# Proportions and allometry

## Jeremy Kahn

### Gulliver's scaling

Gulliver's Travels includes a rather lengthy (half the novel) discussion of Gulliver's voyages to the lands of great and small; to Lilliput, where humans grow only to six inches, and to Brobdingnag, a nation where the humans grow to 144 feet and higher. Swift chose those sizes because they were a handy factor of twelve, and no doubt this convenience made it easier for him to do the mathematics, having worked out that the Lilliputian foot corresponds to the Gulliverian inch, while the Gulliverian foot is merely a Brobdingnagian inch. Swift, a British citizen of the 17th and 18th century, had no doubt spent most of his life doing conversions of twelves because of the English systems for measuring length and money (pence, too, were 12 to the shilling? Does anyone know this one?).

### Swift's Dilemma

Though Swift's ideas for a simple scaled-up human (the Brobdingnagians) and a scaled-down one (the Lilliputians and Blefuscites) are interesting, we know from the scaling principles of surface-area-to-volume ratios that these creatures would have radically different life problems to face than humans in our size range do. The Brobdingnagians, for example, would confront the very problems that elephants confront--how to keep one's body from crushing one's legs.

### The Elephantine Solutions

As the layman's biology text Diatoms to Dinosaurs: the size and scale of living things (Christopher McGowan, Island Press, Washington DC: 1994) explains at length, there is one obvious solution to this mass vs. cross-sectional area of legs difficulty: make one's legs grow thicker faster than one's body grows longer. This is called positive allometry in biology. Allometry can be used to describe this pattern in everything from the relationship between body part sizes as a child grows (ontogenetic allometry), body part sizes between members of the same species in a single age bracket (intraspecific allometry), or members of different species in the same lineage (interspecific allometry).

Allometries are expected to follow the simple pattern of y = b(x**n), that is, a logarithmic relationship. Biologists have a tradition of plotting allometric or suspected-allometric relationships on log-log graphs so that anything that falls within this pattern comes out as a straight line. Anomalies in a log-log plot mean that the organism falls outside the expected proportions for one characteristic even when scaled proportional to another one. The vertebrate eye, for example, scales with negative allometry, meaning that in general, the eye gets smaller relative to the body weight as the organism gets larger. The blue whale, for example, has an eye that though large (basketballish size?) is only a tiny fraction of its total body weight, whereas a small mouse's eye is a relatively large fraction of its body weight. Eyes, apparently (and obviously when one learns the associated biology) don't get much better when they get bigger, and it isn't "worth it" to the animal to try to grow a bigger eye as the animal gets larger. On the other end of the spectrum, eyes deteriorate quickly if they were to remain as small as human eyes do relative to their body weight but shrink down to the size of the Lilliputians. The Lilliputians would all have been blind (or at least have very grainy vision!).

But the elephant's bones are not as thick as this logarithmic relationship would lead you to expect. For an elephant's femur to support the same fraction of its body weight in the manner of one of the medium-sized mammals (cats or rabbits, for example), it would have to use about 70 or 80 percent of its body mass in pure bone. Elephants obviously don't do that, so how do they survive? By changing the way they behave. They are slow, ponderous animals who refuse to pounce, run, or scamper--daily requirements of survival for your average mammal. But the great pachyderms have no natural predators, so they can afford to become larger, even without strengthening their bones, because they can modify their behavior instead. This leads to some interesting questions about how the Lilliputians and Brobdingnagians might have used behavioral modifications to keep themselves alive.

### Thoughts on the binomial and its extensions:

To change topic rather dramatically...

#### Pascal's Tetrahedron?

I understand that Pascal's Triangle is the extension of the binomial theorem to express (among other things) the coefficients of the various powers of a and b in the expression (a + b)**n. I realize that it is fairly late at night, and this may be why I cannot immediately see it, but how does this correspond to the Trinomial Theorem? The expression (a + b + c)**n is a perfectly reasonable way to think about dimensionality, as well, and the resulting shapes, at least in the division of a 3-cube, leave coefficients that match up with what appears to be the top of what I will call Pascal's Tetrahedron. Each face (well, each face descending from the 1; there technically is no bottom face since it extends forever) is an imitation of the Triangle, but when sectioned from above (leaving n fixed and looking only down on the triangle formed, it is more complicated than the Triangle.

#### Wild Speculation

Conjecture: the n-nomial Theorem's coefficients of exponents would be determined by Pascal's n-Simplex.

A problem to sleep on... Good night to all.

mail Jeremy Kahn with comments or questions

or call x6753