Caterpillars rely on a flexible hydrostatic skeleton for support and locomotion.
The same dependence on different aspects of geometry holds for functional relationships. The forces that can be produced by a muscle or the strength of a bone are in each case proportional to their cross-sectional areas; the weight of an animal is proportional to its volume.
Subjected to the forces of Brownian motion,
(1958) is only half the size of the Amazing Colossal Man, but she also pushes the skeletal safety factor beyond reasonable limits.
Understanding scale is key to the problems raised in another classic of life on the small side. Dr. Cyclops (1940) is a tale of a mad scientist who retires to a remote island to perfect his secret machine, a device that emits atomic rays (five years before the bomb!), shrinking anything in their path. When his solitude is disturbed by interlopers (Im convinced he mistook them for a granting agencys site-inspection team), he shrinks them; the rest of the movie follows the battles between the giant doctor and his miniaturized visitors.
I am, by training, an invertebrate zoologist, and virtually all of my research has focused on the biomechanics of marine invertebrates. Thus I note with considerable joy that Hollywood has not forgotten our slimy relatives. My all-time favorite for quality of special effects isIt Came from Beneath the Sea(1955), in which a giant deep-sea octopus, unable to capture its normal prey after it becomes radioactive from eating fish contaminated in an atomic bomb test, invades shallow waters looking for lunch. After snacking on a couple of freighters, the monster discovers San Francisco, where he adds a few police cars, railroad box cars, and the clock tower of the Ferry Building to his diet. The most famous (and visually striking) scene occurs as the monster reaches up out of San Francisco Bay, entwines its tentacles around the Golden Gate Bridge, and pulls the bridge down.
For high camp, you cant do better thanMothra(1962). Two six-inch tall women are kidnapped from a Pacific island by a showman who plans to make his fortune by exhibiting them. The dastardly deed causes a giant egg to hatch into a giant caterpillar that swims across the Pacific, devouring everything in its path, and getting bigger by the minute. After reaching Japan and crushing a large portion of Tokyo, it crawls up a radio tower, spins a cocoon, and a few days later emerges as a moth with the wingspan of a couple of 747s. The downdraft of those wings completes the destruction of Tokyo: buildings are blown down, cars fly through the air. The authorities admit defeat and the tiny women are brought to the airport where the giant moth lands; after the ladies climb aboard, the monster flies off over the Pacific, never to be seen again.
Of course, as an old gem of black humor notes, its not the fall that hurts you, its the sudden stop at the end. A falling object acquires kinetic energy (KE=1/2mv2). That kinetic energy, proportional to the velocity squared, must be dissipated to bring the object to a halt. Heres where being small is a good thing. Not only do smaller objects fall more slowly but, because of the squared-velocity term in the kinetic energy relationship, there is much less energy to be dissipated on impact and thus less injury. (Those of you who vaguely remember Galileo dropping things from the Tower of Pisa may be bothered by the preceding. Galileo used iron balls, where the drag is trivial compared to the force due to gravity, and the fall was not long enough for the balls to achieve any significant fraction of their terminal velocity.)
The evidence clearly points to the poor cephalopod suffering a sudden and massive cerebral hemorrhage from this excess pressure just as it rips down the Golden Gate Bridge. The subsequent passivity of the giant octopus now makes perfect sense–its higher faculties were gone and the only responses it made were due to peripheral reflexes, grabbing the submarine in response to tactile stimulation, twitching when hit with the divers spear. Rather takes the edge off the humans heroic actions at the end of the movie, doesnt it?
Much of the movie is taken up with the Lilliputian team struggling to climb up onto pieces of furniture and then back down again. For the latter, they need not have invested so much time and effort in securing pieces of string to use as ropes: they could simply have jumped. When any object falls, it accelerates until the drag force equals the force generated by gravity acting on its mass; from then on, the velocity is constant. This speed is known as the terminal velocity; for a full-sized human its about 120 mph and is very terminal indeed. However, the drag on an object is proportional to its cross-sectional area, while the force due to gravity is proportional to its mass (and thus volume, if density is constant). As objects get smaller, gravitational pull decreases more rapidly than drag, so terminal velocity decreases.
Similar but less severe problems afflict giraffes, who have to have taut skin on their legs (a living version of Supp-hose) to minimize edema (fluid being forced from the capillaries into the tissues due to the high pressure). Things must have been even worse for the large sauropod like Apatasaurus and Brachiosaurus. Indeed, some folks have suggested that they couldnt have lifted their heads straight up without passing out; it is doubtful their hearts were strong enough to pump blood that high. The poor dinosaurs couldnt have avoided the problem as the octopus did for most of the movie–by keeping the body submerged. A sauropod standing submerged with only its head sticking out of the water would have no problem with edema in the legs or low blood pressure in the head, but if the lungs were deeper than 8-10 feet down, it is doubtful that it would have been able to breathe; it is unlikely that it would have been able to inflate its lungs against the ambient pressure.
The chart at right describes Bieweners findings. On the horizontal axis is body mass, running from a tenth of a kilogram (about three ounces) on the left to 5,000 kilograms (about 5 tons) on the right. The vertical scale is stress, measured in force per unit area. The strength of bone does not vary from one mammal to another: for all mammals, bone breaks when the stress it carries exceeds about 200 megapascals (Mpa)–the hatched region in the middle of the graph. Say you had an animal the size of a chipmunk (body mass about 0.1 kg). Its bones have been measured to carry a stress of about 50 Mpa during routine locomotion. What if that chipmunk got bigger, either through evolution or the effects of consuming radioactive tomatoes, as in so many of these movies? If the chipmunks bones simply enlarged proportionally with no change in shape, stress in the bone would follow the solid curve on the left, with stress increasing as the cube root of body mass. Note that at about 10-20 kg body mass (about 20-40 pounds), that line intersects the hatched region. The implication is that at that size, our hypertrophied rodent cannot move–even routine locomotion would generate high enough stresses to break its bones.
In the cube on the left, length = 1 and volume = 1 (L x L x L). The cube in the middle, where L=2, has a volume of 8. And in the cube on the right, L=3 and V=27.
. There have been a host of Kong movies, but the best are clearly the original (1933) with Fay Wray, the 1976 remake with Jeff Bridges and Jessica Lange, and a 1949 clone entitled
When the Incredible Shrinking Man stops shrinking, he is about an inch tall, down by a factor of about 70 in linear dimensions. Thus, the surface area of his body, through which he loses heat, has decreased by a factor of 70 x 70 or about 5,000 times, but the mass of his body, which generates the heat, has decreased by 70 x 70 x 70 or 350,000 times. Hes clearly going to have a hard time maintaining his body temperature (even though his clothes are now conveniently shrinking with him) unless his metabolic rate increases drastically.
But my colleague Andrew Biewener (formerly at the University of Chicago, now at Harvards Concord Field Station) has revisited this question with surprising results. At least for the long bones in the limbs of mammals, the changes in shape that accompany evolutionary changes in size are not sufficient to compensate for the increased loads. Since all bone has virtually the same breaking stress, this implies that larger animals increasingly push the limits of their own skeletons strength. However, Bieweners direct measurements of bone deformations as an animal walks or runs show that the safety factor (the ratio of breaking stress to working stress) only ranges from three to five. This is remarkably risky design–most things that humans build have safety factors from ten to several hundred. Biewener has looked at animals from chipmunks to elephants and finds that the safety factor is constant across this 25,000-fold size range–scaling has been sidestepped. This result is achieved by a combination of the shape changes in the bones described by Galileo and changes in the behaviors of the animals, particularly adjustments in posture to ensure that the loads the bone must bear are directed along the bones to minimize bending.
Other movies have played with the theme of tiny people in an otherwise normal world, includingThe Incredible Shrinking Woman(1981) andHoney, I Shrunk the Kids(1989). But none of these movies ever deals with the problem of what happens to the objects mass when it shrinks. I can imagine two ways to shrink an object. One would be to start removing molecules, perhaps halving the number in each cycle of shrinkage. But molecules are integer quantities; sooner or later, this strategy is going to lead to half a molecule, which wont work. (Particularly for biological objects. Remember, each cell in your body only has two copies of your genetic information, one in each strand of the DNA in your chromosomes.)
This moment is the creatures undoing, although the movie seems not to realize this fact. After this point, the creature becomes strangely passive, especially in light of its previous rampages. The octopus grabs an attacking submarine, but simply holds it, making no attempt to crush it or bite it. The octopus ignores a scuba diver who swims directly in front of its eye, even when the diver shoots a spear into its brain (which cant have done much damage even if it did actually penetrate the cartilaginous brain case). Finally, the monster is dispatched with explosives and the movie ends.Pulpo, anyone?
(whose special effects, by Ray Harryhausen, are breathtaking). Yet all underestimate the vulnerability of large animals.
One way around this problem is to change the shape of the bone as size increases, so that the cross-sectional area better follows the increase in the animals mass. This is a widespread trend in biology–larger animals have proportionally stouter, thicker bones. Compare the skeletons of a cat to a lion or that of a deer to a moose. This observation is not exactly hot news–both the trend and its explanation were given by Galileo in
Good posture has its limits. Even standing straight, King Kong could not support his incredibe bulk.
(1957), the hero is exposed to radioactive toxic waste and finds himself growing smaller and smaller. He is lost to family and friends while fending off the household cat and must make his own way in a world grown monstrously large. He forages food from crumbs and drinks from puddles of condensation. In one famous scene, he defends himself against a house spider by using an abandoned sewing needle, which he has to struggle to lift.
Upright posture allows animals to support greater weight.
Indeed, sufficiently small animals cannot be hurt in a fall from any height: A monkey is too big, a squirrel is on the edge, but a mouse is completely safe. The mouse-sized people inDr. Cyclopscould have leapt off the tabletop with a cry of Geronimo! secure in the knowledge that they were too small to be hurt.
The really large terrestrial animals are all extinct, but we still have elephants and rhinos for a bit of insight into this problem. Think of the last time you went to the zoo. True, there was a fence around the elephant compound, but a moments reflection will convince you that the fence cant be meant to keep the elephants in–all they would have to do is lean against the fence to bring it down. No, the fence is there to keep you out. What really keeps the elephants in is the dry moat around their compound; a fall of half a dozen feet would shatter the bones in the elephants legs and the elephants know that very well indeed. One of the major flaws of all the Kong movies is that the giant apes are just too active, leaping and crashing around as if they were monkeys, protected by their small size. Remember the elephants, and look on these antics with a bit more skepticism.
Could sauropods have feasted on treetops? Some scientists have questioned whether their hearts could pump blood to such heights.
A second, more subtle, problem pervades the Kong movies. The strength of a bone is approximately proportional to its cross-sectional area; this is simply another way of saying that there is a maximum mechanical stress, or force per unit area, that a bone (or any other material object, for that matter) can withstand. The load the bone must bear is proportional to the mass of an animal. With an increase in size but no change in shape, the load on the bone will increase in proportion to the increase in volume (length cubed), but the cross-sectional area of the bone will only increase as length squared. Eventually, the animals bones will break under its own weight.
The Octopus is a member of the phylum mollusca. Click here for more on animal diversity.
Tiny man vs. spider is a mismatch, but one that favors the man.
In each example, linear dimensions double, but area increases by four times.
Luckily, his lung area has only decreased by 5,000-fold, so he can get the relatively larger supply of oxygen he needs, but hes going to have to supply his body with much more fuel; like a shrew, hell probably have to eat his own weight daily just to stay alive. Hell also have to give up sleeping and eat 24 hours a day or risk starving before he wakes up in the morning (unless he can learn the trick used by hummingbirds of lowering their body temperatures while they sleep).
There havent been a lot of other molluscs that have starred in monster movies; its enough to make an invertebrate zoologist cry, especially since molluscs are the second most diverse phylum on the planet. (Arthropods are first; chordates, the group to which the vertebrates belong, come in a distant third.) Most of Hollywoods efforts along these lines have focused on arthropods. Here, at least, theres a descent representation of biodiversity with giant crustaceans (
Stop the projector! Time for a little analysis.
If you change the size of this object but keep its shape (i.e., relative linear proportions) constant, something curious happens. Lets say that you increase the length by a factor of two. Areas are proportional to length squared, but the new length is twice the old, so the new area is proportional to the square of twice the old length: i.e., the new area is not twice the old area, but four times the old area (2L x 2L).
Lets turn to one of my favorite giant insect flicks.
. Based on some measurements from stills from the original movie, at the beginning of the movie Kong is about 22 feet tall, but by the time he climbs the Empire State Building, he appears to be 50 percent bigger, presumably because he was allowed bananas
As mentioned above, this is achieved by changes in posture. Small mammals run with their limbs bent; large mammals always run with their limbs straight. If youve ever seen a slow-motion movie of a horse running, you may have noticed that the horses leg is perfectly straight when it contacts the ground and it stays straight as long as its bearing the horses weight. This behavior is even more obvious in elephants.
So what happened at the Golden Gate Bridge to completely change the creatures behavior? I think the answer is pretty simple. Any time you have a column of water extending vertically, a pressure is generated at the bottom of the column; one atmosphere of pressure is produced for each 33 feet of height. Our giant octopus, like all macroscopic animals, has a circulatory system extending throughout its body–in essence, pipes (here, large pipes) filled with water. If the octopus extended its tentacles vertically while it was submerged, nothing would happen; the pressure would increase at the base of his tentacles, but that pressure would precisely match the pressure in the surrounding water (which can also be viewed as a column of water).
But clearly there are mammals larger than 10-20 kilograms–you and I, to name two. Indeed, empirical measurements of the working stresses in bones indicate a very different story. The bars on the graph indicate the working stress levels in the bones of a variety of mammals from mice to elephants; hatched bars are indirect estimates from the measures of the forces the animal exerts on the ground, white bars are direct measurements from strain gauges directly attached to the animals long bones. As is apparent, bone stress does not grow as the cube root of body mass. Indeed, the working stress in the bones seems to be independent of body size, running about a fourth to a third the breaking stress for all mammals. In a sense, what we have here is natures design principle for mainstream skeletons: all have evolved to have a safety factor of three to five.
The biological significance of these geometric facts lies in the observations that related aspects of an organisms biology often depend on different geometric aspects. Take physical forces.The magnitude of surface tension forces is proportional to the wetted perimeter (a length); a water strider needs long feet, not big feet, to skate on the surface of a pond.Adhesive forces are proportional to contact areas; geckos need broad, flat feet covered with millions of tiny setae to walk on the ceiling.Gravitational or inertial forces are proportional to volume (assuming that density is constant); a bird that flies into a window may break its neck, but a fly that flies into a window will bounce without injury.
King Kongs imposing size–about 22 feet tall in this scene–makes him vulnerable to injury.
. At 22 feet tall, Kong is about four to five times the size of your garden-variety lowland gorilla. A fivefold increase in height implies a 25-fold increase in bone cross-sectional area and a 125-fold increase in body mass; the stress on the bones thus should be about five times greater than the stress on a normal gorillas bones. But, remember, according to Andy Bieweners data, a safety factor of five is extreme for mammals; Kongs excessive body size should have exhausted the safety factor. True, Kong stands a bit straighter than the average gorilla so he may gain a bit of the safety factor back, but its clear that hes pushing the envelope. Is that why he has such a short fuse and is always roaring and bashing things? Not only does he continually run the risk of breaking his legs, but undoubtedly his feet hurt.
Size has been one of the most popular themes in monster movies, especially those from my favorite era, the 1950s. The premise is invariably to take something out of its usual context–make people small or something else (gorillas, grasshoppers, amoebae, etc.) large–and then play with the consequences. However, Hollywoods approach to the concept has been, from a biologists perspective, hopelessly naïve. Absolute size cannot be treated in isolation; size per se affects almost every aspect of an organisms biology. Indeed, the effects of size on biology are sufficiently pervasive and the study of these effects sufficiently rich in biological insight that the field has earned a name of its own: scaling.
, 1958), and an array of insects ranging from flies to giant preying mantises.
Miniature people confront Galilean science!
(1957), a 100-foot (but normally proportioned) man menaces Las Vegas. Although, based on his size, we would assume his first step should be his last, somehow he manages to survive a fall off Boulder Dam and return for the sequel,
The relatively slender 50-foot Woman violated traffic laws and the rules of safety factor.
The conceptual foundations of scaling relationships lie in geometry. Take any object–a sphere, a cube, a humanoid shape. Such an object will have a number of geometric properties of which length, area, and volume are of the most immediate relevance. All areas (surface area, cross-sectional area, etc.) will be proportional to some measure of length squared (i.e., length times length); volumes will be proportional to length cubed (length times length times length). Equivalently, lengths are proportional to the square root of an area or the cube root of a volume.
Similarly, volumes are proportional to length cubed, so the new volume is not twice the old, but two cubed or eight times the old volume (2L x 2L x 2L). As size changes, volumes change faster than areas, and areas change faster than linear dimensions.
First, how do they see? The crew spend time enjoying the scenery as they cruise the arterial highways, but even at their largest size their eyeballs are much smaller than the wavelength of visible light. Even hard ultraviolet radiation is too long in wavelength to be useful. Perhaps they are using X-rays, but if so, their hapless host has more to worry about than a blood clot.
Reproduced with permission from The Columbia Electronic Encyclopedia. Copyright © 2000 Columbia University Press. All Rights Reserved.
However, before pulling the bridge down, the monster extends his tentacles about halfway up the support towers. The top of the support towers stand 500 feet above the deck, itself 220 feet above the high-water mark. At one atmosphere for every 33 feet and an elevation of 470 feet, thats a total pressure of about 14 atmospheres (209 pounds per square inch). For the first time in its life, there was no surrounding mass of water to offset the pressure increase and the full load of this pressure would act to distend its arteries.
Kong may be pushing the limits of his bone strength, but other movies have clearly crossed the line. In
In the movie, the giant octopus pulls down the Golden Gate Bridge. But raising its limbs so far out of the water may have been the creatures undoing.
(1966). A famous scientist, vital to the national defense, has an inoperable blood clot in his brain. Luckily, a secret government project has just developed a machine that can miniaturize objects. They shrink a submarine and five crew members down to microscopic size and inject them into the comatose scientists bloodstream to find and destroy the blood clot. Lacking a copy of
This phenomenon, first described by Scottish botanist Robert Brown in 1827, is known as Brownian motion; the pollen grains he observed through his microscope appeared to dance randomly in the water. Our hemonauts inFantastic Voyage, ten times smaller than Browns pollen grains, are going to experience the same random and continuous jostling, rather like an endless journey on a train running on bad tracks. Raquel Welch would have been lucky to keep her hands in the vicinity of the control panel, much less actually operate the controls.
These facts were known to our ancestors, who used this aspect of scaling to gruesome effect–a common strategy during medieval sieges was to take a carcass of a horse, let it ripen for a few days in the sun, and then catapult it over the walls of the besieged town. On impact, the carcass would indeed splash, spreading contagion throughout the city.
Another way to shrink an object would be to decrease the distance between an atoms nucleus and its electron cloud-atoms are, after all, mostly empty space. Im not enough of a physicist to have any intuition about what this would do to basic physics and chemistry, but one result of this strategy would be to leave the objects mass unchanged. If volume decreases but mass does not, then density must increase. The shrinkage is sufficiently limited in these movies that we dont have to worry about dealing with miniature black holes, but an object the size of a cell but the mass of a submarine–as inFantastic Voyage–is going to pass through the table, the floor, and the earths mantle like a hot knife through butter.
Caterpillars are peculiar beasts. Unlike adult insects, which have a rigid external skeleton, caterpillars have a flexible skin and support themselves with what is known as a hydrostatic skeleton–a volume of incompressible fluid that transmits forces and pressures from muscular contraction through the body. In essence, they are animated water balloons with the incompressible fluid (the blood, filling all
Never fear! Dr. Cyclops tiny victim could have jumped without risk of injury.
The other end of the size spectrum–the commonplace become gigantic–is much more the norm in monster movies and is certainly what first comes to mind when you think about the genre. The archetype is, of course,
Rachel Welch bombarded by molecules!
The epitome of the small-fry genre may be
In another scene, Raquel Welch floats in a capillary, controlling the submarine remotely with a panel strapped to her waist. Remember that molecules are in constant, vigorous motion, driven by thermal energy. Trillions of molecules collide with your skin each second; all of these collisions average out to produce what we macroscopically call pressure. As objects get smaller, this random bombardment still averages out over time, but at any instant more molecules may collide with one side of the object than the other, pushing the object momentarily to one side.
Zigzag, irregular motion exhibited by minute particles of matter when suspended in a fluid. The effect has been observed in all types of colloidal suspensions–solid-in-liquid, liquid-in-liquid, solid-in-gas, and liquid-in-gas. It is named for the botanist Robert Brown who observed (1827) the movement of plant spores floating in water.
Physiological relationships are not exempt. The rate at which oxygen can be extracted from the air is proportional to the surface area of the lungs; the rate at which food is digested and absorbed to the surface area of the gut; the rate at which heat is lost to the surface area of the body: but the rate at which oxygen or food must be supplied or the rate at which heat is produced is proportional to the mass (i.e., volume) of the animal. If an animal performs well at any given size, size change alone implies that these related functions must change at different rates, since their underlying geometric bases change at different rates; if the animal is to be functional at the changed size, either functional relations must change or shape must change. Monster movies have extensively explored these scaling relationships, albeit usually incorrectly; knowing the true relationships often puts the entire movie into a new light.
Because of these relatively larger surface areas, hell be losing water at a proportionally larger rate, so hell have to drink a lot, too. We see him drink once in the movie–he dips his hand into a puddle and sips from his cupped palm. The image is unremarkable and natural, but unfortunately wrong for his dimensions: at his size surface tension becomes a force comparable to gravity. More likely, hed immerse his hand in the pool and withdraw it coated with a drop of water the size of his head. When he put his lips to the drop, the surface tension would force the drop down his throat whether or not he chooses to swallow.
Galileo sketched the change in shape necessary for a bone to support a larger animal.
, they have more adventures than they should. The scale of the hemonauts varies from viral to bacterial, depending on the scene, and a host of problems arises.
s hemonauts were in for a very bumpy ride.
Lets start small and work our way up.
As J.B.S. Haldane put it in his classic essay, On Being the Right Size, You can drop a mouse down a thousand-yard mine shaft; and, on arriving on the bottom, it gets a slight shock and walks away….A rat is killed, a man broken, a horse splashes. Haldane was being quite literal.
As for the contest with the spider, the battle is indeed biased, but not the way the movie would have you believe. Certainly the spider has a wicked set of poison fangs and some advantage because it wears its skeleton on the outside, where it can function as armor. But our hero, because of his increased metabolic rate, will be bouncing around like a mouse on amphetamines. He wouldnt struggle to lift the sewing needle–hed wield it like a rapier because his relative strength has increased about 70 fold. The forces that a muscle can produce are proportional to its cross-sectional area (length squared), while body mass is proportional to volume (length cubed). The ratio of an animals ability to generate force to its body mass scales approximately as 1/length; smaller animals are proportionally stronger. This geometric truth explains why an ant can famously life 50 times its body weight, while we can barely get the groceries up the stairs; were we the size of ants, we could lift 50 times our body weight, too. As for the Shrinking Man, pity the poor spider.