Here’s how Roald Dahl describes what happened one morning when the Queen of England invited a large and unexpected guest (“The BFG” ) to have breakfast with her:
There was a frantic scurry among the Palace servants when orders were received from the Queen that a twenty-four-foot giant must be seated at breakfast with Her Majesty in the Great Ballroom within the next half hour. The butler, an imposing personage named Mr. Tibbs, was in supreme command of all the Palace servants and he did the best he could in the short time available. A man does not rise to become the Queen’s butler unless he is gifted with extraordinary ingenuity, adaptability, versatility, dexterity, cunning, sophistication, sagacity, discretion, and a host of other talents that neither you nor I possess. Mr. Tibbs had them all.
The redoubtable Tibbs may have possessed dexterity and sagacity, but his grasp of allometry was sadly deficient. Allometry is the part of biology that studies how the size of a creature relates to other aspects of the creature’s life. Mr. Tibbs, noting that the Big Friendly Giant was four times the height of an ordinary man, decided that the BFG needed a quadruple-sized meal:
Everything, Mr. Tibbs told himself, must be multiplied by four. Two breakfast eggs must become eight. Four rashers of bacon must become sixteen. Three pieces of toast must become twelve, and so on. These calculations about food were immediately passed on to Monsieur Papillion, the royal chef.
To Tibbs’ consternation, eight eggs weren’t enough to put a dent in the giant’s appetite; even seventy-two eggs weren’t enough to satisfy him.
What was Tibbs’ mistake? And how many eggs should he have offered the BFG?
HOW TO FEED A GIANT
If stomachs were cube-shaped, we could answer this question with a simple picture: a 4-by-4-by-4 cube has 64 times the volume of a 1-by-1-by-1 cube, as can be seen from the fact that you can fit exactly 4×4×4=64 unit cubes into a cube with side-length 4. (In fact, that’s why 4×4×4, aka 43, is called “4 cubed”.)
If stomachs were spherical, we wouldn’t be able to draw such a simple picture; there’s no way to pack 64 spheres of radius 1 into a single sphere of radius 4. Still, the formula V = (4/3) π R3 governing the volume of a sphere of radius R tells us that when you multiply R by 4, you multiply V by 43, or 64.
In the case of a stomach-shaped stomach, there’s no way to pack 64 unit stomachs into a stomach 4 times as large in every direction, and there’s no simple formula for the volume of a stomach, but we can nonetheless be certain that if you scale up a stomach by a factor of 4 in every direction, you multiply its volume by 43. Putting it differently: There actually is a simple formula for the volume of a stomach, of the form V = C R3, but C isn’t a mathematical constant like π — it’s a quantity that you’d have to determine empirically (and probably messily).
What we see at work here is the concept of geometric similarity and the way volume scales with size. When two bodies are geometrically similar, so that the larger one is just the smaller one magnified by some scaling-factor s, then the volume of the larger is equal to the volume of the smaller multiplied by s3. If the BFG is proportioned like the average human male but is four times as tall, with femurs that are 4 times as long as ordinary human femurs and 4 times as wide and likewise for all his other bones and organs, then his stomach will have four-cubed times the volume of an adult human stomach, so he’ll want 64 times as many eggs as your typical diner-denizen. That is, 2 × 64 = 128 eggs (which is more than 72).
But is it realistic to assume that the BFG would be proportioned like a regular human, as shown in Quentin Blake’s illustrations? Galileo would have said “no”. In his Dialogues concerning Two New Sciences, Galileo wrote:
From what has been already demonstrated, you can plainly see the impossibility of increasing the size of structures to vast dimensions either in art or in nature … It would be impossible to build up the bony structures of men, horses, or other animals so as to hold together and perform their normal functions if these animals were to be increased enormously in height; for this increase in height can be accomplished only by employing a material which is harder and stronger than usual, or by enlarging the size of the bones, thus changing their shape until the form and appearance of the animal suggests a monstrosity.
The problem with a horse scaled up by a factor of s is that its weight would scale like its volume, that is like s3, whereas its ability to support itself would grow like the cross-sectional area of its legs, which scales like s2. Now s2 is big, but it can’t keep pace with s3; if a horse the size of a Trojan horse tried to run, its legs would break instantly.
The same issues would affect scaled-up human beings. To avoid these problems, an evolutionarily successful giant hominid would have to be very squat and round — more like the Incredible Hulk than Blake’s graceful BFG (let alone Ben Orlin’s stick-figure giant).
ELASTIC SIMILARITY
A precise version of Galileo’s insight was proposed by the engineer-biologist Tom McMahon, one of my favorite teachers back in college. Like Galileo, whose views on scale encompassed both the natural world and the human-made world (or what Galileo calls human “art”), McMahon saw no clear dividing line between the two realms. For him, the body was a precision-engineered machine, constrained by physical law and sculpted by the forces of natural selection in the direction of maximal efficiency. Consider gait: when you plant one foot and swing the other leg forward, the motion of that swinging leg is as much governed by ballistics as it is by the muscular effort involved. As far as gravity is concerned, in that moment you are just an inverted pendulum.
According to McMahon’s theory, if one examines a collection of different-sized species with similar body-plans from an engineering point of view, “only by introducing a regular distortion in shape with increasing size can all the animals be stable. For equal stability under buckling loads in a set of animals forms, the requirement is that the ratio of the cube of the length to the square of the diameter must be kept constant in every element of the body as size increases. … This new scaling theory, which we shall call elastic similarity, uses two length scales instead of one. Longitudinal lengths, proportional to the longitudinal length scale, ℓ, will be measured along the axes of the long bones and generally along the direction in which muscle tensions act. The transverse length scale, d, will be defined at right angles to ℓ, so that bone and muscle diameters will be proportional to d. Furthermore, there will be a rule connecting ℓ and d, the same rule discovered in the previous paragraph, namely, 
That’s from page 124 of On Size and Life, cowritten by McMahon and John Tyler Bonner; here
From page 125 of “On Size and Life” by John Tyler Bonner and Thomas McMahon.
A BFG constructed according to the principles of elastic similarity would have femurs, tibias, etc. that are four times as long as their normal human counterparts but a full eight times as wide. Squatness of individuals limbs translates into overall squatness of the torso, so we might expect the BFG, although four times as tall as the average human, to be fully eight times as wide from side to side and eight times as deep from front to back. So assuming his body is made from the same protoplasm as normal humans — just more of it — he’d weigh 4×8×8 = 256 times as much as a normal human.
So, does that mean that a Hulk-like BFG should have 2 × 256 = 512 eggs for his breakfast?
THE BIGGER THEY ARE, THE SLOWER THEY TICK
So far we’ve paid attention to shape and size, ignoring the aspect of time. But what’s geometrically similar isn’t necessarily dynamically similar. Consider that a small pendulum swings faster than a big one. (I’ll say more about precisely how much faster next month!)
In 1932, veterinary scientist Max Kleiber noticed that when you compare different species of mammals, ranging from mice to elephants, metabolic rate grows like the 3/4 power of mass. (Here metabolic rate refers to the entire organism’s energy throughput; when we divide by the mass, we find that the energy throughput of the average cell actually shrinks as organisms get larger.) Small animals live short, hasty lives; large animals live at a leisurely pace and digest their food slowly. As McMahon writes (on page 292 of his book Muscles, Reflexes, and Locomotion), “any variable related to metabolic rate, including the food required per day, should scale in proportion to W3/4, rather than W1.0.”
Let’s put all the pieces together. Under McMahon’s theory of elastic similarity, a hominid who’s 4 times as tall as us would be not 43 times as heavy (as would be the case for geometric similarity) but 44 times as heavy. At the same time, by Kleiber’s rule, the hominid’s metabolism would run slower than ours, so the weight-factor of 44 would correspond to a food-intake factor of only (44)3/4 = 43. Hence the BFG needs just 2 × 43 = 128 eggs after all!
DOWNSIZING
The preceding section owes a large debt to the short section of McMahon’s book (less than a page long) captioned “Feeding Gulliver”, in which he brings a biomechanical eye to the passage of Jonathan Swift’s Gulliver’s Travels that depicts the Lilliputian scientists’ debate about how much food and drink to give their guest. McMahon points out that, by assuming geometric similarity and ignoring metabolism, the Lilliputian scientists made two mistakes that happened to cancel each other out.
The idea of tiny people was certainly not original with Swift, and other authors have put their own spin on the theme; I remember from my own childhood, with great fondness, Mary Norton’s book The Borrowers and its sequels. Most creators of these works ignore the issue of scale, though the Ant-Man franchise (with numerous comic books and a movie) at least tips its hat to the biomechanical fact that ants are capable of feats of strength impossible for the lumbering likes of us.
The most recent exploration of the idea of tiny people just came to a cinema near you in the form of the new Matt Damon movie Downsizing. I haven’t had a chance to see it yet, but you can view the trailer to get a sense of the premise: those wacky scientists (what will they think of next?) invent a device that can send you on a one-way trip to Smallsville, reducing you to .0364 percent of your old volume. (You can tell that it’s Science because the percentage is given to three significant figures.)
If you think I’m about to score easy points by pedantically pointing out ways in which the movie ignores what it would really be like to be tiny, well of course I will. For instance, why aren’t the tiny people’s voices higher-pitched? Vocal chords are oscillators, and their frequency should get higher when they’re smaller. For that matter, the tiny people’s hearts should beat incredibly fast; that should be the first thing Damon’s character Paul Safranek notices when he wakes up, before he even has a chance to worry that one particular part of his anatomy might no longer be scaled geometrically relative to the rest of him.
Paul thinks that by shrinking his body and stretching his dollars he can spend the rest of his years in retirement, but what kind of leisure activities does he have in mind? If he wants to swim, he might be distressed by the increase in water’s apparent viscosity. If he wants to play golf (presumably miniature golf, played with a mini driver), he’ll find that the feel of swinging a club is totally different from what he remembers. He might enjoy sports that involve running, because his new itsy-bitsy-ness would make it easy for him to sprint at speeds that, relative to his height, appear super-human. But sooner or later it would occur to him that this kind of turbo-charged sprinting, viewed from above, resembles nothing so much as scurrying. And in washing up after his exertions, he’d find it harder to get clean because of the increased effect of surface tension.
Maybe Paul doesn’t notice these changes because he’s become stupider. After all, he’s got fewer neurons now. Or does he have smaller ones? It’s unclear that down-sizing individual cells makes sense, given the nature of cellular machinery. But if there were smaller neurons, they’d probably be stupider too.
DINOSAURS
I haven’t used the word in this essay so far (except in the title), but Death has been lurking in the background all the way through, via the theory of natural selection. If your progenitors a million years ago had not used the minimum possible muscular effort when running away from predators, thereby enabling themselves to keep running longer, they would ended up as cougar-kibble, leaving the planet to more calorically frugal hominids. Whenever you think about how well-suited the human body is to the human mode of life, you should offer silent thanks to your ancestors’ less well-endowed rivals who unintentionally did you the favor of getting out of your ancestors’ way. The non-survival of the not-fittest is a key part of the evolutionary picture. And of course the poster children for non-survival are the giant dinosaurs who died off by the millions at the end of the Cretaceous period, leaving the field free for the small mammals who gave rise to us.
The morbid side of biology has been given its due in the field of biomechanics. Consider this quote by J. B. S. Haldane, from his essay “On Being the Right Size”: “You can drop a mouse down a thousand-yard mine shaft; and, on arriving at the bottom, it gets a slight shock and walks away, provided that the ground is fairly soft. A rat is killed, a man is broken, a horse splashes.”
The juxtaposition of the two preceding paragraphs might lead us to the preposterous idea that what killed off the big dinosaurs and let small mammals survive was an event of exactly the kind that Haldane described: big animals died, and small animals survived, because they all fell down.
Fell down? How? From where? Why? Before I answer that question (or coyly refuse to), let me confess that the all-fall-down hypothesis is a deeply silly one. Fortunately for me, it turns out there’s a venue where such silliness is welcomed and even celebrated: the Festival of Bad Ad Hoc Hypotheses, or BAHFest. I gave a talk at BAHFest in April 2017, presenting (100% in jest) my own theory of what killed the dinosaurs. Part of what makes for a good BAHFest talk is the inclusion of actual science amidst the silliness. In my case, some of the science was biomechanics. I touched upon the fact that small animals can survive a fall better than large animals, as explained by the fact that the kinetic energy of impact grows roughly like the cube of an animal’s length while bone strength grows only like the square of the length. Likewise, I cited, in support of my false theory, the true fact that large birds require disproportionately large wings compared to small birds, because load grows like the cube of an animal’s length while lift grows only like the square of the length. I only wish my old teacher Tom were still alive to see me put his ideas to comic use. (As the author of several comedic novels involving science, he had his own way of combining his love of science with his goofy sense of humor.)
Oh, did you want to know how it could possibly have happened that all creatures great and small fell (some falling to their deaths) at roughly the same time, millions of years ago? You’ll just have to watch the video of my talk.
This essay is dedicated to the memory of Thomas McMahon. Thanks to Sandi Gubin, Brian Hayes, Mike Lawler, Brent Meeker, Mike Stay, and James Tanton.
Next month: Roasting a Dodo and Biking on Mars: The Magic of Dimensional Analysis.
ENDNOTES
#1: Roald Dahl knew that sometimes the dictates of story take precedence over the dictates of science. Consider his book James and the Giant Peach. At one point in the book, five hundred seagulls carry the Giant Peach (“The GP”?) over the ocean. But even as a child I knew five hundred ordinary seagulls wouldn’t be enough. I’m not sure how many it would take. What do you think, readers?
One might also consider the possibility of Birds Of Unusual Size (cf. Rodents Of Unusual Size), of which the horse-sized duck is a famous example.
#2: Perhaps Mr. Tibbs, as a mere butler, should have passed the question of portion-size on to the chef, Monsieur Pappillion. At the very least, the chef would probably have been aware that some allometric relationships are non-obvious. The example of cooking a bird springs to mind; as I’ll discuss next month, the cooking time for a bird is proportional neither to the length nor the volume of the bird but rather to its surface area.
#3: One feature of elastic similarity is that it predicts an upper limit on how big an animal adhering to a specific body-plan can be. Consider what happens when limb-diameter d exceeds limb-length ℓ. Do you get a pancake-shaped creature with pancake-shaped arms and legs? At some point, the spatial relationships between one body part and another make it impossible for each of them to change shape in the fashion dictated by elastic similarity. I like to imagine the sorts of creatures that brush up against the limits of elastic similarity — Galileo’s “monstrosities”. I’m reminded of the old song “Mister Five by Five”, as well as the famous spherical cow and the less famous spherical horse.
#4: Mike Stay writes: “The scaling of brains is at least as complex an issue as the other scaling laws you’re addressing in the essay, and the different extremes are really interesting.” He provided three links:
Wikipedia’s explanation of the relationship between brain-size and body-size in its article on the Encelphalization quotient.
The Frontiers in Neuroanatomy article “When larger brains do not have more neurons: increased numbers of cells are compensated by decreased average cell size across mouse individuals” by Herculano-Houzel, Messeder, Fonseca-Azevedo, and Pantoja.
The Discover article “How tiny wasps cope with being smaller than amoebas” by Ed Yong.
I found the third one quite amazing.
#5: Brent Meeker writes: “Are you aware of the somewhat tongue in cheek explanation by Isaac Asimov of the demise of the dinosaurs? He noted that their nerves were not myelinated and therefore had a fairly low transmission speed. On the other hand as dinosaurs grew larger their speed scaled as the square root of size (cf. Haldane). So there came a size at which a long necked dinosaur would have his head emerge from the vegetation and see a cliff but the signal from his brain to stop would travel back more slowly than his body traveled forward and he would plunge to his death.”
#6: On page 14 of Why Size Matters, Bonner says that the Lilliputian scientists didn’t make the right call when they scaled up Gulliver’s food and beverage rations by a factor of height-cubed. Specifically, he says that the amount of wine they gave him would would have made him extremely drunk. This seems to contradict what McMahon writes. Can anyone explain this to me? I’ve asked Bonner, but haven’t received an answer as of this writing. And McMahon, alas, is no longer among the metabolizing.
Perhaps intoxication is governed by a different allometric exponent than the slower process of digestion? An experiment in which pizza and beer are fed to undergraduates of various different sizes might do much to resolve the question. It could also win the experimenter an IgNobel Prize, or at the very least the adoration of the students at her institution.
#7: After you’ve seen my BAHFest talk you might enjoy the text of an interview in which I answer questions about my presentation.
REFERENCES
John Tyler Bonner, Why Size Matters. This elegant little book covers a lot of the same ground as the longer book with McMahon.
John Tyler Bonner and Thomas McMahon, On Size and Life. Great writing and beautiful pictures.
Roald Dahl, The BFG. If you’ve read the book, you’ll recall that the BFG isn’t even the largest hominid on our planet; he is called “Runt” by bigger (and not at all friendly) giants. Also, the BFG has the ability to fly by magical means that lie wholly outside the purview of speculative biomechanics.
Roald Dahl, James and the Giant Peach. You have to love a book that kills off the protagonists’ parents in such a bizarre and off-handed fashion in the first two pages.
Galileo Galilei, Dialogues Concerning Two New Sciences. Most scientists content themselves with founding at most one new science per book, but not Galileo.
D’Arcy Wentworth Thompson, On Growth and Form. The chapter “On Magnitude” is a masterpiece.
Geoffrey West, Scale: The Universal Laws of Growth, Innovation, Sustainability, and the Pace of Life in Organisms, Cities, Eonomies, and Companies. I haven’t had a chance to read this ambitious book yet, but at least I’ve borrowed it from the library, which is a start.
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How drunk you get depends on the rate of alcohol absorption rather than the absolute amount of alcohol consumed, and absorption happens through the wall of the small intestine, which scales only as s². What I’m not sure about is what happens to alcohol elimination, which I would think would scale according to the size of the liver, presumably s³. So maybe Gulliver would get very drunk but sober up fast.
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