Growth and Proportion
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Text length: 1,795 words
Excerpted from On Growth and Form, Chap. II
by D'Arcy Thompson
, published 1917
Forces shaping growth and form do not scale proportionally - a change in overall size usually requires a change in structure to accommodate these increased pressures
There are important differences in kind between the forms of large and small things because the same forces affect them differently
Growth requires trade-offs - in order to maintain basic operations, enormous design innovations and increased complexity are required to support continued growth
Keywords:
Growth, allometry, shape, form, force, scale, development, proportion, gravity, surface, volume, large, small, organism, biology, engineering, material, strength, mass, similitude
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Summary
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rowth is a natural process of change in a system in response to forces acting on the system. While non-living systems can grow (e.g., a puddle in a rain storm), it is the growth of living organisms that is most mysterious. At any stage of growth, a system has a specific form, whether it’s the shape of the puddle or the dimensions and structure of an organism. This form provides clues to the growth of the organism.
D’Arcy Thompson, the great naturalist and polymath, explores this connection in the enduring classic On Growth and Form. As long as something is growing, changes of form are likely. In this excerpt Thompson shows that not all changes are possible; there are strong constraints imposed by the forces acting on the organism and by the space in which the growth occurs. The ‘Principle of Similitude’ says that forces acting on a system can vary in different ways with the linear dimension of the system. For example, the ability of a tree leaf to gather light varies with the square of the linear dimension while the total mass of the tree varies with the cube of the linear dimension. This simple fact imposes limits on how big trees can get. The same forces do not act identically on large and small organisms – e.g., bacteria vs. whales – so the ways in which these organisms can grow is quite different. Scale constrains growth and, as a result, form.
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On Growth and Form
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he terms Growth and Form, which make up the title of this book, are to be understood, as I hardly need say, in their relation to the study of organisms. 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 such as are in conformity with physical and mathematical laws. And while growth is a somewhat vague word for a very complex matter, which may depend on various things, from simple imbibition of water to the complicated results of the chemistry of nutrition, it deserves to be studied in relation to form: whether it proceed by simple increase of size without obvious alteration of form, or whether it so proceed as to bring about a gradual change of form and the slow development of a more or less complicated structure…
The form, then, of any portion of matter, whether it be living or dead, and the changes of form which are apparent in its movements and in its growth, may in all cases alike be described as due to the action of force. In short, the form of an object is a ‘diagram of forces’, in this sense, at least, that from it we can judge of or deduce the forces that are acting or have acted upon it: in this strict and particular sense, it is a diagram – in the case of a solid, of the forces which have been impressed upon it when its conformation was produced, together with those which enable it to retain its conformation; in the case of a liquid (or of a gas) of the forces which are for the moment acting on it to restrain or balance its own inherent mobility. In an organism, great or small, it is not merely the nature of the motions of the living substance which we must interpret in terms of force (according to kinetics), but also the conformation of the organism itself, whose permanence or equilibrium is explained by the interaction or balance of forces, as described in statistics.
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The principle of similitude
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o terms of magnitude, and of direction, we must refer all our conceptions of form. For the form of an object is defined when we know its magnitude, actual or relative, in various directions; and Growth involves the same concepts of magnitude and direction, related to the further concept, or ‘dimension’, of Time. Before we proceed to the consideration of specific form, it will be well to consider certain general phenomena of spatial magnitude, or of the extension of a body in the several dimensions of space.
… We see that an understanding of the correlation between length and weight in any particular species of animal, … enables us at any time to translate the one magnitude into the other, and (so to speak) to weigh the animal with a measuring-rod; this, however, being always subject to the condition that the animal shall in no way have altered its form, nor its specific gravity. That its specific gravity or density should materially or rapidly alter is not very likely; but as long as growth lasts changes of form, even though inappreciable to the eye, are apt and likely to occur. …
A common effect of scale is due to the fact that, of the physical forces, some act either directly at the surface of a body, or otherwise in proportion to its surface or area; while others, and above all gravity, act on all its particles, internal and external alike, and exert a force which is proportional to the mass, and so usually to the volume of the body…
In short, it often happens that of the forces in action in a system some vary as one power and some as another, of the masses, distances, or other magnitudes involved; the ‘dimensions’ remain the same in our equations of equilibrium, but the relative values alter with the scale. This is known as the ‘Principle of Similitude,’ or of dynamical similarity, and its consequences are of great importance. In a handful of matter cohesion, capillarity, chemical affinity, electric charge are all potent; across the solar system gravitation rules supreme; in the mysterious region of the nebulae, it may haply be that gravitation grows negligible again.
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