Wizards, Aliens, and Starships: Physics and Math in Fantasy and Science Fiction

by Adler, Charles L.

  One response to this entire chapter is to say, “Well, they’re magic candles, so they burn brighter than ordinary ones, and don’t drip, and cost nothing.” This is perfectly acceptable, although J. K. Rowling never writes that they are. It doesn’t detract anything from the story to take this view. However, another response is to say that this is one detail of a fascinating book, and it is part of “the Game” to examine such details and see where they lead us. I feel that the latter approach is far more interesting and more fun than leaving the books unexamined.

  NOTES

  1. We will usually round this to c = 3 × 108 m/s, which is almost always close enough for any calculation we make.

  2. Of course, tungsten bulbs themselves aren’t ideal illuminators, which is why people are switching to compact fluorescent bulbs. The fluorescents have higher effective temperatures, so even more of their light is in the visible spectrum.

  CHAPTER FOUR

  FANTASTIC BEASTS AND HOW TO DISPROVE THEM

  4.1 HIC SUNT DRACONES

  One of the chief hallmarks of a fantasy story is the presence of otherworldly creatures. This is true of science fiction as well. In science fiction they are called aliens rather than monsters, but in numerous works the two are treated in a very similar manner. The troll at the bridge judges whether the party is worthy to cross or whether he must eat them; the aliens from Alpha Centauri judge humanity and decide whether to destroy us or reward us with advanced technology.

  The chief difference, at least between “hard” science fiction—that based on known scientific principles—and fantasy, is that aliens in science fiction need to “work” somehow; that is, their biology must function along known and understood scientific lines. This does not mean they must be dissimilar to monsters or other creatures in a fantasy period. For example, in Robert Heinlein’s novel Starman Jones, the aliens attacking the human settlers on an unknown world look very much like centaurs, hybrids between humans and horses [122]. Another Heinlein novel, Glory Road, features beasts that are very much like dragons (i.e., large firebreathing lizards) except they cannot fly [116]. Of course, this novel has very strong fantastic elements. There are countless examples from other stories in which alien creatures are based on mythological or fantastic animals.

  This raises an interesting question: which fantasy animals are possible ones and which aren’t? That is, if we took a census of fantastic beasts from various fantasy novels, which could we then place in a science fiction story without having to bend the rules too much? Here’s a very small list of fantastic creatures from the Harry Potter novels (not, of course, unique to them): centaurs, giants, unicorns, giant spiders (acromantula), hippogriffs, and dragons. Which of these are possible beasts and which are impossible? To answer the question we need to look at the issue of scaling and how it affects the properties of animals and their metabolism.

  4.2 HOW TO BUILD A GIANT

  The recent movie Avatar is set on the world of Pandora, a moon of a gas giant circling one of the stars in the Alpha Centauri system. The surface gravity on Pandora is about 80% Earth normal; the Pandorans are portrayed as large, willowy beings standing about eight feet tall and possessed of great physical strength compared to humans. Is this accurate? Given the facts above, how large should a Pandoran be?

  There is a lot to this question, and many aspects of biology enter into the answer. However, the field of biomechanics can guide us in a general consideration of this problem. If we want to build a giant, how do we go about it? Can we simply take a human and, say, double him in all dimensions? Or is it more complicated than that?

  In the early 1920s, J. B. S. Haldane, the great mathematical biologist, wrote the fascinating essay “On Being the Right Size” [106]. In one part of it he examined whether giants such as Giant Pagan from The Pilgrim’s Progress could stand. This is an interesting point: if we look at organisms from the small (say, insects) to the largest land animals, such as elephants, we realize their shapes are not particularly similar but do change in a systematic way. The legs supporting insects are very narrow and spindly compared to their body size, while the legs supporting an elephant or other giant creature are very thick compared to its body. Also, there are no creatures that stand on two legs, even for short periods of time, that are larger than humans. This is an aspect of a famous law originally formulated by Galileo and usually referred to as the square-cube law: the weight an animal must support is proportional to the cube of the size of the animal, but the supporting structure is proportional only to the square of the size. There are a lot of ramifications to this law, and it is not the only scaling law that applies to biology, so we will go through it slowly and in detail.

  Let’s model an upright biped (human, Pandoran, or giant) as a large load standing on two pillars. This will do for a static model; what happens when the thing walks I’ll defer until later. How much weight can be put on the pillars before they collapse? As we will see, this determines the overall structure of the biped.

  Let’s model a human as a cylinder (torso and head) standing on two cylindrical pillars (the legs). We’re ignoring the arms and head, but I’m assuming that their weight is not significant compared to the weight of the torso. I’m also going to ignore the weight of the legs. It’s an oversimplified model but should be good to start us out. I’m going to assume that the length of the torso (L) is the same as the length of the legs, and that the radius of the cylinder is about one-sixth the length, meaning that the circumference of the cylinder is very roughly equal to its length; this seems reasonable, given average belt lengths. This is a very rough sketch of a human being, obviously.

  The volume of the torso is given by

  and the mass is M = ρV, where ρ is the average density of the human body, very roughly equal to the density of water (1,000 kg/m3). This seems a reasonable assumption for the overall shape of a human torso: a simple calculation of the mass for a human whose overall height 2L = 1.8 m gives a torso mass of 60 kg, or 130 lb, which seems about right.

  What we are interested in here is the width of the support pillars, the legs. The big question is one of scaling: as we increase the mass from that a normal man to that of a giant, will the overall shape stay the same? That is, will the width of the legs with respect to their length stay the same, or should we make them thicker, as Haldane implied? And can we estimate what the maximum height for such a giant could be, both on Earth and on other planets (such as Pandora), with different values of g?

  There is one other question one can ask: why did we make the assumptions that we did, that is, that the overall leg length is equal to torso length, and that the torso diameter is proportional to torso length? We’ll address that below as well.

  4.2.1 Biological Scaling

  The principle of allometric scaling is an extremely important one in biomechanics. Allometry is any relation that can be written between two biological variables in the form of a power law:

  Quite often the independent variable is the mass, mainly because it is extremely easy to measure. For example, let’s consider the relation between the mass and the length of the “cylinder torso” we modeled above. We can model the dependence of the length of the cylinder on its mass as

  We expect a relation of this sort between body length and mass, as mass is proportional to volume and volume has dimensions of (length)3. Another way to put this is that if we take two objects of the same shape but different overall scale, the ratio of their volumes is the cube of the scaling factor: as the size doubles, the volume and mass increase by a factor of 23, or 8.

  This relation holds exactly only for objects that are the same shape. Animals, of course, come in all shapes and sizes. However, studies relating animal mass to overall length show that over a very wide range of sizes and species, the law that overall length is proportional to the cube root of mass holds relatively well. The surface area is related to the mass by the scaling parameter

  A ∝ M0.67,

  again, consistent with geometrical sca
ling.

  Back to our giant. A column such as a leg bone will buckle if the weight it supports exceeds a critical value. The great and prolific mathematician Leonhard Euler was the first person to investigate this problem, which is consequently referred to as Euler buckling: a column with a circular cross section of radius r and length L will buckle if the weight atop it exceeds a critical value, given by

  Here, E is the elastic modulus of the material, which is a measure of how “stretchy” or “bendy” the material is. I am going to assume that the leg proportions are set by this relationship: animal support-bone lengths and widths will be set by the criterion that they need to be able to support the weight on top of them safely and without buckling. Instead of solving the problem exactly, I’ll discuss some scaling issues here. Since the weight is proportional to the volume, which scales as L3, we can write out a proportionality,

  Solving this,

  r ∝ L5/4.

  Another way to put this is that the ratio of radius to length, r/L, is proportional to the fourth root of the length L1/4. As the being increases in size, the width of the legs will grow out of proportion to the rest of the structure. Giants should appear squat compared to normal human beings because they need thicker supports in relation to their height to hold up their weight. One can see that this is true by analogy with quadrupeds: elephants don’t look like scaled-up horses; their legs seem a lot thicker relative to body size than a horse’s legs. Grawp the giant, first introduced in Harry Potter and the Order of the Phoenix, is described as being 16 feet tall, or about three times the height of a normal human, if we round up [204]. His legs should therefore appear 31/4 = 1.3 times thicker than a human’s in relation to their length. To be blunt, this calculation surprised me, as I would have expected the legs to be much thicker. Of course, this model is simple-minded in that I have assumed that the length of the leg is simply proportional to the overall body length, which is questionable.

  There’s an interesting supporting piece of evidence for this model. Bone mass is proportional to bone volume, which scales as r2L ∝ L14/4 ∝ M1.17. That is, the bone mass (at least for leg bones) should increase more rapidly than the overall mass of the creature. This is in fact measured: bone mass increases as M1.08. This is less than our estimate indicates, but perhaps not too far out in left field. This may simply indicate a failure of geometrical scaling for our overly simplified model of a biped.

  One often sees the assertion that giant humanoids (at least on Earth) are impossible because of square-cube arguments, that is, because the weight increases so much more rapidly than the surface area supporting it. A recent paper in Physics Education states (in discussing the maximum height reached by an animal):

  What would be the minimum required thickness of the bones of animals? To answer this question, let us examine the resistance of the bones and the weight-bearing limbs. Compressed solid materials may undergo a maximum tension, Tbr, before breaking. [70].

  The authors then go on to state that because of this rule, bone diameter should scale as bone length to the 3/2 power. This is simply untrue: the scaling law quoted above for bone mass implies that bone diameter scales at a lower power (5/4 = 1.2) with bone length, and again, studies of bone mass in relation to body mass bear out a lower scaling power. However, there may be a nugget of truth in the statement. The model we have put together assumes that the scaling between bone length and bone diameter is caused by the need to avoid Euler buckling of the structure. At low masses, the force needed to cause the structure to buckle is less than the force needed to break the bones in compression, but at some critical value of the height/mass ratio, this criterion is reversed. This is worked out in detail in one of the problems on the book’s website (press.princeton.edu/titles/10070.html).

  We can do a few other interesting calculations. For example, the walking speed of a biped is set by the pendular frequency of the swing of its legs. Detailed arguments can be found in Steven Vogel’s textbook, Comparative Biomechanics, but essentially, the period of a pendulum is proportional to , where L is the length of the pendulum and g is the acceleration of gravity [243]. Viewing the moving leg as a swinging pendulum of length L, the stride length will be proportional to L, so the speed will be proportional to . If we assume that a typical human walking speed is about 4 mph (about 1.8 m/s) and that L is about 1 m, we arrive at the formula

  A few things to note:

  • Larger animals have greater walking speeds than smaller ones. Interestingly, the time it takes to make a stride increases with size, but this is more than compensated for by the increased length of each stride.

  • Higher-gravity planets will have faster walkers because of the factor g in the equation.

  Grawp, being three times the size of a human, should be able to move 70% faster than one, although the movements will seem labored because of the time it takes for each stride.

  Now, back to the Pandorans: they are giants compared to humans. The males’ average height is 3 m, or roughly 10 feet. By comparison, a typical human height of 6 feet is about 1.9 m. Their planet has only 80% of the surface gravity of Earth. Because of this, human walking speed on Pandora should be about 10% less than on Earth because of the lower gravity. This should be noticeable, but the movie doesn’t show people walking more slowly. However, Pandorans on Pandora should move 30% faster than humans because of the size difference. One should note that these results are dictated by simple physics and by the fact that Pandorans seem to walk the same way humans do.

  Are these scaling laws correct? One must be a bit careful when applying them. The results should be taken with a grain of salt, for several reasons:

  • First, these considerations are dictated by general physical principles, and may not be valid under local, specific conditions.

  • Second, technically speaking, these scaling laws should apply only when comparing different species with similar shapes. That is, we might expect these laws to apply when comparing humans to (hypothetical) giants (i.e., two different species of bipeds), but not when comparing humans to horses or horses to eagles. However, no two species are ever exactly scaled versions of one another, so the results above have to be taken with a grain of salt.

  • Finally, all of these scaling laws have to be assessed against actual experimental data, which can be tricky. It is hard to measure these scaling parameters with precision; usually they are quoted with error bars of about 10% or more.

  Actual scaling parameters from studies done on real animals indicate that the arguments given above are within the bounds of possibility, but not proven. There is simply too much smear in the actual data. In particular, body surface area scales with mass to a power somewhere between 0.63 and 0.67. The higher exponent is what one would expect if geometric scaling were the only issue involved, and we did not need to worry about the support forces on the structure.

  4.3 KLEIBER’S LAW, PART 1: MERMAIDS

  “The whale is not a fish, you know—it’s an INSECT.”

  —MONTY PYTHON: THE SECRET POLICEMAN’S BALL

  4.3.1 Mermaids Aren’t Fish!

  I have made perhaps one of the most important discoveries in the field of cryptozoology in the last century: mermaids aren’t fish. Perhaps I should say “merpeople” instead.… This actually seems a fairly obvious conclusion, at least when one looks at almost all depictions of mermaids in popular culture. The big clue is that merpeople, in almost any depiction you see them in, have horizontal flukes. This means they are actually oceangoing mammals, like whales, and not fish at all. Perhaps I should explain.

  Herman Melville is best known for his massive fantasy novel, Moby-Dick. There are no mermaids in the story, but boy are there plenty of descriptions of whales. In the chapter “Cetology” he enters into a long discussion of whether the whale is a fish:

  To be short, then, a whale is A SPOUTING FISH WITH A HORIZONTAL TAIL. There you have him.… A walrus spouts much like a whale, but the walrus is not a fish because he is amphibious. But the last
term of the definition is still more cogent, as coupled with the first. Almost any one must have noticed that all the fish familiar to landsmen have not a flat, but a vertical, or up-and-down tail. Whereas, among spouting fish the tail, though it may be similarly shaped, invariably assumes a horizontal position.

  Melville places whales among the fishes, but he cites a number of reasons why biologists don’t: the whale has lungs, not gills; the whale lactates and gives birth to its young, rather than laying eggs; and the whale spouts. And, as in almost all depictions of mermaids, it has a horizontal tail (or fluke), whereas all true fish have vertical tails. This implies a vastly different method of swimming for them: sharks, for example, swim by wriggling their fins and bodies back and forth, in a motion almost impossible to emulate by a swimming person. A whale, on the other hand, moves its tail up and down, which can easily be done by any swimming human. This is because the whale is a seagoing mammal. Millions of years ago the ancestors of the whale left the land and went back into the oceans. Their hind legs fused together to form the tail, which kept its skeletal structure more or less intact, meaning that the joints in it are hinged for up-and-down motion. Indeed, the cartoonist Chuck Jones, when trying to teach animators how to animate seals swimming, took one of his grandsons and tied his forearms to his torso and his two legs together, then put flippers on hands and feet and tossed him into the pool [131]. The natural motions the boy made in keeping afloat were very similar to how seals and all other sea mammals swim.