A TRIBUTE TO D'ARCY THOMPSON R.J.C.Wilding. BDS, Dip Pros. M.Dent. Ph.D

It makes a refreshing change in reading, to come across some old fashioned, leisurely science. Just after the turn of the century, D'Arcy Thompson wrote "On Growth and Form" (it was revised and reprinted in 1942). His attention was drawn to the unity of life and the relationship between form and function. His many drawings not only reflect his love of nature, he was diverted by shellfish, frogs, snowflakes and bubbles, but his drawings also reveal a perception for mathematical pattern and mechanical design which were well ahead of his time. It is just as well he was not dependent on the approval of a committee for his research funds, as he would surely have had all his proposals turned down. Firstly, because there were too many of them, and secondly, because his work is quite free of statistical analysis, an icon of modern science. His genius lay in making order out of the great diversity he recorded. All his observations seem to have been collected for the contribution they could make to his search for a unifying design in nature. He was looking for something which would tie all the repeated biological phrases together, a unity of purpose amongst all the diversity. And he found it, in mathematical expressions for the helix of a shell or the shapes of growing cells; he found it in the structure of snow flakes, the cracks in basalt and the stripes of a zebra.

He argued that there were common laws at work." The search for differences or fundamental contrasts between the phenomena of organic and inorganic, of animate and inanimate things, has occupied many mens minds, while the search for commonality of principle or essential similarities, has been pursued by few; the contrasts are apt to loom too large, great though they be".

Thompson knew there were common principles among quite diverse forms of life, but he felt it necessary to be cautious about debating the philosophical or religious basis for this unity. The rigour of 20th Century science would not tolerate a meaning or purpose to nature. Scientists at the turn of the century had only just shaken off the cosy ideas that Providence had indeed provided. In an essay called "Huttons Purpose" Stephen Gould recalls the joy of a geologist, who in 1870 had written that coal was a wonderful gift Providence had made, so that we might keep warm. Yet more joy; Providence, with characteristic consideration, had arranged that this buried treasure, too deep for us to discover, occasionally rose to the surface and came peeking through, as if to say "Here I am"! So, Thompson was understandably wary of giving a purpose to the form and function of living things. But he felt reasonably certain that that the universality of the principles of physics and mathematics could be upheld. He wrote "in general, no organic forms exist, save as are in conformity with physical and mathematical laws". Here he was on safer ground.

He decided that the laws most commonly found to operate as unifying principles were those concerned with static and dynamic forces. "The form of any particle of matter, whether it be living or dead, and the changes in form which are apparent in its movements and in its growth, may in all cases be described as due to the action of force". Forces of tension, compression and shear occurred in all living structures and influenced both growth, function and form.

He observed that the bones of a museum skeleton would lie in a heap on the floor without the clamps and rods pulling them together. In the living animal, he argued, tension holds the skeleton together as much as weight does. We have difficulty in remaining upright while we go to sleep, and so do most large mammals, because we have to keep our muscles active, in literally, pulling ourselves together. But tension is not only developed by muscle pull. The ligaments of our joints are also vital structural elements, less obvious in man because our joints allow an unusual freedom of movement. The forelegs of a horse lock when bent slightly backwards, preventing the leg from bending, thereby allowing the horse to sleep while standing up, without expending much energy in keeping postural muscles active. In this case it is the ligaments of the joints of the foreleg which stop the skeleton from collapsing.

Man-made structures are also held together by the same tension. The towering medieval cathedrals were in fact heaps of stone without any significant tensile stresses. They were built on the Roman principal that arches are better able to support weight than lintels. But steel provided a material which resisted tension. So instead of bridges resting on piles of stones, they could be suspended on steel cables, hung between concrete abutments; in the suspension bridge, the abutments are pulled inwards. Victorian bridges gave Thompson the analogies he wanted to make with living structures. He observed that the fore quarters of a buffalo, follow the design principles of the Forth Bridge. The sturdy front legs support the shoulder girdle from which the head and chest are suspended. The shoulders hang under an arch of high vertebral spines tied across by ligaments. The use of bone and tendon in nature was surely related, he urged, to the use of stone and steel by engineers.

Based on this and many other examples, D'Arcy Thompson believed that there was purpose in design in living structures which obeyed engineering principles.

There were common principles to be found at other levels. A useful structural element of bridges, plants and animals is the tube. We are reminded of a reed, the quill of a feather, an antler, a long bone of an limb. Tubes are light and provide stiffness, although they do have a tendency to collapse and kink in the middle if they have to support any weight, especially if the tube is long. The weight a tubular column will support, is in fact inversely proportional to the square of its height. The tall stem of a bamboo would collapse were it not divided into a series of short units by periodic thick bands; each unit being short enough not to collapse. The hollow shaft of the femur gets progressively thicker towards the middle to prevent collapse at this weak point of the tube.

Tubular structures can be strengthened by longitudinal fibres. Plant stems are responsive to the loads applied and can increase the amount of fibre if necessary. A peach stalk gets stronger (not thicker) as the peach grows heavier. Thompson found that young sunflower stalks broke if loaded with 160 gms. If loaded, but not fractured, by say 150 gms for 2 days the load on the stalk increased to 250 gm without causing fracture. Thompson concludes that "strain, the result of stress is a direct stimulus to growth itself. Growth may be coordinated with the structural adaptations required by mechanical requirements." This principle is, a fair prophesy of work which has only emerged towards the end of this century which identifies internal bone strain as a stimulus for deposition. If lines of stress are studied by engineers they can be used to predict the design required to relieve them. "The skeletal form, as brought about by growth, is to a very large extent determined by mechanical considerations and tends to manifest itself as a diagram, or reflected image of stress".

Structural adaptations to stresses may occur within a material, without change in its exterior size and shape. Shearing strain has the effect of displacing fibrous elements away from lines of stress. Examples are combing shanks of wool, drawing out wire, repeatedly folding a Samurai sword blade. The result of these shearing strains is that the fibres becomes aligned at right angles to the shear stress. So Thompson concludes, " there is a tendency for a material to be laid down just in lines of stress thereby avoiding the disruptions due to shear". The trabeculae of the femoral head are indeed arranged along lines of stress; the pattern of one set crossing the other at right angles in accord with theoretical stress design.