Size and Shape

The immutable laws of design set limits on all organisms.

When the kindly entomologist of Them discovered that the giant queen ants had left for their nuptial flight, he quickly calculated this simple ratio: a normal ant is a fraction of an inch long and can fly hundreds of feet; these ants are many feet long and must be able to fly as much as 1,000 miles. Why, they could be as far away as Los Angeles! (Where, indeed, they were, lurking in the sewers.) But the ability to fly depends upon the surface area of the wings, while the weight that must be borne aloft increases as the cube of length. We may be sure that even if the giant ants had somehow circumvented the problems of breathing and growth by molting, their chances of getting off the ground would have been far worse than that of the proverbial snowball in hell.

Other essential features of organisms change even more rapidly with increasing size than the ratio of surface to volume. Kinetic energy, for example, increases as length raised to the fifth power. If a child half your height falls unsupported to the ground, its head will hit with not half, but only 1/32 the energy of yours in a similar fall. A child is protected more by its size than by a “soft” head. In return, we are protected from the physical force of its tantrums, for the child can strike with, not half, but only 1/32 of the energy we can muster. I have long had a special sympathy for the poor dwarfs who suffer under the whip of cruel Dr. Alberich in Wagner’s “Das Rheingold.” At their diminutive size, they haven’t a chance of extracting, with mining picks, the precious minerals that Alberich demands, despite the industrious and incessant leitmotif of their futile attempt.

This simple principle of differential scaling with increasing size may well be the most important determinant of organic shape. J.B.S. Haldane once wrote that “comparative anatomy is largely the story of the struggle to increase surface in proportion to volume.” Yet its generality extends beyond life, for the geometry of space constrains ships, buildings, and machines, as well as animals.

The great range of designs among medieval churches can be attributed partly to size. The twelfth-century parish church of Little Tey, Essex, England, was only 57 feet long and had a simple floor plan (lower right corner). The floor plan of Norwich Cathedral, also twelfth century, shows adaptations—transept, chapels—required for the 450-foot-long building. The need for light and support dictated complex cathedral layouts.

Medieval churches present a good testing ground for the effects of size and shape, for they were built in an enormous range of sizes before the invention of steel girders, internal lighting, and air conditioning permitted modern architects to challenge the laws of size. The tiny, twelfth-century parish church of Little Tey, Essex, England, is a broad, simple rectangular building with a semicircular apse. Light reaches the interior through windows in the outer walls. If we were to build a cathedral simply by enlarging this design, then the periphery of the outer walls and windows would increase as length, while the area that light must reach would increase as length times length. In other words, the size of the windows would increase far more slowly than the area that requires illumination. Candles have limitations; the inside of such a cathedral would have been darker than the deed of Judas. Medieval churches, like tapeworms, lack internal systems and must alter their shape to produce more external surface as they are made larger.

The large cathedral of Norwich, as it appeared in the twelfth century, had a much narrower rectangular nave; chapels have been added to the apse and a transept runs perpendicular to the main axis. All these “adaptations” increase the ratio of external wall and window to internal area. It is often stated that transepts were added to produce the form of a Latin cross. Theological motives may have dictated the position of such “outpouchings,” but the laws of size required their presence. Very few small churches have transepts.

I have plotted periphery versus the square root of area for floor plans of all postconquest Romanesque churches depicted in Clapham’s monograph of English ecclesiastical architecture. As we would predict, periphery increases more rapidly than the square root of area. Medieval architects had their rules of thumb, but they had, so far as we know, no explicit knowledge of the laws of size.

Like large churches, large organisms have very few options open to them. Above a certain size, large terrestrial organisms look basically alike—they have thick legs and relatively short, stout bodies. Large Romanesque churches are all relatively long and have abundant outpouchings. The invention of the flying buttress strengthened later Gothic buildings and freed more wall space for windows. Churches could then become relatively wider and simpler in outline (as in the Cathedral of Bourges).

The “invention” of internal organs helped animals retain the highly successful shape of a simple exterior enclosing a large internal volume; and the invention of internal lighting and structural steel has helped modern architects design large buildings with simple exteriors. The limits are expanded, but the laws still operate. No large Gothic church is higher than it is long, no large animal has a sagging middle like a dachshund.

I once overheard a children’s conversation in a New York playground. Two young girls were discussing the size of dogs. One asked: “Can a dog be as large as an elephant?” Her friend responded: “No, if it were as big as an elephant, it would look like an elephant.” I wonder if she realized how truly she spoke.


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