crustaceans or
fishes could relate to each other by a geometric transformation ‘will
of itself constitute a proof that variation has proceeded on definite
and orderly lines, that a comprehensive “law of growth” has
pervaded the whole structure in its integrity’ (p. 727, Thompson,
1917a). To master evidence, he combined techniques learned from
Dürer and Descartes, used proportional drawings of biological
forms on a Cartesian grid and subjected them to increasingly
complex mathematical transformations.

Mathematical experimentation with biological shapes allowed
Thompson to reveal and visualize connections between biological
shapes whether during development, such as the growing and
maturing skulls of primates or growing plant leaves (Fig. 2A,B),
or through evolution (Fig. 1C). One of the most striking (and
famous) of Thompson’s examples of phylogenetic change in

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shape as a geometric transformation is evolution of the skull shape
from an early horse ancestor Hyracotherium (which lived in
Eocene) to the modern horse Equus, represented as a coordinate
transformation and to the same scale of magnitude (Fig. 1D-F).
The principles and methods he used for this analysis are at the
foundation of the modern field of geometric morphometrics (the
analysis of 2D and 3D shapes using Cartesian geometric
coordinates) (Klingenberg, 2010; Adams et al., 2013). The
entire concept of ‘morphospace’ – mathematical space within
which multiple shapes could be placed and compared – was,
arguably, invented by Thompson. The diagram of the Cartesian
coordinates projecting shape outlines of skulls representing steps
in horse evolution onto the same grid (Fig. 1F) allows one to
capture and visually demonstrate the continuous and gradual
nature of this particular evolutionary story. Later in the 20th
century, a more definitive set of approaches was developed, in the
form of geometric morphometrics, by paleontologist David Raup
and developmental biologist Pere Alberch (Raup, 1961, 1966;
Alberch et al., 1979; Oster and Alberch, 1982; Adams et al.,
2013). These allowed researchers to address a wide and ever
growing range of biological problems from coiling shell shape
distributions in extinct gastropods to the developing digits of
amphibians and other tetrapods (Oster and Alberch, 1982; Oster
et al., 1988; Klingenberg, 2010; Adams et al., 2013).
Using geometric transformations on related organisms,

Thompson believed that he was demonstrating that biological
forms were, indeed, related and that their shapes were produced by
what he called the ‘laws of growth’. What are these ‘laws’ and why
might this concept still be important today? Thompson’s extensive
modelling with biological shapes led him to conclude that the
growth of an individual organism can be generalized to all of the
individuals within a species, or even a group of related species.
Thompson’s ‘laws of growth’ referred to empirically derived and
theoretically envisaged principles, which applied to all patterns of
biological growth with the resulting shapes molded by development
and influenced by the physical environment surrounding the

growing organism (Fig. 2C). Although Thompson never attempted
to explicitly explain the ultimate causes for the transformations he so
carefully observed, it is clear that he thought every biological shape
to be a reflection of the ‘diagram of forces’ that acted upon it and he
believed that these generative forces were largely internal.
Thompson wrote: ‘The deep-seated rhythms of growth which, as I
venture to think, are the chief basis of morphological heredity, bring
about similarities of form which endure in the absence of conflicting
forces; but a new system of forces, introduced by altered
environment and habits, impinging on those particular parts of the
fabric which lie within this particular field of force, will assuredly
not be long in manifesting itself in notable and inevitable
modifications of form’ (p. 717, Thompson, 1917a). He was well
aware of the science of embryology, its main postulates and
advances made by key embryologists of the time, such as Karl Ernst
von Baer, Wilhelm Roux and Ernst Haeckel. Embryology (better
known as developmental biology today) was already an important
scientific field in his day, making a great impact on the minds of
biologists and providing clues about the mechanisms generating
morphological diversity. However, although the ‘laws of growth’
were conceptually inspired by these new exciting studies, the exact
mechanisms controlling individual development in the early 20th
century remained largely unknown. Today, we give a much more
mechanistic explanation to Thompson’s ‘laws of growth’,
associating them with the entire panoply of developmental
processes at all hierarchical levels from gene sequence to cell
proliferation, differentiation and signalling, tissue- and organ-level
phenomena to full organismal complexity (Fig. 2C).

‘I suppose everyone must admit that there are “laws of growth”…
but after all one does not feel sure how far this is really admitted’,
Thompson mused in a letter to a friend (Thompson, 1889). He
realized that his ‘theory of transformations’, despite numerous
explained case studies, directly accounted for only a very small
fraction of the existing biological diversity. It was also far from clear
in Thompson’s times how the ‘laws of growth’ could be related to
the Darwinian evolutionary process.

Genes

Environment

Developmental mechanisms
(‘laws of growth’)

Morphological variation

Fig. 2. Biological shapes,
transformations and the ‘laws of growth’.
(A) Geometric shape changes during
hyacinth leaf growth, which follows a very
specific set of spatial rules, e.g. following
particular ratios of radial and tangential
growth velocities (Fig. 359 from Thompson,
1917a). (B) Geometric morphometric space
describing shape alterations in hyena skulls
during development from juvenile to adult
(Tanner et al., 2010). (C) The ‘laws of
growth’ interpreted as a broad set of
developmental mechanisms translating
genetic information and physical forces of
the environment into individual biological
shapes and contributing to their diversity.
Images in C created by Keith Chan;
copyright Alexmit (www.fotosearch.com);
animal images are illustrations from Seton
(1898).

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Pecking for the origins of morphological variation

‘I have stated, that in the thirteen species of ground-finches, a nearly
perfect gradation may be traced, from a beak extraordinarily thick, to one
so fine, that it may be compared to that of a warbler.’

Darwin (1839)

Thompson’s work was well ahead of its time in many respects. As
already mentioned, it provided a powerful paradigm for the field of
geometric morphometrics and it remains the most celebrated attempt
to quantify the morphological diversity observed in the natural world
(Adams et al., 2004; Arthur, 2006; Slice, 2007; Pappas and Miller,
2013; Polly and Motz, 2017). There have been numerous studies that
have successfully applied Thompson’s ideas to a variety of biological
forms (e.g. Garnier et al., 2005; Depecker et al., 2006; Bhullar et al.,
2012; Drake et al., 2017; Klein and Svoboda, 2017; Fabbri et al.,
2017). However, only relatively recently have we begun to connect
the ‘theory of transformations’ to phylogenetic studies and
developmental genetics and to explain the origins of morphological
diversity (Weston, 2003; Larson, 2005; Klingenberg and Zaklan,
2000; Klingenberg, 2010). To be informative, the geometrical
transformations related to morphological variation in different
species must themselves be related to each other in a way that is
meaningful in terms of both phylogeny and the underlying
developmental genetics of morphogenesis. Can such connections
be shown using modern methods and approaches on specific
illustrative case studies? What do they tell us about the role of
development in morphological evolution?
A classical textbook example of morphological diversity is

Darwin’s finches (Thraupidae), much of whose success can be

attributed to beak shape variation (Darwin, 1845; Lack, 1947;
Bowman, 1961; Grant, 1986). These birds inhabit the Galápagos
Islands and comprise a monophyletic group of 15-16 closely related
species that have been described as case studies on adaptive
radiation, niche partitioning, and rapid morphological evolution
(Grant and Grant, 2008). In fact, Darwin’s finches occupy
ecological niches normally occupied by different families of birds
on the mainland, such as warblers, finches, thrushes, grosbeaks and
woodpeckers (Grant, 1986). Molecular phylogenies suggest that all
members of this group retained and exploited the beak shape that
they inherited from the last common ancestor, echoing Charles
Darwin, who first speculated that: ‘From an original paucity of birds
in this archipelago [Galápagos], one species had been taken and
modified for different ends’ (Darwin, 1845). After more than
100 years, morphological and ecological studies have identified key
components of bill morphology of Darwin’s finches and established
their adaptive significance (Lack, 1947; Bowman, 1961; Grant,
1986, 1999; Herrel et al., 2005; Foster et al., 2008). Recently, their
diversity was examined from a different perspective to understand
whether there was a mathematical structure underlying divergent
bill shapes that can be connected both to their phylogenetic relations
and bill developmental genetics (Campàs et al., 2010). Darwin’s
finch beaks are known to differ in overall size as well as depth, width
and length, so it was hypothesized that bill shapes in these species
might differ simply by their scales, and thus it might be possible to
superimpose their bill shapes onto a single common shape after
normalizing each axis with its corresponding scale. Mathematically,
this normalization is equivalent to a scaling transformation, in
which each axis is stretched by a constant scaling factor (Fig. 3).
When beak shapes of Darwin’s finches were analyzed to determine

Scaling Shear

Group A

Group B

Group C

Depth

Length

Fig. 3. Geometric relations among the beaks of Darwin’s finches. (Left) Lateral profiles of beaks of Darwin’s finches. (Centre) Group structure under scaling
transformations focusing on the upper beak profile: untransformed shapes and shapes collapsed onto a common shape via scaling transformations.
(Right) Collapse of all group shapes onto a common shape via a composition of shear and scaling transformations suggesting a two-tier morphological variation
(adapted from Campa ̀s et al., 2010).

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whether they are related by a scaling transformation, species
clustered according to the similarity of their collapsed profiles
(Campàs et al., 2010).
The geometric analysis clearly identified three morphological

groups, within which the bill shapes are related through scaling
transformations alone. The first group (group A) corresponded to
the genus Geospiza in addition to the black-faced grassquit (Tiaris
bicolor), representative of a group basal to Darwin’s finches; the
second group (group B) corresponded to the tree (Camarhynchus),
Cocos (Pinaroloxias inornata) and warbler (Certhidea) finches;
whereas the third group (group C) consisted of a single species, the
vegetarian finch (Platyspiza crassirostris). Thus, scaling
transformations accounted for a substantial part of the variation
observed in the beak shapes of Darwin’s finches by reducing the
complexity from 15 original beak shapes to three different (group)
shapes. However, the differences between ‘group shapes’ could
not be explained by scaling (Fig. 3). Mathematically, scaling
transformations form a subgroup of affine transformations (linear
mapping methods that preserve points, lines, curves and planes,
allowing one to map points in one shape to points in another), which
also includes shear transformations (Campàs et al., 2010). When
shear transformation along the bill depth axis was added, all beak
shapes of Darwin’s finches collapsed onto a single common shape.
Thus, remarkably, not only could the diverse beak shapes of
Darwin’s finches be transformed into each other pairwise, all of
them could be transformed into the ancestral shape provided by a
molecular phylogeny (Campàs et al., 2010). This geometric analysis
demonstrated that the (species- and genus-level) beak shapes of all
Darwin’s finches are related by affine transformations, characterized
by precisely three parameters: the depth and length for the scaling
transformation and an additional parameter measuring the degree
of shear.
The general significance of such two-tier variation detecting the

same pattern of hierarchical collapse of shapes was further
confirmed by performing a pairwise comparison of the beak
shapes in about 200 bird species spanning a significant section of
Passeroidea, including most tanager and cardinal relatives of
Darwin’s finches (Thraupidae and Cardinalidae) (Fritz et al.,
2014). Such a precise morphological pattern echoes the
observations of Goethe, who commented that: ‘The laws of
transformation according to which nature produces one part
through another and achieves the most diversified forms through
the modification of a single organ’ (Goethe, 1790); and suggests a
highly structured and versatile generative developmental process.
To explain such morphological diversity at a mechanistic level

requires an explicit connection between the genes involved in
shaping the beak during development and the parameters
characterizing the observed geometric transformations. Analysis
of naturally occurring hybrids between species of Darwin’s finches
and the more recent comparative analysis of genes expressed in the
developing beak primordium suggests that avian beak morphology
is a polygenic character regulated by multiple developmental genes
(Grant, 1993; Grant and Grant, 2002, 2015; Grant and Grant, 2008;
Lawson and Petren, 2017). For example, changes in the expression
of Bone morphogenetic protein 4 (Bmp4) control the depth and
width in the embryonic cartilage component of the upper beak
skeleton, which forms first and establishes species-specific beak
shape during embryonic development. The Bmp4 expression level
and pattern in the frontonasal mass (upper beak primordium)
display a strong correlation with the scaling factors and quantify a
large portion of the adult beak morphological diversity (Abzhanov
et al., 2004). Other developmental genes, such as CaM, TGFβIIr,

β-catenin and Dkk3 form a regulatory network and together explain
other scaling parameters, such as beak length or width (Abzhanov
et al., 2004, 2006; Mallarino et al., 2011, 2012) (Fig. 4B-D). The
necessity of including shear transformations to explain the full
morphological variation in the beaks of Darwin’s finches suggests
the involvement of more significant and as yet unknown
developmental changes controlling the exact curvature of beak
profiles characteristic of different ‘group shapes’. The uncovered
hierarchical morphological structure is likely to be related to the
hierarchical structure of developmental regulatory networks, which
are thought to be the proximate cause of evolutionary changes in
morphology. These could include modifications at different stages
of embryonic development and the involvement of different types of
developmental mechanisms, such as diffusible morphogenetic
signals, the pre-patterning of skeletal condensations resulting in
distinct morphogenetic maps using transcription factors, and
regulation of the planar polarity of cell division during growth.

Mechanisms specifically controlling shear transformation are
suggested by the developmental strategy by which the beak grows
(Fig. 5A). When the primordial beak forms, it contains a large zone of
actively proliferating cells, termed a growth zone. As the beak grows,
this group of diving cells is gradually depleted and the growth zone
decays at a constant rate until it completely disappears at the end of
beak development (Fritz et al., 2014). Thus, the conical shape of
the beak (a sectioned parabolic cone to be exact) is produced by
an ‘envelope’ of all of the growth zone shapes observed over
developmental time (Fig. 5B). The behaviour of this growth zone
determines both the scaling-based and shear-based transformations.
While scaling can be explained by diffusing molecules,such asBMP4,
signalling to the growth zone from the outside (Campàs et al., 2010),
the distinct beak curvatures that distinguish the different beak ‘group
shapes’ may rely on the precise and coordinated internal alignment of
the cell division planes of proliferating cells relative to the main axis of
growth (Fritz et al., 2014).

Modularity of the cranial skeleton and evolution of diverse
skull shapes

‘We tend, as we analyse a thing into its parts or into its properties, to
magnify these, to exaggerate their apparent independence, and to hide
from ourselves (at least for a time) the essential integrity and individuality
of the composite whole.’ (p. 712, Thompson, 1917a)

Studying the vertebrate skull in its full complexity is particularly
challenging and intriguing. The vertebrate head is a fascinating part
of the body, with its intricate organization and multifunctional
design. With a host of taxon-, age- and sex-specific features and
highly adaptive characteristics, an image of the vertebrate head with
all its attributes can often instantly tell us what species the animal
belongs to and provide information about its ecology. The exact
anatomy of the head, the overall shape of the skull and shape of the
individual skeletal parts and how they are integrated with the brain,
eyes and muscles all reveal the animal’s natural history. There are
now many morphological studies showing how geometric
transformations of even anatomically complex multi-part
structures – such as tetrapod limbs, vertebral columns and bony
crania – allow for direct comparisons in the most morphologically
diverse clades, such as cichlid fishes, anole lizards and phyllostomid
bats (Cakenberghe et al., 2002; Clabaut et al., 2007; Monteiro and
Nogueira, 2011; Muschick et al., 2012; Sanger et al., 2012, 2013;
Wilson et al., 2015).

Early in embryonic development, the faces of all amniote
vertebrates (mammals, reptiles and birds) are remarkably similar

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despite the substantial phenotypic differences observed in adults
(Hall, 1996; Raff, 1996). To understand the evolutionary
significance of such conservation at the developmental level,
geometric morphometric analyses have been conducted on the
entire face during the whole developmental trajectory for all major
amniote groups (Young et al., 2014). The faces of early embryos
appear as an assemblage of protruding tissue buds filled with cells,
which need to expand and fuse in a particular order to form the

cranium. This comparative analysis found that the most conserved
period of amniote facial shape coincides with the time when fusion
of prominences occurs. Next, comparative morphospace was used to
infer principles of facial growth and to predict potential but
unrealized shapes. Surprisingly, deviations from the conserved
early embryonic trajectories resulted in increasing mismatch in the
shape and size of craniofacial prominences, which increased the
likelihood of clefts (incomplete fusions) and later abnormalities.

*

Fig. 4. Evolution of beak shape diversity in songbirds and developmental mechanisms for beak morphogenesis. (A) Exceptionally high diversity of beak
shapes in Tholospiza (Darwin’s finches on Galápagos and endemic Caribbean relatives, marked with an asterisk) and Hawaiian honeycreepers as compared with
the much more limited morphological diversity in Galápagos mockingbirds (and a single Caribbean endemic Bahama mockingbird Mimus gundlachii, also marked
with an asterisk). Tholospiza and honeycreepers show much higher degrees of sympatric distribution (related species co-inhabiting the same geographic areas),
while Galápagos mockingbirds (and other Galápagos land birds) display allopatric speciation with just four morphologically similar endemic species for the entire
Galápagos archipelago. (B) Highly modular multigenic beak developmental programs are found in Darwin’s finches and close Tholospiza relatives but not in the
more basal species (modified after Mallarino et al., 2012). Darwin’s finches and the closely related Caribbean species Loxigilla noctis deploy multiple regulatory
genes to control both prenasal cartilage and premaxillary bone parts of the developing beak, whereas the more basal species use a much more limited set of
genes to regulate the shape of premaxillary bone (genes expected to be deployed based on Galápagos species are not expressed). The red arrow in the
Tholospiza phylogeny points at the phylogenetic position within this group beyond which higher morphological diversification is observed and needs to be
explained (phylogeny from Burns et al., 2002; modified after Mallarino et al., 2012). (C) Within Darwin’s finches, species with deep beaks have strong expression
of Bmp4, TGFβIIr, β-catenin and Dkk3, whereas expression of CaM is upregulated in species with elongated beaks, such as Geospiza scandens and Geospiza
conirostris. BMP4 and CaM act independently to alter the growth of the prenasal cartilage and TGFβIIr, β-catenin, and Dkk3 regulate the premaxillary bone.
(D) Darwin’s finches and the closely related Caribbean L. noctis share beak developmental programs, which are highly modular in terms of both tissue
and molecular composition, while the more basal species Loxigilla portoricensis and Loxigilla violacea have simpler beak developmental patterns and
represent a simpler starting condition (also found in outgroup species such as zebra finches). Illustrations of the head of each bird species were derived from
del Hoyo et al. (2017) with permission from Elisa Badia. Illustrations of Hawaiian honeycreepers are by H. Douglas Pratt (Pratt, 2005).

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Moreover, experiments on live chicken embryos that aimed to alter
the signalling environment in the early face forced the
developmental trajectory into the normally unoccupied
morphospaces and resulted in abnormalities, such as clefts of the
primary palate (Young et al., 2014). It appears that fusion of
craniofacial prominences is a strong selective filter against
developmental shape variation, explaining morphological
conservation at the early stages. Once this critical stage is passed,
phenotypic diversity increases sharply. This, in effect, is an example
of the powerful, if constraining, role of the intrinsic developmental
mechanisms in generating morphological diversity.
What is the role of modularity in the generation of diversity?

Recently, a detailed comparative geometric morphometrics analysis
was reported of 3D skull shapes in both Darwin’s finches and
Hawaiian honeycreepers within the same morphospace using X-ray
microcomputed tomography (µCT) scans of their cranial skeletons
(Tokita et al., 2017) (Fig. 6A-D). The Hawaiian honeycreepers and
Darwin’s finches have both evolved remarkable levels of adaptive
cranial morphological variation, and this analysis demonstrated that
cranial shapes in both groups are much more diverse than in their
respective outgroups (Fig. 4A) (Tokita et al., 2017). The Hawaiian
honeycreepers as a group displayed the highest skull shape diversity
and disparity of all the bird groups studied. Interestingly, Darwin’s
finches showed strong covariation between the shape of the whole
skull and those of the upper beak, orbit, palatine and adductor
chamber. By contrast, in Hawaiian honeycreepers, the parts of the
skull are less strongly coupled with the shape of the whole skull
(Fig. 6D). Such results suggest that the high level of disparity in
skull morphology observed in Hawaiian honeycreepers is
associated with changes in modularity and integration of
individual skull elements, allowing for more evolutionary
flexibility to explore the morphospace. Modularity here refers to
the ability of a biological system to organize individual and discrete
units that can increase the overall flexibility of the system. This tends
to facilitate selective forces, whereas integration exerts an opposite
effect as it works to match and bind the modules together (Hall and
Olson, 2001). Similarly, studies on mammalian and fish skulls
indicated a significant role for changes in skull integration and

modularity in both the extent and directionality of skull shape
changes (Goswami, 2007; Drake and Klingenberg, 2010; Goswami
et al., 2014). The exact developmental mechanisms controlling
integration and modularity of the skull skeletal elements are not yet
known but further studies should reveal how the ‘laws of growth’
actually operate as they shape vertebrate cranial diversity.

Geometry of life on a large scale: transformation from reptile
to bird

‘The many diverse forms of Dinosaurian reptiles, all of which manifest a
strong family likeness underlying much superficial diversity, furnish us
with plentiful material for comparison by the method of transformations.’
(p.754, Thompson, 1917a)

As predicted by D’Arcy Thompson, comparisons based on
geometric transformations can be successfully applied to both
small and large evolutionary scales with reasonable success. For
instance, modern birds represent a surviving group of theropod
dinosaurs and their unique skulls are morphologically radically
different from those of their reptilian relatives and ancestors and any
other vertebrates. Among major innovations of the avian head are
the toothless beak derived from the fusion of premaxillary bones,
uniquely shaped palatines (bones that form the roof of the mouth),
highly reduced face/snout, and a hugely expanded brain and
overlying domed cranial roof (Fig. 7A,B). The nature of the reptile-
to-bird transition became clearer when a geometric morphometric
study was performed integrating developmental, neontological and
palaeontological data, which revealed that paedomorphosis, by
which descendants resemble the juveniles of their ancestors, was
responsible for several major evolutionary transitions in the origin
of birds (Bhullar et al., 2012). The same set of variable skull
landmarks was analyzed across extant and extinct members of
Archosauria (‘ruling reptiles’), from the basalmost taxa such as
Euparkeria to the early dinosaur Herrerasaurus, non-avian
theropods (e.g. Guanlong), crocodilians, primitive birds
(Archaeopteryx, Confuciusornis and Yixianornis) and modern
birds (Fig. 7B) (Bhullar et al., 2012). Adult skull shapes were
analyzed together with those of juveniles and embryos, wherever

‘ ’

Fig. 5. Principles of beak development in songbirds. (A) Snapshots of growing beak for embryonic day (E) 5-9, showing developing beak outlines (black), the
size of the growth zone (red), its centroid (blue) and the relevant length scale for shaping the upper beak profile (yellow). All measures of the growth zone
are derived from midsagittal cross-sections of zebra finch embryo beaks, stained to show cell nuclei (blue) and dividing cells (green). Areas with a high density
of dividing cells are defined as the growth zone (red outline). (B) The final conical shape of the beak is given by an ‘envelope’ of the growth zone observed
during developmental time, which decays at a constant rate until it shrinks to size zero generating a tip (adapted and modified from Fritz et al., 2014). All
beak shapes are well fitted by equations, as shown, that describe beak shape and the profile [pu(x) and pl(x)] generated by a fixed growth law and growth zone
decay, where ϕ is a vector parameterizing the shape and all other free parameters in the problem. The midsagittal sections of songbird beaks are all calculated to
be conic sections.

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available, within the same morphospace. Basally branching bird
relatives and primitive birds clustered with embryos and early
juveniles of other archosaurs, indicating paedomorphosis (Fig. 7B).
This paedomorphosis is most evident in their relatively shorter face,
enormous eyes and enlarged brains (Bhullar et al., 2012). The
paedomorphic trend holds even after body size and phylogeny are
statistically accounted for: non-avian dinosaurs such as
Compsognathus are small, yet its skull shape falls within the
ancestral adult cluster, while ostrich and emu (Dromaius) are very
large but their skulls group with other birds. This morphometric
analysis revealed at least four paedomorphic episodes (marked I-IV
on Fig. 7B) in the history of birds, which connect ancestral skull
shapes of early archosaurs through a series of morphological
transitions to modern birds (Fig. 7B). This evolutionary story covers
over 240 million years and represents one of the most dramatic

morphological transitions in animals yet can be addressed entirely
within the framework provided by Thompson’s ‘theory of
transformations’.

Despite the overwhelming imprint of paedomorphosis on the
early evolution of the bird skull seen in the collapse of the face and
the enlargement of the brain, localized peramorphosis, a trend which
is the reverse of paedomorphosis where the descendants mature past
the ancestral adulthood condition and evolve previously unseen
traits, also occurs later in the bird lineage and explains, for example,
the origin of the distinctive avian beak structure (Bhullar et al., 2012,
2015). The avian beak was an evolutionary innovation, formed
from the fusion and elongation of premaxillary bones and uniquely
shaped palatines, which allow for kinesis (upper beak movement
relative to the skull). This ‘new snout’ of birds, the bill, permitted
vast diversification in avian ecology. A combined approach

Fig. 6. High level of morphological disparity in
avian skull shapes is associated with
increased modularity. (A) Microcomputed
tomography (µCT) scan of a bird skull showing
geometric morphometric landmarks (above, red
dots on skull scan) and defined modules
(beneath, coloured lines). (B) The phylogeny of
bird species used for mapping shape data by
squared-change parsimony. Representative,
more basal, species were used as outgroups.
(C) Morphospace occupied by skull shapes. The
polygons show parts of the morphospace
occupied by different groups revealing that
Hawaiian honeycreepers (polygon shaded blue)
have some of the most diversified and divergent
skulls of all species studied. (D) Darwin’s finches
show strong covariation between the shape of the
whole skull and several skull modules (red thick
lines), suggesting strong integration within the
skull. By contrast, Hawaiian honeycreepers show
strong covariation only between the shape of the
whole skull and that of the upper beak (blue thick
line), indicating much greater flexibility of
evolutionary change. Such a unique pattern of
morphological integration in their skulls is likely to
reflect an altered and much higher state of
developmental modularity within the skull, which
contributed to the outstanding level of disparity of
skull morphology observed in Hawaiian
honeycreepers. Cranial modules in D are colour-
coded in the same way as in A. All images adapted
and modified from Tokita et al. (2017).

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bridging paleontology, comparative anatomy and experimental
developmental biology showed that the avian beak occupies a
distinctive part of the facial morphospace within the archosaurs. To
expose underlying developmental mechanisms, developmental gene
expression was studied in the embryonic face (Bhullar et al., 2015).
This study revealed a novel mid-facial expression domain of Fgf8 in
early bird embryos and the downstream mid-facial WNT-responsive
region at later stages. Altering Fgf8 expression in chicken embryos
using specific antagonists altered later WNT responsiveness to the
ancestral pattern. The resulting experimental skeletal phenotypes
clustered geometrically with ancestral fossil snouted forms instead of
with beaked birds (Bhullar et al., 2015). Other studies have
highlighted the roles of other signalling molecules, such as SHH
and BMP, in the origin and large-scale evolution of beak shapes (Hu
et al., 2015a,b; Smith et al., 2015). These studies are good examples
of how the ‘laws of growth’ can be understood in more precise and
mechanistic terms.

Thompson’s ‘laws of growth’ versus Darwin’s natural
selection

‘I have tried to make it as little contentious as possible. That is to say
where it undoubtedly runs counter to conventional Darwinism, I do not
rub this in, but leave the reader to draw the obvious moral for himself.’

D’Arcy Thompson on On Growth and Form (Thompson, 1958)

The reception of D’Arcy Thompson’s 1917 book (edited in 1942
and 1961) was extremely wide-ranging as it drew different reactions
from a very diverse audience. Developmental biologists were eager
to understand the ‘laws of growth’ and were mostly inspired or
otherwise influenced in a positive way. Mathematicians and applied
physicists found it fascinating, and attempts to perfect Thompson’s
approaches continue to this day. It had a great impact on the
immunologist Sir Peter Medawar and theoretical biologist Joseph
Henry Woodger, philosopher Michael Ruse, architects Philip

Fig. 7. Geometric morphometrics analysis of skull shapes in Archosauria reveals paedomorphosis (retention of juvenile features) at the origin of birds.
(A) Photographs of fossil skulls of the basal theropod dinosaur Coelophysis (early juvenile and adult) and first bird Archaeopteryx ( juvenile and adult). (B) Several
transitions are revealed during evolution of birds from non-avian theropod dinosaurs associated with heterochrony (changes in developmental timing). I-IV refer
to heterochronic (changes in developmental timing) transitions during evolution of modern avians. Transition I is the most dramatic paedomorphic transition,
during which skulls of birds broke the existing trend towards more elongated snouts and acquired juvenile-like shapes. Other transitions (II-IV) produced
features found in modern birds, such as toothless beaks, larger brain and eyes. Grey spots show positions of all other archosaur taxa analyzed in this study; only
the most important taxa are highlighted. All images modified from Bhullar et al. (2012).

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Beesley and Sarah Bonnemaison, as well as countless designers and
artists who were mesmerized by the idea of organismal
transformations (Medawar, 1945; Woodger, 1945). Both Julian
Huxley and Stephen J. Gould, famous evolutionary biologists, were
motivated by Thompson’s ideas to produce their own theories
(Huxley, 1932; Gould, 1977, 2002). However, many contemporary
evolutionary biologists found it a challenge to accept the concepts
explaining biological forms by physical or largely internal
biological processes. They thought that the book was arguing
against Darwinian evolution, and many believed that by using his
famous transformation diagrams, D’Arcy was arguing that ‘laws of
growth’ rather than evolution could be used to explain the different
forms of related species. At the British Association meeting in 1894,
Thompson gave a lecture titled Some Difficulties in Darwinism. It
was never published in a journal or a book but was briefly
summarized by a Nature correspondent: ‘He doubts the efficacy of
the struggle for existence in the case of humming-birds, etc., and in
these cases he regards the profusion of forms, colours, and other
modifications as due merely to laws of growth, and thinks that
growth may be more exuberant in the absence of struggle and
hardship’ (Anon, 1894). In other words, the general perception was
that Thompson saw natural selection more as an impediment to
creating diversity rather than as a generative force, and he was
considered a ‘stubborn opponent’ to Darwinism (Gould, 1971). Was
Thompson completely wrong in his conviction?
A century later, this dispute still might not be completely settled.

Consider how adaptive radiations are explained in a standard biology
textbook (Simpson, 1953; Futuyma, 2009; Zimmer and Emlen,
2012). Adaptive radiation, or the rapid evolution of morphologically
and ecologically diverse species from a single ancestor, usually
implies two coincidental processes: multiplication of species number
(increased taxonomic diversity defined as the ‘species richness’) and
increased phenotypic disparity (defined as ‘morphological
diversification’). Alternative scenarios include increases in species
richness without major morphological diversification, and dramatic
morphological changes in few isolated lineages. The importance of
adaptive radiation as an evolutionary phenomenon was first
recognized by George Gaylord Simpson in The Major Features of
Evolution (Simpson, 1953). Much of the current species richness,
ecological and morphological diversity in such clades as birds,
angiosperms and mammals, is the result of multiple radiation events
(Moen and Morlon, 2014; Gill, 2007). As already mentioned, some
of the best-studied textbook examples of adaptive radiations are
found in birds: Darwin’s finches, Hawaiian honeycreepers and
Madagascar vangas. All of these avian clades display an unusually
high degree of beak morphological variation, which is matched by a
diversity in diets and feeding behaviours, increased species richness
and sympatric distribution of their species (Fig. 4A). Molecular
phylogenies suggest that Darwin’s finches are part of a larger clade
called Tholospiza and that this entire clade should be considered as a
single adaptive radiation (Fig. 4A,B) (Burns et al., 2002, 2014).
Compared with other related lineages, this group has undergone
extensive bill evolution in a relatively short time frame, especially
among the Darwin’s finches, which demonstrate ‘exceptional’ rates
of diversification (Burns et al., 2002, 2014). These dramatic changes
in bill morphology have occurred with little genetic divergence
among the species (Burns et al., 2002).
There are two possible explanations for Tholospiza diversity

(which are not necessarily mutually exclusive). A model for
adaptive radiation developed by Simpson (1953) involved
simultaneous divergence of multiple lineages as a consequence of
entering new adaptive zones, such as isolated islands or new and

unusual habitats. Thus, the canonical explanation for diversity
among Darwin’s finches and their Caribbean relatives is that their
radiation occurred in response to the niches available to their
ancestors invading new island habitats (Losos and Ricklefs, 2009).
However, their diversity is both exceptional and unique, as the
Galápagos and Caribbean islands are inhabited by representatives of
many other families of land birds whose ancestors had the same
opportunities for diversification but show much more limited
morphological variation and a high degree of allopatry (no more than
one species per island). Darwin wrote, about the finches now bearing
his name: ‘These birds are the most singular of any in the
Archipelago’ (Darwin, 1839). Indeed, he was right, on Galápagos
there are 28 endemic species and 6 endemic subspecies of birds:
martins, mockingbirds, owls, hawks, flycatchers, doves, warblers,
cuckoos and others. These locally adapted species evolved from 18
different ancestral bird speciesthat colonized the islands over the last
5 million years. All of these other birds display allopatric speciation
and low levels of morphological variation, including beak shapes,
when compared with Darwin’s finches on the same islands. In
particular, Galápagos mockingbirds enjoy diverse diets on different
islands, which they have colonized for as long as the Darwin’s
finches, but they remained morphologically conservative and have
similar beaks to each other and their American and Caribbean
relatives (Fig. 4A) (Arbogast et al., 2006). When Tholospiza family
members were compared with the similarly aged and sized groups of
tanagers, it was concluded that this clade demonstrated unusually
high levels of morphological diversity, even considering their island
habitats (Burns et al., 2002). Asthis studyconcluded:‘An alternative,
more structuralist interpretation isthat the ancestor to all of these birds
possessed a developmental-genetic architecture (passed on to its
descendents) that included a greater variety of regulatory genes
controlling nasiocranial development’ (Burns et al., 2002).

A similar comparative study with similar results was performed on
Hawaiian honeycreepers (Lovette et al., 2002). To understand the
origin of their morphological diversity, beak shapes of Hawaiian
honeycreepers were contrasted with those of the endemic Hawaiian
thrushes. Phylogenetic analyses indicated that the ancestral thrush
colonized the Hawaiian islands as early as the common ancestor of
the honeycreepers. The similar timing of colonization suggests that
the observed differences in diversity between the Hawaiian
honeycreeper and thrush clades did not result from the length of
time that they spent on the archipelago (Lovette et al., 2002).
Morphometric analyses of the clade-specific morphological
characteristics associated with rates of diversification demonstrated
that the relatives of Hawaiian honeycreepers (Carduelini) displayed
significantly greater variation in bill size and shape among species
than the Hawaiian thrushes and their non-Hawaiian relatives
(Turdinae). In fact, Hawaiian honeycreepers have diversified to fill
much of the beak shape morphospace occupied by New World
passerine birds in general. Like the study on Darwin’s finches, this
report concluded that: ‘The greater morphometric variation of the
carduelines suggests that the ancestor of the honeycreepers may have
had an intrinsically higher capacity for morphological change than
did the ancestor of the Hawaiian thrushes’ and that such ‘high bill
shape lability in the Carduelini… might therefore represent a key
innovation of morphological versatility’ (Lovette et al., 2002).

In summary, all of these comparative investigations related to
extensive and rapid morphological evolution combined suggest a
rather striking picture. First, Darwin’s finches and Hawaiian
honeycreepers (and their diverse relatives) are the classic examples
of adaptive radiation and morphological diversification but their beak
and skull diversity cannot be explained by the external factors driving

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natural selection alone. For both clades, novel intrinsic mechanisms
providing higher capacity for generating morphological diversity
have been suggested. Second, as discussed above, beaks of Darwin’s
finches are controlled by a multigenic modular developmental
program, which allows for independent shape changes along
individual axes. Third, the follow-up comparative developmental
analysis indicated that such a sophisticated beak developmental
program is present in Darwin’s finches and their closest and
morphologically diverse Caribbean relatives but it is not found in
the more basal Tholospiza species (Fig. 4B-D) (Mallarino et al.,
2012). This later observation suggests that the most important
innovation in the Tholospiza clade was an intrinsic one: the formation
of the modular multigenic program that enabled more flexible control
over various parameters of beak shape and size. It is not yet clear
when this regulatory network expressed in beak cartilages and bones
emerged, and further phylogenetically guided developmental studies
are needed. In particular, it will be interesting to analyze how the
novel gene regulatory network emerged, perhaps with the help of
hybridization between rapidly evolving species exchanging
beneficial mutations, a phenomenon called ‘adaptive introgression’.
Finally, the study of diversity in entire skull shapes, described earlier
in this Review, indicated that it was high in Darwin’s finches and
extremely high in Hawaiian honeycreepers and was linked to altered
patterns of integration and elevated levels of modularity of skull
skeletal elements (Fig. 6D) (Tokita et al., 2017). The developmental
mechanisms controlling development of the vertebrate skull are still
in their infancy; for instance, little is currently understood about how
different parts of the skull interact during development and how such
interactions change during evolution.
Taken as a whole, evidence suggests that the internal mechanisms,

the famed Thompson’s ‘laws of growth’, indeed exert a huge
influence over morphological diversity and may explain much,
perhaps most, of the increased generative capacity in certain avian
and other animal clades to produce variation. In particular, more
research is needed to understand the role of novel developmental
changes in clades that are unexpectedly and exceptionally diverse for
the conditions that they inhabit. For such groups, the presence of the
novel ecological niches would be an important, indeed probably
required, condition, but far from sufficient to promote variation at
the observed levels. Natural selection would play an optimizing and
sieving role for the morphological variants produced at elevated
levels. Within this structuralist framework, one can better understand
why D’Arcy Thompson was hesitant to explain the ‘profusion of
forms’ in hummingbirds and other birds by the means of natural
selection alone.

Concluding remarks

‘This book [On Growth and Form], at once substantial and stately, is to
the credit of British Science and an achievement for its distinguished
author to be proud of. It is like one of Darwin’s books, well-considered,
patiently wrought-out, learned and cautious – a disclosure of the
scientific spirit.’

Thompson (1917b)

Charles Darwin is credited with having recognized natural
selection as the fundamental process driving adaptive
evolutionary change of organisms. However, it is less well
known that his true position was much more nuanced. Darwin
wrote a number of passages showing that he did recognize the
‘laws of growth’ as being a more important evolutionary process
than natural selection. In 1872, he wrote about the role of these
‘laws’ in biological diversity: ‘We may easily err in attributing

importance to characters, and in believing that they have been
developed through natural selection. We must by no means
overlook the effects…of the complex laws of growth, such as
correlation, compensation, of the pressure of one part on another’
and ‘we thus see that…many morphological changes may be
attributed to the laws of growth and the interaction of parts,
independently of natural selection’ (Darwin, 1872). Finally, in a
letter to Alpheus Hyatt, Darwin sends an important message: ‘I
should be inclined to attribute the character in both your cases to
the laws of growth and descent, secondarily to Natural Selection. It
has been an error on my part, and a misfortune to me, that I did not
largely discuss what I mean by laws of growth at an early period in
some of my books. I have said something on this head in two new
chapters in the last edition of the Origin…Endless other changes of
structure in successive species may, I believe be accounted for by
various complex laws of growth…Therefore I should expect that
characters of this kind would often appear in later-formed species
without the aid of Natural Selection, or with its aid if the characters
were of any advantage’ (Darwin, 1872).

Thus, the morphology of biological structures is first created,
shaped and transformed by novel alterations in the organism’s
development by the ‘laws of growth’, and then resulting
modifications become both the source and subject of evolution.
Thompson and Darwin might have been singing a similar tune after
all. The ‘laws of growth’ in modern interpretation include
developmental genetics and should embrace all interacting
ontogenetic processes from the molecular to organismal levels.
Many evo-devo studies currently focus on dissecting the relative
roles for intrinsic factors in triggering and modulating
morphological evolution. Future investigations should test the
hypothesis that some of the most famous examples of adaptive
radiation coupled with morphological diversification and dramatic
large-scale evolutionary transitions are driven primarily by the
propagative capacity of novel developmental genetic programs. The
origin and evolution of the underlying developmental innovations at
the genetic, epigenetic and genomic levels deserve a detailed study
as well. It will be important to combine evidence from multiple
fields using different methods – geometric morphometrics,
comparative embryology and functional experimentation
(Mallarino and Abzhanov, 2012). There is substantial benefit in
blending such efforts to solve the great puzzle that is the evolution of
biological diversity.

As many new research publications show, the legacy of D’Arcy
Thompson’s greatest book lives on and its ideas and approaches
remain surprisingly relevant. The current synergy between
morphology/morphometry, developmental genetics and
evolutionary biology fields is thriving on many of the ideas first
touched on by Thompson. After 100 years, On Growth and Form
continues to inspire experimentalists and theorists alike and makes
us all wonder what the next century of studying biological
transformations will bring.

Competing interests
The author declares no competing or financial interests.

Funding
The author gratefully acknowledges previous support on research described here
from the National Science Foundation (10B-0616127 and 1257122) and from the
Templeton Foundation (RFP-12-01).

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