How Big is Big and How Small is Small
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Figure 2.3 A collection of Littorina saxatilis (periwinkle) collected in Scotland. Courtesy of D. C. Smith.
A recent collection of nearly a thousand periwinkles from across northern Scotland gives us a nice data set, some of which are shown in Figure 2.3. After measuring all these shells it was determined that the length of the spiral of Littorina saxatilis is 10.5 ± 2.2 mm. But in that simple statement we have lost a lot of other interesting information. We only know the length of the spiral—just one number. The smallest periwinkle in the collection was 5 mm long. Is this as small as they get, or is it just a limit due to the method of collection? Their size would clearly vary with age. It also varies with the environment. In the town of Ham the average shell was about 14 mm long where as in Inverewe it was only 9.5 mm. What number should we quote in the end? For our purposes it is enough to say that a periwinkle is bigger than a dust mite or flea, yet smaller than a pygmy shrew. It is about 1 cm in size.
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Just a bit larger than the periwinkle, at the scale of a few centimeters, a lot of new structures appear within animals. For instance, the passive trachea, which served insects so well, is replaced with lungs and gills. These respiratory systems are active in that they move air through an animal and, with the aid of blood and a circulatory system, they can get oxygen deep within a bigger creature.
A few centimeters is also the size of the smallest mammal, the Etruscan pygmy shrew and the smallest bird, the bumblebee hummingbird, both weighing only a few grams. The Etruscan pygmy shrew is also the smallest warm-blooded animal. The term warm-blooded is out of favor among biologists who instead use the term homeothermic, which means that maintaining a constant internal temperature. Smaller animals are cold-blooded or poikilothermic, because they have such a high body surface-to-volume ratio that they cannot keep themselves warm. So once again here is a boundary set by the surface-to-volume ratio. An organism below the size of the Etruscan pygmy shrew has so much surface area for its small volume that it will cool and heat quickly and its body temperature will follow that of the air or water where it lives.
Scale really does affect what can happen. There is this idea, popular in fiction, that we could have fully functioning humans of extreme sizes, like the Lilliputs from Gulliver’s Travels, “not six inches high” (15 cm), or the Brobdingnag, the giants from that same book. Better yet, science fiction likes mutant beasts such as Godzilla or King Kong. However, in reality these beings just would not work. One of the first people to realize this was Galileo Galilei (1564–1642) who wrote about the effects of scale in his book, The Discourse and Mathematical Demonstrations Relating to Two New Sciences, often simply called Two New Sciences.
Galileo starts out with a series of observations. Why can we build a small boat with no supports, yet at the arsenal shipyard of Venice, large ships need stocks and scaffolding to hold them together while under construction or they will rip themselves apart? Or, as he noted, “a small dog could probably carry on his back two or three dogs of his own size; but I believe that a horse could not carry even one of his own size.” A small stone obelisk or column can be stood up on end without difficulty, but a great obelisk like the one in St. Peter’s Square or Cleopatra’s Needle will go to pieces when lifted unless a special cradle and scaffolding are used to stand it up on end. Cats can fall distances without harm that would injure a man or kill a horse. Finally, he observed,
Nature can not produce a horse as large as twenty ordinary horses or a giant ten times taller than an ordinary man unless by miracle or by greatly altering the proportions of his limbs and especially his bones, which would have to be considerably enlarged over the ordinary.
I was told once that an elephant the size of a cathedral would crush its own legs. It is a curious calculation that I could not resist following through. The largest elephant is Loxodonta african, the savanna elephant. The largest savanna elephant ever measured was shot in Angola in 1956. It stood 4.2 m at the shoulders and weighted 12,000 kg, but this was an extreme case. An average savanna elephant is about 3 m tall at the shoulders and weighs about 6000 kg. Let us scale this up to a cathedral. The main vault inside of Notre Dame de Paris is 34 m high, whereas York Minster is 31 m. Let us scale our 3-m elephant to 30 m at the shoulders. If its height increases by a factor of 10, then its length and width would also increase by a factor of 10, so its volume and mass would increase by a factor of 1000. That means that our Loxodonta cathedralus will weigh 6,000,000 kg (see Figure 2.4). How thick will its leg bones need to be to hold this up? Curiously enough, four bone pillars of 30-cm diameter will suffice, for bone really is tough under compression. Bone has seven times the compression strength of concrete. But an animal is different because it can move, and a single step of this 6,000,000-kg elephant will shatter a 30-cm diameter leg bone.
Figure 2.4 Savanna elephant Loxodonta african (3–4 m), Loxodonta cathedralus (30 m) and Notre Dame de Paris (34 m).
If we simply scaled all the length dimensions of a real elephant the bones would have one hundred times the cross section, but they would need to carry one thousand times the mass, so they are not really thick enough for an active elephant. However, all we really need to do is increase the strength of the bones by a factor of ten, which means increasing the cross section by an additional factor of ten. If we do this it should be about strong enough to walk, except … a normal elephant is 13% bone by mass. Our reinforced Loxodonta cathedralus will have 10 times as much bone, ten times that 13%, which is a real problem. So in the end, if we scale our elephant to the size of a cathedral it can stand there because bones are strong under compression. But either we increase the percentage of bone, until it is only a skeleton, or as soon as it takes a step, it will break a leg. Super-monstrous creatures are not going to happen in nature.
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So how big can an animal grow to be? The largest creature to ever walk the face of the earth was the Argentinosaurus. It has been estimated that it weighed 80,000–100,000 kg. Other dinosaurs were taller (the 18-m Sauroposeidon) or longer (the 40-m Supersaurus), but none was more massive. The Argentinosaurus was about eight or nine times the mass of the largest modern elephant (12,000 kg). That means it is like a savanna elephant with the lengths scaled up by a factor of two. These prehistoric giants were only a factor of two longer, taller and wider than our modern elephants. And like their modern counterparts, they were vegetarians!
Why is it that the great predators, the top of the food chain, the ultimate hunters and killers are not the biggest creatures out there? The Tyrannosaurus rex (~10,000 kg) was dwarfed by the Brachiosaurus (~50,000 kg) and the Argentinosaurus. The largest tiger (Panthera tigris; ~280 kg) and the lion (Panthera leo; ~250 kg) are much smaller than the elephant. Even the largest terrestrial hunters, the polar bear (Ursus maritimus; ~500 kg), is less than half the size of the elephant. The reason has to do with the energy density—the calories per hectare of land—available in the biological world.
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The largest creature ever to exist does not worry about crushing its legs—it does not have any. The largest creature ever seen on Earth is alive today and swimming in the oceans: the great blue whale (Balaenoptera musculus). Longer and sleeker looking than most whales, the blue whale measures over 30 m from the tip of its flat, U-shaped head
to the end of its great tail fluke. The longest whale ever measured was 33.6 m long. The heaviest was 177 tons (177,000 kg). These leviathans have solved the structural problem of crushed legs and ever-growing bones by floating and not trying to support their mass on four brittle pedestals. The blue whale is not a vegetarian, but has solved the “calories per square kilometer” problem by eating krill, a type of tiny shrimp, which is plentiful in the ocean. This may in fact be the largest difference between the size of the predator (30 m) and the prey (0.01 m), and one of the shortest of food chains: phytoplankton (10−5 m) to krill to whale.
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But why limit our discussion of living beings to only the animal kingdom? If we are looki
ng for the largest living thing we must also consider flora, the kingdom of plants. The tallest tree on earth is the California redwood, Sequoia sempervirens, a member of the cypress family. For many years it was thought that the tallest was the Stratosphere Giant, measured at 112.8 m and living in Humboldt Redwood State Park in northern California. However, recently (2006) a new tallest tree, the 115.5-m Hyperion, has been found in nearby Redwood National Park. I like the idea that there may still be undiscovered giants someplace in those forests. The largest redwood by volume is the Del Norte Titan, with an estimated volume of 1,045 m3. A number of studies predict that under ideal conditions a redwood may grow to 120–130 m in height. The limiting factor is that the tree cannot move water up any higher. But it takes 2,000 years to grow a tree to these heights, and it is unlikely for conditions to be ideal for such a long period.
I have stood in Humboldt Redwood State Park at the bases of some of these trees. Before arriving I remember thinking how odd it is to name trees in the same way we name people. However, these are not just ordinary trees growing in the forest. These giant trees define the space and allow the other trees and plants of the forest into their presence. The towers of York Minster are only 60 m tall, about half the height of the Hyperion. The towers of the Brooklyn Bridge are just over 80 m above the water.
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While the redwoods may be the tallest trees, they are not the biggest. The giant sequoia Sequoiadendron giganteum owns that title. Although the largest sequoia, the General Sherman, is only 84 m tall, it is estimated to contain 1,500 cubic meters of wood, nearly 50% more than the Del Norte Titan. These trees, measuring over 30 m around the trunk, are truly giant. Their soft, cork-like bark seems to muffle any sound these stately groves may have. The giant sequoia grow on the shoulders of the Sierra Nevada mountains in the eastern part of central California. It is not as moist there as in the redwood groves near the coast, but the conditions must be right for longevity. These trees are not only large because genetics and design allow them to be, but also because they survive. They are all marked with scorches from forest fires, but they keep growing. The oldest giant sequoia that was cut down and had its rings counted was 3200 years old, which means it had lived since the time when the Greeks took to their thousand ships and sailed to Troy. There seems to be no fungus or fire that will kill them, they have no natural predators. But one physical constraint remains. Their normal fate is to succumb to gravity and fall under their own weight.
For two or three thousand years they have grown in these quiet cathedral-like groves. Yet “cathedral-like” is wrong. They tower over, and are more ancient than the stone handiwork of man. Perhaps instead one should say that cathedrals have towering heights like a redwood and massive, ageless presence like a sequoia.
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There are other living things that some argue qualify as the largest on Earth. For example, all the trees of an aspen grove can be part of the same organism, for each tree can be a sprout off a single underground root system. There is also a giant fungus, Armillaria ostoyae, which is measured in square kilometers. But these are clonal colonies. To me, they do not qualify as a single living entity because they do not reproduce by growing another grove, and their parts are not mutually dependent. Finally, when I wrote down the characteristics of life I added the ability to die. If you cut a grove or a fungus in half it does not die; each part does not depend on the rest.
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Bacteria to sequoia is an amazingly wide range of sizes for life to span. From a micrometer to a hundred meters is a size factor of one hundred million. Likewise, from the lifetime of a bacteria at 20 min (103 s) to the lifetime of ancient trees at nearly 5000 years (1011 s) is also a factor of one hundred million.
If you stand in a grove of giant sequoias and pick up a handful of dirt you can be assured that with a microscope you would see bacteria. In the pond water next to you you will find amoebas and rotifers. There are worms and mites underfoot, flies, bees and birds in the air. Measuring in at 0.5–1.7 m you will see Homo sapiens—humans—wandering slowly in awe. There are white fir and sugar pines growing there, as well as the dominant sequoia. All of these can live together because they each occupy their own unique niche, their own place in the environment. Usually a niche is described as a constraint of the environment. We recognize two separate niches because this place is wet and that place is dry, this area is warm and that region is cold, and certain plants and animals occupy certain niches. But we can also think of a niche, not in terms of the constraints or problems for living, but in terms of solutions to those problems. The problems of life may span over eight orders of magnitude, but the solutions do not. Trachea can get oxygen into an insect, whales need lungs and a circulation system. A nervous system works in an elephant, but not in a bacterium. The solutions have ranges given by the laws of nature, and it is the many combinations of these solutions that make each and every living species unique.
3
Big Numbers; Avogadro’s Number
When I was a boy I was once in a quiz bowl contest. Our team was asked the question “Why is it easier to become a billionaire in America than in Britain?” My hand shot up and I answered that it was because a British pound sterling was worth more than an American dollar, and so it was harder to acquire a billion of them. It was 1974 and I was an American living in Britain and I knew that the exchange rate was about £1 to $2.40. But that was not the answer that the quiz master was looking for. He rephrased the question, “Why is a billion pounds in Britain more than a billion pounds in America?” None of us got the question. If it was not the exchange rate, what could it be?
What I had failed to realize is that the word “billion” meant different things on opposite shores of the Atlantic. “English” is not simply English, and we are reminded that, “America and England are two nations divided by a common language.” I think that if the word billion had been written in French, or better yet in Chinese, I might have guessed that the confusion lay in the word itself, for there we had a very simple English word, a noun, just the name of a number. What could have been more precise than a number?
I can fill a ship in New York harbor with 1,000,000,000 nails, write on the ship’s manifest “one billion nails,” and send them across the ocean to London. There, the nail counter and customs officer can count out “1,000,000,000 nails” or “one thousand million nails” and, unless we know something about the translation of numbers, we may have a falling out. Is it a breach of contract to order a billion nails, but to receive only a thousand million, and think that you ordered a thousand times as many nails as you received?
The crux of the problem is that there are two conventions in the world for naming large numbers: the long scale and the short scale. The long scale is the older of these two and dates back about 500 years to Jehan Adam and Nicolas Chuquet, who wrote large numbers something like
1, 234567, 890123, 456789, 012345.
In the long scale the second group of digits from the right (456789) are millions, the third group (890123) are billions, the fourth group (234567) are trillions and the left most digit (1) is a quadrillion. The convention for naming these different parts of a large number is straight forward: bi-, tri-, quad- are all prefixes that get stuck onto the front of -llion, but -llion itself is without meaning, except as invented for “million.” The word million is a curiosity dating back only six or seven hundred years, and means a “great mille,” or a great thousand. (Remember that a mile is a thousand steps.) The prefixes continue beyond our example to quin-, sex-, sept-, oct-, nef-, and so forth, following the Latin for five, six, seven, eight and nine (see Table 3.1). So let us apply this to a big number like the length of a light year expressed in meters,
1 light year = 9460 730472 580800 m
Table 3.1 Names of numbers.
This figure can be written in words as nine thousand four hundred sixty billion, seven hundred thirty thousand four hundred seventy-two million, five hundred eighty thousand eight hundred meters or, m
ore fully, nine thousand-billion four hundred sixty billion, seven hundred thirty thousand-million, four hundred seventy-two million, five hundred eighty thousand, eight hundred meters. The words thousand-billion and thousand-million need to be there to make the long scale work. I wonder if it was this repeated use of the word thousand that eventually led to the short scale.
The short scale was also invented in France, but much later, in the eighteenth century. At this time American commerce and culture was being established, and this is the system presently used in the US. In the short scale we would read
1 light year = 9, 460, 730, 472, 580, 800 m
as nine quadrillion, four hundred sixty trillion, seven hundred thirty billion, four hundred seventy-two million, five hundred eighty thousand, eight hundred meters. Is it a better system? Well, you only need to deal with three digits at a time, but you need a lot more prefixes.
So what system does France, the birth place of both systems use? It originally used the long form. In the eighteenth century it used the short form, but then later reverted to the long form. Actually, I expect that most people never even thought about these sorts of numbers. Even kings and the national treasury would be satisfied with millions as a big enough number.
So if I meet my old quiz master, how should I answer the question, “Why is it easier to become a billionaire in America than in Britain?” Well, I should now answer just the way I did then. “It is easier in America because one pound is presently worth more than one dollar.” In fact, in 1974 the same year that this question was asked of me, the British prime minister, Harold Wilson, announced that when government official used the word billion, then it meant 1 with nine zeros. America has evolved as the world’s banking center, so the way to write a national debt tends to follow the American usage or the short scale, at least in the English speaking world. An alternative is to just write out the number. For example, a few years ago the world’s gross annual product was estimated to be,