Thanks to Alan Mason for this excellent piece on D’Arcy Wentworth Thompson.


D’Arcy Wentworth Thompson is most famous for his book, “On Growth and Form”, published in 1917, when he was fifty-seven. He was a populariser of science before that term had been invented, and in a time when radio and television broadcasting lay in the future; his medium had to be the printed word. Unfortunately, his book appeared during the economies of wartime, and before high quality colour photography had developed, and before cheap, high quality colour printing was available. He and his readers had to be content with a few black and white photographs, and simple line drawings. In this essay, I should like to complement D’Arcy Thompson’s ideas with some extra illustrations.

What he achieved, scientifically, was to draw together a very wide range of random experimental observations, and he managed to unite them in terms of the basic principles of physics, and to describe them mathematically. For the general reader, he attempted to describe the current scientific ideas in clear language. His other great achievement was to apply mathematics to biological processes at the macroscopic level. The sunflower (3) and succulent plant (4) heads are not only aesthetically pleasing, but they embody mathematical principles, as well. Today, in an era of computerised graphics, many of his insights seem distinctly “old hat”, but that is the fate of pioneers; to be rapidly overtaken by the people coming along behind.

We know that all animals and plants are derived from a single fertilised cell, the zygote, and their form or shape, at any stage, depends upon the developmental processes through which they pass. D’Arcy Thompson was very interested in what these processes might be, and the underlying physical principles. This essay does not try to summarise the whole of “On Growth and Form”, but four of his topics have been selected, which are interesting, and lend themselves to a graphical presentation of his ideas.


1. Surface Tension

D’Arcy begins by looking at a simple fluid cylinder, and the way that it gradually breaks up into spheres under the influence of surface tension (5). He draws parallels with microscopic tubular plants, like Spirogyra, and the way it forms ovoid and spherical structures (6) during its reproductive processes

Human beings are relatively large animals, and surface tension is a physical process that we can largely ignore, but for some small animals like ants or little spiders, the surface tension of a single raindrop can imprison or even kill them. For other insects like pond-skaters, (7) surface tension is a valuable property of water, enabling them to move around with great rapidity.

2. Drop Splashing

The splashing of drops into milk is a much faster illustration of the dynamics of surface tension. The initial drop impact produces a crater whose rim breaks up into droplets (8) and subsides into a rising column (9). D’Arcy presumed that the complex physics of the “splash process” in fluids was comparable to the development of the calyx of a flowering plant (10). The function of the calyx, at the base of a flower, is to protect the petals and immature parts before the flower opens.

Although D’Arcy had established that the form of the calyx and water splashes were similar, he had not proved that the developmental processes were the same. This would have been difficult to do, and would need some delicate experiments on flower development.

3. Falling Droplets in Fluids

The Coelenterates are primitive animals comprising corals (11), sea anemones, hydroids, and jellyfish. Medusae are the floating distributive phases of the group. In a similar way to the surface tension effects, D’Arcy draws attention to the remarkable similarity between the form of falling droplets in fluids (12), and the form of mature Coelenterate medusae (13).

Once again, although D’Arcy had shown the parallels of the two different forms, it is seems clear that the growth or development of the medusa produces shapes that are slow to sink in water, as the organisms are pelagic, or free-floating in the oceans. They have a simple musculature which contracts the bell, expelling water downwards and raising it upwards, enabling these animals to avoid sinking.

D’Arcy admits, “It is hard indeed to say how much or how little these analogues imply.”


1. The Cannon-Ball Problem

The idea of packing standard units into a larger container has been a practical issue for traders for millennia, and it is likely that early Greek philosophers may have investigated it, as it is such an obvious problem with a mathematical basis. The first published record we have of a mathematical analysis is in 1588, by the Englishman, Thomas Harriott, 1560 – 1621 (14). He was a graduate of Oxford University, and in 1580 was employed as a mathematics tutor by Sir Walter Raleigh (15). He was able to use his knowledge of astronomy to advise on navigation and ship design for the great seafarer.

Harriott accompanied Raleigh on an expedition to the Americas in 1585 – 86, where they spent some time on Roanoke Island off the coast of what is now North Carolina. Apart from Harriott’s knowledge of astronomy and navigation, he also acted as interpreter to meetings with the Native Americans. He had learned the Algonquian language from two ex-patriated Algonquin’s, while in England, before he left on the expedition. In 1588, Harriott published, “A Briefe and True Report of the New Found Land of Virginia” in which he also explained that Raleigh had asked him to work out the best way to stack cannon-balls aboard a warship. It was imperative for cannon-balls to be closely secured, so they did not roll about, but also to be readily accessible. Ships’ captains used a stout rectangular, or triangular wooden base to stack the balls (16).

2. The Mathematical Theory of Close-Packing

He produced a theory of what is now known as “close-packing” and he later corresponded with the astronomer Johannes Kepler, 1571-1630 (17), who subsequently formulated his own ‘Kepler Conjecture’ which concerns the “packing density” of spheres. This concept was taken up by the brilliant German mathematician, Carl Friedrich Gauss (18), (1777-1855) who proved that the maximum value was given by a simple formula, П/3√ 2.

Studies of the geometry of the close-packing process had been a fascinating and purely abstract part of formal mathematics for centuries, until suddenly, at the close of the nineteenth century, it assumed considerable importance in physics and chemistry. Now, all that abstract theoretical work was vital, to explain the packing of atoms within the structure of crystals.

3. The Development of Crystallography

The new science of crystallography began with simple examples, like sodium chloride, which only has four kinds of atoms; the sodium ion, chloride ion, as well as hydrogen and oxygen from water molecules. The crystals are cubical in shape, as shown in the upper picture of natural rock salt in Figure 19. The lower picture is a photomicrograph of tiny crystals separating from a saline solution. Crystallographic studies have shown how the elements are arranged, in cubical form, even at the atomic level, (20), where the green spheres are chloride ions and the smaller blue spheres are sodium ions.

Newly-discovered X-rays were used in the late nineteenth century to analyse crystal structure, because their wavelength is so short that they readily penetrate the spaces between atoms. During the twentieth century experimental methods were gradually refined until by the 1950s Maurice Wilkins, at King’s College in London, had perfected the new technique of X-ray fibre diffraction. This was an enormous step forward, because until that point crystallographers could only work with crystals. Wilkins’ method now enabled them to determine the structure of long fibrous biological molecules, such as the collagen of tendons and connective tissue.

In 1953, the evidence provided by the research of Wilkins, and Rosalind Franklin, enabled Francis Crick and James Watson to propose a molecular structure of DNA (21). Following on this success, it became possible to solve the much more difficult problem of the molecular structure of globular proteins, which are long chain molecules coiled into a ball. The British crystallographer, Dorothy Hodgkin, established the molecular structure of the hormone insulin (21) in 1969, and this, together with the elucidation of DNA, were seen as the crowning achievements of twentieth-century X-ray crystallography.

4. Close-Packing Two Dimensional Deformable Units

D’Arcy briefly discusses close-packing, and then passes to hexagonal symmetry. A key aspect of the previous discussion, is the rigidity of the units, whether they are cannon-balls, atoms or ions. However, in many other systems, from bubbles of foam, to biological tissues, the basic units are deformable, and D’Arcy illustrates how, in a two dimensions, circles deform to hexagons (22A). His relatively simple diagram was based on work published by the Italian, A.P.P. Bonanni in 1681. He also showed how the individual elements of two dimensional structures can be altered to polygons, in, for example, the insect wing (22B).

5. Close-Packing in Three Dimensions

The honeycombs of bees can be seen simply as a collection of sets of two dimensional structures. The regularly hexagonal nature of the cells is not a piece of clever engineering on the part of the bees, but simply a natural outcome of the construction method. D’Arcy quotes Rasmus Bartholin, (bar too lin) 1625-1698, the Danish doctor and mathematician, who said “the hexagonal cell was no more than the necessary result of equal pressures, each bee striving to make its own little circle as large as possible.” (23)

In fully three dimensional structures, basic spherical units are modified into regular polyhedra. The commonest example of this process is found in the tissues of animals and plants. This was difficult to observe directly, in the early days of microscopy, and scientists relied on thin sections which showed polyhedra as regular polygons. In the section of the parenchyma or packing tissue of a potato tuber, (24) the circles are only slightly distorted towards polygons, because there are still many triangular spaces at the corners. The purple spots are stored starch grains that have been stained with iodine solution.

Cork, because of its texture, easily lends itself to sectioning with a sharp knife. The first microscopist to publish drawings of cork sections was the Englishman, Robert Hooke, 1635 – 1703, in his book, “Micrographia”. He was also first to describe the spaces he observed as “cells”, because they looked to him, like regular rows of monks’ cells in an architectural plan of a monastery.

What he had not realised was that the cork cells were dead and the original contents had gone. Most cells are not empty, but his word stuck and has now developed an international usage. In Figure 25, the cells at top left are rectangular but those at top right are more polygonal. The lower illustration is a far superior image, showing variations in the cell shape and wall thickness.

6. Close-Packing a Variety of Sizes

Mathematicians were concerned with packing identical shapes, because this is the easiest problem to investigate. As microscopical research advanced it became clear that cells exist in a wide variety of shapes and sizes, and uniform cellular structure in tissues is the exception, rather than the rule. Sometimes the arrangement of different kinds of cells can appear random (26), but in most cases it is highly organised as in the soft and hardwoods illustrated (27).

7. Complex Polyhedra

D’Arcy illustrated (28) how subsequent research revealed that many cells have a complex polyhedral shape. Since that time the academic study of polyhedra has advanced, as shown in Figure 29. The object below the blue chart is a heavy steel ball which appears to have the beginnings of a polyhedral structure. Its size can be gauged from the fingers holding the ball. I have used it as a mystery object to promote discussion, particularly when I explain that it was originally spherical in shape.

8. The Steel Ball

This object was collected by me in the late sixties, from a cement works at Pitstone, on the edge of the Chiltern Hills. Cement is made by heating chalk and clay together in a furnace. In this case, the chalk was dug directly from quarries around the works, as the Chiltern Hills are composed entirely of chalk from the Cretaceous Period. The furnace, or kiln, was rotatory (30), and about fifteen feet in diameter, (5 m) and about 300 feet long (100 m). Apparently, the maximum length of these rotatory kilns is around 750 feet (230m) and the maximum diameter is about 20 feet (6m).

The kiln slopes slightly downwards from the starting position. It is designed to work continuously, so that as crushed chalk and clay is fed in at the upper end, it gradually moves down toward the exit as the kiln slowly rotates. Heating is provided by a forced draught of town gas and air, converting the calcium carbonate in chalk to calcium oxide, and dehydrating the aluminium silicates of the clay. The long flame can either be, with the movement of the chalk and clay, (co-current) or as is more usual, counter-current. This was nearly fifty years ago, so the Pitstone kiln was probably an old-fashioned design. The action of the flame could be seen through a hand-sized blue glass window.

The resulting material from the furnace is a hard clinker, which needs to be pulverised before it can be bagged and sold for building construction work. The pulverisation is carried out by feeding the clinker into a ball mill. This is a heavy steel drum (31), about fifteen feet in diameter, and 30 feet long, (5m diam. by 10m long). While the furnace rotated quite slowly, perhaps twice per minute, the ball mill turned much faster, about once in five seconds, or 12 times a minute. The clinker was mixed with a mass of steel balls, each about 2 inches (5 cm) in diameter, and as the mill turned it was ground into a fine powder, the familiar cement. The noise inside the mill room was quite deafening.

Eventually, the steel balls wear out and they are emptied out into a spoil heap, to be sent to a steelworks for reprocessing. When I collected a couple of the balls from the spoil heap, (with permission from the management) they were slightly rusty, so I cleaned them up with abrasive emery paper. What fascinated me was the fact that the ball was a totally man-made object, subjected to a wholly random artificial process, and yet it appeared to be developing into a gently rounded, almost biological form, which might easily have grown from a spore or seed. This is why my steel ball seems appropriate to an essay on Growth and Form.

9. Regular Polyhedra

The ball has ten “facets” which makes it a decahedron. What I call a “facet” is not flat surface, but a roughly circular shape, with a deep circular concavity, as shown in the sketches for Figure 32. We know, (References F) that there are only five REGULAR POLYHEDRA, that is, solids with identical faces that are regular polygons. These are the tetrahedron (4 faces, all triangles), cube (6, squares), octahedron (8, triangles), dodecahedron (12, pentagons), and icosahedron (20, triangles), as in Figure 33.

In addition the UNIFORM POLYHEDRA have identical polyhedral angles at all the vertices, and all the faces are regular polygons, but not necessarily of the same kind. The decahedron falls into this category.

I discovered, from background reading for this essay, that there are 32, 300 topologically distinct kinds of decahedra, so it would not be easy to link my steel ball to one of these.


D’Arcy devotes an entire chapter of his book to this topic, and perhaps because it is well-illustrated with fascinating pictures (34), it has become the most well-known aspect of his work.

Essentially, a plane spiral is a line which moves in a circle from a central point, with a gradually increasing radius. Mathematicians, technically define a spiral as “a plane curve whose equation in polar co-ordinates is written r = f (θ)”, (References F). In plainer English, the radius of a point on the line of the spiral is a function of the angle, θ. The popular and commonly used computer program, “Excel”, has a sub-routine, “Radar” which can be used, with modifications, to plot a limited Cartesian (normal) graph in polar co-ordinates.

1. The Archimedian Spiral

The simplest spiral type to begin with is the Archimedian spiral. Again, in plain English, the width of the space between the lines of the spiral is constant. In Figure 36, the straight line OP runs from the centre to the outer edge and the curved line cuts it at regular intervals.

In mathematical terms, “the Archimedian spiral is defined by r = aθ,” (References F). Real-life examples of this type of two-dimensional Archimedian spiral is the flat coil of rope on a ship’s deck, a rolled-up dressmakers’ tape, the surface of an old gramophone record (37)and the photomicrograph of the myelin sheath of axon of a nerve cell (38)

In the “Excel” generated graphs, 1 and 2 below, the basic equation is r = (1/5) θ or r = θ/6 and the distance between whorls is 60 units. The reason is, that a complete circuit is 360 degrees, and 360/6 = 60. In the “Radar” sub-routine, the program closes a complete sequence of values, which is not required in a spiral, so this is avoided by deleting the final values at the end of each row of data.

2. The Equiangular Spiral

There are a variety of other mathematical types of spiral, but D’Arcy concentrated on what he called, “The Equiangular Spiral”, but which is also known as the Logistic Spiral and the Logarithmic Spiral. His interest lay in its association with the world of nature. “As the spiral of Archimedes may be looked on as a coiled rope, so a long cone coiled upon itself, is a rough simulacrum of an equiangular spiral.”

D’Arcy’s diagram (39) of the equiangular spiral, demonstrates one of its most interesting properties. In the diagram, O, is the centre of the spiral, and two lines intersect it at right angles enabling successive widths of the whorls to be identified. D’Arcy claimed, “The whorls continually increase in breadth in an unchanging ratio. Figure 39 is used to show successive widths along a given line, namely OH, OG, OF, OE on the vertical; and OB, OD, OC, on the horizontal.

The ratios to be calculated are thus; OH/OG, OG/OF, OF/OE and OB/OD, OD/OC. What emerges is that the ratios give a constant value that can be determined more and more accurately as the width of the spiral increases. The value of this ratio is approximately 1.618, and the present convention is to symbolise the ratio by the Greek letter f, φ (Phi) pronounced “fy”. This ratio, φ, is another incommensurable, like П.

Mathematicians define this spiral, in polar co-ordinates, by r = a exp (θ cot b) or r = ae (θb) where a is the numerical value 5, and b is the angle of a tangent to a point on the curve (0.1759). (References F)

3. The Fibonacci Series

The numerical value of the ratio, φ, from the equiangular spiral, also appears in a different context. The Fibonacci (fee bon atch ee) Series is a run of numbers, which is quite simply defined. Each term of the series is the sum of the two preceding terms. Hence the series begins, 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, etc. as shown in Graph 5. If the ratio of consecutive terms is calculated, it gradually oscillates towards 1.618, as a limit (Graph 6).


SEQUENCE n 1 2 3 4 5 6 7 8 9 10 11
TERMS T 0 1 1 2 3 5 8 13 21 34 55
RATIO f 0 0 1 2 1.5 1.6667 1.6 1.625 1.6154 1.6190 1.6176

This sequence of numbers is named after Leonardo Bonacci (c.1170 – c.1240), who was the first mathematician to publish an account of the series for European readers; in his 1202 Liber Abaci (Book on Calculating). He did not produce the series himself but learned about it from the Indian sub-continent. He was also the first mathematician to introduce the Hindu-Arabic number system to Europe. This is the system we use today for calculations, and it swiftly replaced the cumbersome Roman numeral system, which is now only used for listing items or sections in a text.

4. The Golden Section

The third area of mathematics in which the value 1.618 or φ appears is what is known as the “Golden Section” or “Golden Number”. Compared with the other two themes it is easier to define. In one-dimensional terms, a straight line is divided in the Golden Section when, the ratio of the longer piece to the shorter, is the same as the whole to the longer. It is easier to understand diagrammatically, as shown in Figure 40, where the two sets of ratios equal 1.618 or φ. Here, the black line is divided so that a/b = (a + b)/a = 1.618 and in two dimensions both the green and pink rectangles have sides in the Golden Section. These are seen as “ideal” or most aesthetically pleasing proportions, and are widely used for books, paper sizes and paintings.

D’Arcy spends some time in showing how the equiangular spiral can be constructed on rectangles in the Golden Section. He explains how, (40) by constructing a square (yellow) on the larger side of a Golden Section rectangle (pink), a new and larger Golden Section rectangle is created. In Figure 41, starting with rectangle A and square B, a succession of increasingly larger rectangles is produced. He also demonstrates how the equiangular spiral can be drawn on an hexagonal array.

5. Phyllotaxis

The word, phyllotaxis, (fil o tax iss) is the technical term for the arrangement of leaves on the stem of a plant, and it is of interest here, because it involves both the Fibonacci Series, and the Equiangular Spiral. Curiously, the word does not appear within the index of “On Growth and Form”, and D’Arcy devotes no more than a rather lacklustre page to “Spirals in Plants”. The term, phyllotaxis, was coined as early as 1754, by the Swiss naturalist, Charles Bonnet. Given that D’Arcy was a biology Professor, it puzzles me that he did not draw on any of the wide range of beautiful and fascinating examples from the plant world.

The main constraint, on the arrangement of leaves on a plant stem, is to avoid having the upper leaves shading the lower ones from sunlight. Plants have evolved a wide range of strategies to manage this problem and I do not want to describe them all here, nor do I want to go into the known hormonal and other mechanisms for controlling the process. There are, however, a limited number of examples which I think are relevant to the general flavour of D’Arcy Thompson’s book.

The apex or growing point of a plant consists of a mass of rapidly dividing unspecialised cells. The primordia are special regions which will develop into leaves. In the example illustrated, (43), the primordia are arranged in a spiral fashion, with the oldest near the start of the alphabet, and the youngest at the top towards the end of the alphabet.

The angle between successive leaf stems is called the divergence angle. When the number of leaves is relatively small the divergence angle is usually expressed as a simple fraction of 360 degrees, so that plants with alternate leaves have a 1/2 divergence. Figure 44 is a special case when the plant has 2/5 divergence, showing how the primordia are distributed around the spiral. The oldest primordia have the lowest red number.


N = numerator, d = denominator, D = divergence ratio as a decimal

Deadnettle, Mint 1 2 0.500
Beech, Hazel 1 3 0.333
Oak, Apricot 2 5 0.200
Sunflower, Poplar, Pear 3 8 0.375
Willow, Almond 5 13 0.385

Readers may well have noticed that the numbers of the fractions are terms in the Fibonacci Series. To be more precise, the numerator and denominator consist of a Fibonacci term and its second successor. In the discussion of the Fibonacci Series in Section 3, the ratio φ is greater than one, because the numerator is always the larger number.

In these fractions concerned with divergence, the numerator is always the smaller number, because they describe a part of the 360 degree circle. The 2/5 divergence illustrated in Figure 44 means 2/5 of 360 or 144 degrees, so that each primordium is 144 degrees from its neighbours. It is clear from Table 2 that the decimal value, D, is oscillating towards a limiting value. When the number of leaves becomes very large, such as the growing head of the succulent Sedum, (45), D approaches this limiting value at 0.382 which turns out to be another incommensurable.

The Relation between φ and D

We know the ratio between successive terms of the Fibonacci Series is 1.618, the Golden Number, φ. Its reciprocal, is often represented by capital phi, Φ, which may be confusing. The value of Φ is 1/φ = 0.6180, and we find that the ratio Φ/φ = 0.6180/1.618 = 0.38195 which is the value of D.

If D = Φ/φ = 1/φ divided by φ/1 = 1/φ X 1/φ = 1/φ2

The purpose of these calculations is just to demonstrate that D is not a new incommensurable, but merely the simple function of φ, so D = 1/φ2.

When the number of leaves in the growing head of a plant becomes large, they become arranged in equiangular spirals (46). This figure is drawn directly from Figure 45, and shows the thirteen distinct spirals present. The reason for these rather technical structures is due to developmental processes. Returning briefly to Figure 44, with 2/5 divergence it is clear that some leaves lie directly above others, so that primordia 1, 6, and 11 all lie on the line of zero degrees.

These repeating patterns may be successful, in an evolutionary sense, when there are few leaves, but when there are many it is essential to avoid them. The best way to avoid this is by incorporating an irrational number into the placement of the primordia. That is why the incommensurable, D, has its place in the evolution and development of plant growing points and flowers.

In some species such as Aloe, (47) the spirals are extremely pronounced, and in some flower heads, like Helianthus (48), there are criss-crossing equiangular spirals.

6. Spiral Shells and Gnomons

D’Arcy spends some time discussing and illustrating “gnomons”, which I must confess, is a geometrical term I had never met before. I assumed it must be down to my ignorance of mathematics, but I was relieved to discover that it does not appear in my small mathematical dictionary either, (References F), so it must be a rather antique concept. (The word is also used to describe the metal plate and shadow triangle of a sun-dial.) Fortunately, the term “gnomon” is simple to define, and understand.

A gnomon is a particular geometrical shape, that when it is added to a different shape results in an increase in size, but the proportions of the new shape are the same as the original one. This is more easily shown diagrammatically, in Figure 49.

D’Arcy points out that gnomons can be used to describe the growth lines of the Ormer, (50), a bivalve mollusc, which lives in the southern waters of Britain. The holes in the shell aid water flow. As his book is concerned with growth the concept of gnomons is, for him, a very fruitful one.

In discussing the multiplicity of beautiful shapes in molluscan shells, which make them such collectable items, D’Arcy argues that there are three particular angles controlling growth and form. These are signified by the Greek letters, alpha, beta and gamma, α, β and γ, which appear in his diagrams (51) explaining growth patterns, but these are too involved to warrant description here.

7. Spiral animal horns

D’Arcy devotes an entire chapter to “The Shapes of Horns and of Teeth or Tusks”, which is a fascinating topic, but he admits, “These horn-spirals are less symmetrical, less easy of measurement, and less easy of investigation (than shell-spirals). Let us dispense with mathematics; and be content with a very simple account of the configuration of a horn.” This is a surprising response from a man whose whole raison-d’etre was to place structural biology on a firm foundation of physics and mathematics.

He recognises three types of horn; (a) the hollow horns of the horned ruminants like sheep, goats, oxen and the various kinds of antelope (52/53), (b) the solid antlers of deer, shed annually, and (c) the rhinoceros horn which grows from the calf’s bud, into the equiangular spiral curve of the adult horn (54). The chapter is principally concerned with the first category.

Given the great variety of horn types, D’Arcy has to be content with just five illustrations. He explains that the horns have a bony core, covered by a dermal layer, richly supplied with blood-vessels, while the outer epidermal layer develops the keratinous material of horn, which is chemically similar to hair or nails. The zone of active growth at the base of the horn, adds, ring by ring, as the ‘generating curve’ of the horn’s shape. He tries to analyse the growth process by considering the cross section of the horn. This is often an isosceles triangle in antelope, goats and sheep, but may be an equilateral triangle, as in Ovis ammon, the Argali sheep (55).

D’Arcy says, “In the great majority of horns we have no difficulty in recognising a continuous logarithmic spiral. In the gemsbok the spiral angle is very small, or the horn is very nearly straight”, but in others there may be a tight spiral, with the whole horn describing a distinct curve as in the sable antelope (53).

The concluding example is an unusual one, the male narwhal. This is a white whale, whose head carries a single, long, straight, tightly-twisted horn. “It is the only tooth in the creature’s head to come to maturity; it grows to an immense size, 8 or 9 feet long” (up to 3 m). Occasionally, male narwhals develop two horns, and the fact that “they are identical screws, with both threads running the same way,” gives D’Arcy a valuable clue to the causation of this feature.

In swimming forward, the narwhal is inclined to roll slightly by the action of its tail, and it needs to continually adjust for this. The horn too experiences this roll, but it is attached by its growing root, and as a “torque of inertia” is continually applied, it creates a twist as it grows. The inertial torque will be the same for twin horns and therefore they will develop identical screw twists.


1. Changes in Comparative Anatomy

The subject of Comparative Anatomy, the study of differences of shape, size and details, in related animals (57), has been a staple of biological science since the eighteenth century. It was given a new impetus in the mid-nineteenth century, when Darwin’s “Origin of Species” was published, because it was realised that Comparative Anatomy illustrated evolutionary changes. In D’Arcy Thompson’s day, mathematics, and particularly transformational geometry, was beginning to be applied to comparative anatomical studies.

In the mid twentieth century, Comparative Anatomy, having been eclipsed by exciting progress in biochemistry and genetics, for a few decades, began to benefit from the application of computer programs to its raw materials, like fossils (58), X-rays of existing animals, and extensive statistical measurements of form in populations. D’Arcy’s book drew together a range of recent research and ideas on transformational geometry in Comparative Anatomy, making them available to a wider audience.

2. Cartesian Transformations

He begins by explaining the then laborious process of Cartesian Transformation, and he would be delighted by modern computer programs that carry out the work in, literally, the twinkling of an eye. The graphics program “GIMP” has four different kinds of transformations, (59), but the widths of the grid also changes. One of D’Arcy’s early illustrations, (60), is on the canon bone at the end of the leg, showing transformation along one axis only, like the second row of Figure 59.

Having embarked on the fascinating topic of transformation in the skeletons of vertebrates D’Arcy turns aside to discuss plant leaf shape in some detail, before returning again to the leg of ungulate mammals. I find this surprising, given that leaf anatomy is a far less interesting topic than phyllotaxis which he ignores completely. On re-visiting the three ungulates (61), he makes their lower leg length a standard value, to demonstrate the change in the proportion of the canon bone.

Following some very questionable evidence from the German artist, Albrecht Dürer, (1471-1528) on the supposed proportions of the human head, D’Arcy returns to safer ground with the shapes of the crab carapace (back shell), and the way in which the original roughly circular outline has been transformed in five different ways (62). In Figure 63 he shows how a basic geometric transformation could convert one named genus of fish into a different one. The scientific names have been deleted from 62 and 63 for simplicity.

3. Transformations in the Great Apes

As a final dramatic flourish, at the very end of his book, D’Arcy presents a human skull in a grid of Cartesian co-ordinates, and shows how a rather complex curved transformation can convert it into the shape of a chimpanzee or baboon skull. He explains that it has not been possible to present a “series of successive and continuous gradations through Mesopithecus, Pithecanthropus, Homo neanderthalensis and the lower or higher races of modern man. The failure is not the fault of our method. It merely indicates that no one straight line of descent exists.”

This final example points up the problem of these grids and transformations. What do these relationships, which D’Arcy describes and illustrates, actually prove? Not that one species evolved into another one. We know, from other evidence, that the baboons, chimpanzees and humans have a common ancestry. Baboons separated from the line of chimps and people a very long time ago, tens of millions of years. Chimpanzees and humans have a much more recent common ancestor, but the two lines separated around ten millions years ago. None of the three species is the ancestor of the other, so what do the transformations mean? This leads conveniently to the next section.


1. Lack of Scientific Hypothesis

Perhaps the great weakness of the book is that D’Arcy describes fascinating relationships, but offers little in the way of theory or hypothesis to explain them. The essence of science is the creation of testable hypotheses, which can either be modified and extended, or abandoned altogether, in the light of the research evidence. When tackled on the lack of a preface to his book, D’Arcy replied that the whole book was a preface, suggesting that what was needed was another book, probably by a different author, containing some central, unifying hypothesis.

Returning to the subject of primate skulls (65), there are at least two processes at work. One is the evolutionary changes brought about by natural selection, and the other is changes in embryological development. The chart in Figure 65 has embryological development as the x-axis, and evolutionary change as the y-axis. The skulls of all newborns tend to be rounder than their adult forms.

2. Neoteny or “Peter Pan” Evolution

The evolutionary trends in the newborn are towards a shortening of the muzzle, and an increase in size of the back of the skull. In maturing, the newborn of the lower apes on the chart, show an increase in muzzle length. By contrast, the newborn human skull matures by an increase in the prominence of the chin. In comparison with all the other apes, human beings are rather unspecialised.

Their arms and legs are of much the same length as each other, as are the fingers and toes. The dentition is also unspecialised, lacking prominent canine teeth. These are the characteristics of an anthropoid embryo; put simply human beings are apes which have not properly grown up. More technically, human beings have evolved to achieve sexual maturity at what was previously an immature stage. This is a widely observed phenomenon in evolutionary biology, and is called neoteny.

3. Rejecting Evolutionary Theory

D’Arcy’s Cartesian grids needed to be based on the newborn (66) of the common ancestor of his three example species, so we can demonstrate both embryological development, and evolutionary change. To be fair to D’Arcy, this specimen is not available, but it is possible to make the attempt, because it has a bearing on the mechanism of modern aspects of theory of evolution. This precisely points up the weakness of D’Arcy’s philosophical position; he ignores the research on embryological development which was current in his day, as if it was irrelevant, and he never uses any explanations based on evolutionary theory because he does not really believe in it.

4. Phylogenetic Trees

The family tree of the great apes, in Figure 67, is a modern one, based on a study of the nuclear DNA of the different species. It demonstrates the closeness of the relationships, in terms of the number of nucleotide substitutions needed in separating the various lines of descent. Such phylogenetic trees were widely used in D’Arcy’s day, but they were based mainly on comparative anatomy. The DNA work has largely confirmed the theories of the anatomists and classifiers, but it has been of most use in explaining odd or unusual groups whose classification position was uncertain.

5. Absence of Causal Explanations

In case any reader thinks I am being unfair to D’Arcy, let me quote from the foreword of the 1961 edition of his book by the biologist, John Tyler Bonner.

“Since he was clearly concerned with the explanation of biological growth and form in physico-mathematical terms the reader must often be prepared for disappointment if he expects to find immediate causes. Most experimental scientists are only satisfied if they can understand a particular form by the configuration of its immediate precedents, and these in turn are analysed in the same way so that an epigenetic chain is exposed; this is the basis, for instance, of ‘causal’ embryology.

D’Arcy Thompson, on the other hand, was quite satisfied with a mathematical description or a physical analogy. No doubt this condition of mind is closely tied in with the fact that he himself was in no way an experimenter. He even refused, in his 1942 edition, to recognise experiments that bore on the facts of his 1917 edition.”

6. Glaring Omission of Chemistry

“The other aspect of his approach is that chemistry (68) is almost wholly absent. In biology, biochemistry has become the most potent source of new knowledge, the most rapidly growing forest of new information, yet the idea that form and its change is related to the reactive properties of chemical substances is almost completely ignored.

But with all his impressive armoury of talents he was, in the world of science, a lonely figure. He himself was well aware of this and was even proud that he did not run with the pack. His views on evolution are constantly creeping to the surface throughout the book. They are ideas that were heretic in 1917 and they remain so today. He does not deny natural selection, but suggests that it operates only to eliminate the unfit and does not serve as a progressive force in evolution.

The idea that every structure is an inherited adaptation arising through selection he considers particularly unjustified and unreasonable; in many cases he would consider the structure to have arisen by direct physical forces, molecular in very small structures and mechanical in larger ones (that is, ‘direct adaptations’).”

7. Omission of Genetics

“Heredity and the activity of genes in development are wholly missing in the book except for a few passing references which seem to imply that they do not fit into his scheme of things. (Again we have an instance here, as we did in the case of biochemistry, of a complete omission of one of the most important trends in biological thought in the past fifty years.)

Phylogeny, the study of animal ancestry and relationships, which was the central concern of the comparative anatomists at the turn of the century, is pushed aside and replaced by the idea that the functional aspect of form is far more important than blood relationships and family trees.” (Bonner, References A)

Remember that Bonner was writing his foreword in 1961, over fifty years ago. His criticism of D’Arcy’s omissions and blind spots are not only still valid today, but the case he made is even stronger now. Bonner saw clearly the importance of genetics (69), based on detailed biochemical knowledge, as the basis of understanding evolutionary changes.

8. Inheritance of Trends

I have some sympathy with D’Arcy’s reluctance over evolution, because at that time, it was widely believed to proceed by a series of short steps, produced by random mutations. There seemed to be no good reason why randomness should create successful evolutionary adaptation. D’Arcy’s studies of continuous physical processes made him believe that these were the driving forces, and natural selection merely removed the unsuccessful individuals.

There is, perhaps, a way of reconciling the two opposing views. It may well be that organisms inherit not single small mutations but TRENDS. For example, a TREND to increasing size, or a TREND to an increasing length of the legs, or a TREND for one of D’Arcy’s geometrical transformations. Hence, organisms would not inherit a lot of separate, disconnected genes for various parts of the body, but a TREND which simply affected all parts in a defined way. Natural selection would remove individuals that had taken the trend too far.

As an illustration of this I have chosen the way that the pectoral girdle (chest and shoulders) has evolved in the four-legged vertebrates. In the amphibian and many reptiles, the limbs sprawl to the sides (70), and the belly drags along the ground as they move. There was a TREND for the scapula, (shoulder blade), to rotate downwards and backwards. This had the effect of moving the forelegs under the body and raising the belly clear of the ground. This TREND could have been one of D’Arcy’s transformations.


D’Arcy Wentworth Thompson (71) was born in Edinburgh in 1860, the son of a well-to-do academic family. His father was Professor of Greek at Queen’s College, Galway, in Ireland, which was then part of Great Britain. He attended the Edinburgh Academy from the age of ten until seventeen. He entered Edinburgh University in 1878, aged eighteen, to study medicine. He never qualified as a doctor, because in 1880 he moved to Cambridge University, in order to study Natural Sciences, and three years later, obtained his BA degree in 1883.

Academic Posts

Aged twenty-four, in 1884, he was appointed Professor of Biology at University College, Dundee, in Scotland. Modern readers may well be surprised that such a young man was given a Professorship without first serving a period as a junior academic. At that time, Scotland had four full universities, (Edinburgh, Glasgow, St Andrew’s and Aberdeen) dating from the sixteenth century, but the College at Dundee was not a university in its own right, but an “outpost” of Aberdeen.

D’Arcy remained Professor at Dundee for 32 years, from 1884 to 1916, until he was appointed Professor of Natural History at St Andrew’s University, in 1917 when he was aged 57. He remained there (72) as Professor until his death in 1948 at the age of eighty-eight.

Arctic Studies

At Dundee, D’Arcy created a new Zoological Museum as part of his department, for teaching undergraduates, and also for research. It grew to be a very large museum, as one of its specialisms was Arctic Studies, and the whale exhibits were inevitably very large too. The reason for this specialisation was the port of Dundee’s links with the whaling industry. His expertise in this area, led him to be appointed as a representative of the British Government to an international enquiry into the fur seal industry. At the age of forty-six, he went on two expeditions to the Bering Straits, in 1896 and 1897. These Straits (73) lie between, what was then Imperial Russia, and the USA, close to the Arctic Circle.

D’Arcy was fascinated by the wild, mountainous landscapes of the Straits, (74, 75) and he seized the opportunity to collect a wide range of specimens for his museum in Dundee. He produced a rather dull report for the Government in 1897, on his investigations into fur-sealing (76), which can now be read directly on the internet, (https://archive.org/details/reportbyprofesso00thomrich).

The Decline of D’Arcy’s Influence

Many modern zoology students would be surprised that D’Arcy had left it so late before he went on an international expedition at the age of 46, given that he was a university professor, with influence nationally, and in local industry. Following the trips to the Bering Straits, he went on no further expeditions.

Given that D’Arcy was a professor for 63 years in total, he produced few books. Apart from, “On Growth and Form”, his major work, he translated Aristotle’s “History of Animals” from ancient Greek to English, and wrote a compendium of birds mentioned in classical Greek works.

Although D’Arcy had created a zoological museum for Dundee, after he left for St Andrew’s in 1917, it went into decline, and forty years after its creation, the building had been cleared for demolition, in 1956. Most of the specimens were dispersed to other museums, but a small central teaching collection was retained, and today, this is the Dundee University D’Arcy Thompson Zoological Museum. Although this sounds rather grand, the museum is in the basement, and is only open to the public on Friday afternoons in the summer. This compares poorly with public access to the Pitt-Rivers, or the Ashmolean Museums in Oxford.

The reasons for this decline have been referred to earlier, when John Tyler Bonner said of D’Arcy, “he was, in the world of science, a lonely figure”. Exactly. He created no “school”, and had no acolytes around him, as most brilliant researchers do, because he had closed his mind to biochemistry, genetics, embryology and evolution. Most of his brighter students would have realised that these four areas represented the future of biology, and this was a world from which they would be excluded, if they followed D’Arcy’s example.


I have made use of the rather cramped, black and white line illustrations from D’Arcy’s book, and added colour and given them more room, hence the description, “After a diagram in D’Arcy, etc. Further illustrations have been selected from a variety of reference works.

A. “On Growth and Form”, by D’Arcy Wentworth Thompson, abridged edition, edited by John Tyler Bonner, Cambridge U P, 1961

B. “Maritimea”, editor, Charles F Gritzner, Millennium, 2009

C. “The Discovery of North America” by Michèle Byam, Hamlyn, 1970

D. “The Wordsworth Dictionary of Biography”, Wordsworth, 1994

E. “Heaven and Earth”, editor Amanda Renshaw, Phaidon, 2002

F. “The Penguin Dictionary of Mathematics”, edited by J Daintith and R D Nelson, Penguin, 1989

G. “Phyllotaxis” wikipedia article

H. “The Encyclopedia of Animals”, Per Christiansen, Amber, 2008

J. “The Ultimate Book of Mammals”, Jonathan Metcalf, Dorling Kindersley, 2010

K. “Whales, Dolphins and Man”, by Jacqueline Nayman, Hamlyn, 1973

L. “Evolution” by Douglas Futuyma, Sinauer, 2006


1. D’Arcy Wentworth Thompson (google image)

2. A modern reprint of the original book (google image)

3. Florets in a Sunflower (google image)

4. Leaf mosaic of a Succulent rosette (google image)

5. The disintegration of a cylinder of oil in water (After a diagram in D’Arcy, op. cit)

6. Tubular filaments of green alga Spirogyra and its spore stages (google image)

7. Pond skater resting on the surface of the water (google image)

8. Initial splash (Photo Harold E Edgerton, MIT)

9. Subsiding splash (Photo Harold E Edgerton, MIT)

10. The calyx of several different species of Campanula Bellflower (After a diagram in D’Arcy, op. cit)

11. A range of corals (Gritzner, op. cit)

12. Falling ink in water, l. and falling oil in paraffin, r. (After a diagram in D’Arcy, op. cit)

13. The medusoids of three different species of Coelenterates (After a diagram in D’Arcy, op. cit)

14. Thomas Harriott, 1560 – 1621 (google image)

15. Sir Walter Raleigh (Byam, op. cit)

16. Stack of spheres in a regular tetrahedron (google image)

17. Johannes Kepler, 1571-1630 (google image)

18. Carl Friedrich Gauss, 1777-1855 (google image)

19. Sodium chloride (Smithsonian Institution)

20. Sodium chloride crystal at the atomic level (google image)

21. Detailed structure of complex linear and globular molecules (google image)

22. Packing in two dimensions (After a diagram in D’Arcy, op. cit)

23. Honeycomb of bees (D’Arcy, op. cit)

24. The parenchyma of Solanum potato tuber (google image)

25. Cork cells from Robert Hooke’s drawings, top, and modern microscope, below (google image)

26. Parenchyma cells of the plant, Geum urbanum, “Wood Avens” (google image)

27. Pine charcoal, top, Mahogany, below (Renshaw, op. cit)

28. Polyhedra from, “On Growth and Form”, (D’Arcy, op. cit)

29. Chart of complex polyhedra, and steel ball (google/Robb image)

30. Rotating kiln to make cement (google/Romero image)

31. Large ball mill, top, and a cutaway view of the interior while in motion (Break-day/google image)

32. Facets of a steel ball (Author)

33. Five regular polyhedra (google image)

34. The sectioned shell of a Nautilus (D’Arcy, op. cit)

35. Live Nautilus on a coral reef; its spiral shell is internal (Gritzner, op. cit)

36. The Archimedian Spiral from “On Growth and Form”, (D’Arcy, op. cit)

37. Spiral groove in gramophone record (Can stock photo)

38. Section of the spiral myelin sheath of axon of the axon of a nerve cell, (in an electron photomicrograph) (google)

Graphs 1- 2 (Author)

39. The Equiangular Spiral from “On Growth and Form”, (D’Arcy, op. cit)

Graphs 3 – 6 (Author)

40. Lines and Rectangles on the Golden Section (Author)

41. Equiangular Spiral built on the Golden Section (After a diagram in D’Arcy, op. cit)

42. Construction of the equiangular spiral on an hexagonal array (D’Arcy, op. cit)

43. Plant stem apex with spirally arranged leaf primordia (Author)

44. Arrangement of the leaf primordia with 2/5 divergence (Author)

45. Phyllotaxis with a large number of leaves in equiangular spirals

46. Thirteen equiangular spirals in a rosette of Sedum leaves (Author)

47. The five equiangular spirals of the cactus, Aloe (google image)

48. The flower head of Helianthus (Sunflower) (google image)

49. Three types of gnomon (the coloured portion) (Author)

50. Exterior of the shell of the Ormer (Haliotis) where the growth lines are boundaries of gnomons of the original shell (D’Arcy, op. cit)

51. Ranges of values for α β γ (D’Arcy, op. cit)

52. Kudu buck on the African savannah (google image)

53. Sable Antelope (Christiansen, op. cit)

54. The rhino horn’s equiangular spiral (Metcalf, op. cit)

55. Ovis ammon, the Argali sheep (D’Arcy, op. cit)

56. Male narwhal, top, and a hornless female on the Greenland ice shelf, below (Nayman, op. cit)

57. Comparative anatomy of six species of related antelopes (Christiansen, op. cit)

58. Comparative anatomy of the ancestors of whales (After Futuyma, op. cit)

59. Four transformations (Author)

60. Transformation along x-axis only (After a diagram in D’Arcy, op. cit)

61. Increase in canon bone proportions (After a diagram in D’Arcy, op. cit)

62. The carapace of six different genera of crabs (After a diagram in D’Arcy, op. cit)

63. Three different types of transforms in fish (After a diagram in D’Arcy, op. cit)

64. Cartesian co-ordinates of human skull projected on to other primates (After a diagram in D’Arcy, op. cit)

65. Skulls (Author)

66. Two trends in great ape embryology and evolution? (Author)

67. A modern phylogeny or family tree of the great apes (After a diagram in Futuyma, op. cit)

68. “Doctor Boucard”, 1929, by Tamara de Lempicka, Taschen, 1990

69. Human chromosomes (from a scanning electron micrograph)

70. Evolutionary rotation of the scapula (Author)

71. D’Arcy Wentworth Thompson 1860 – 1948 (google image)

72. The Castle and shore at St Andrew’s (google image)

73. The Bering Straits (google GraphicMaps.com)

74. The mountains on the Bering Straits (google, kenpics)

75. Helicopter over the Bering Straits, 2008 (google, rusalca)

76. The Northern Fur Seal of the north Pacific Ocean (Christiansen, op. cit)

77. D’Arcy Thompson, 1860-1948, a 1950 painting by David Shanks Ewart (google image)

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