3
Sand and Imagination I
Very Large Numbers of Very Small Things
The Walrus and the Carpenter
Were walking close at hand;
They wept like anything to see
Such quantities of sand:
“If this were only cleared away,”
They said, “it would be grand!”
“If seven maids with seven mops
Swept it for half a year,
Do you suppose,” the Walrus said,
“That they could get it clear?”
“I doubt it,” said the Carpenter,
And shed a bitter tear.
THE LOOGAROO’S OBSESSION
If you live or travel in the Caribbean and are of a superstitious disposition, or if you simply believe in the precautionary principle, take enough sand from the beach to make a good-sized pile outside your door. This will prevent the loogaroo from entering. The loogaroo is a vampire, most commonly a female. Each night, she rids herself of her skin, hides it under a tree, and flies off in search of blood, flames shooting from her armpits and orifices, leaving a luminous trail through the sky. She can take on different forms and gain entry through the slightest crack, but she has, fortunately, a weakness: she is an obsessive counter. On finding the pile of sand you have left, she will be compelled to interrupt her bloodthirsty quest and count the grains. If you have left enough, this will take her till dawn, when she must rush off to reunite with her skin. If you are concerned that there might not be enough grains of sand, mix in some nails of a ground owl—whenever she encounters one of these, she is so distressed that she drops the sand she has counted and has to start over again.
The loogaroo is clearly related to the loup-garou, the French werewolf, and is probably the result of an interbreeding of that creature and voodoo (in which a pile of sand is the home of the spirits and the symbol of the Earth). The loup-garou enjoys the bayous of Louisiana, where a sifter placed outside your door will cause it to compulsively count the holes. It would seem likely that selecting a sifter with small holes and at least trying to fill it with sand would prove doubly effective.
Compulsive counting appears to be a traditional failing of vampires in a wide range of cultures—presumably Count von Count from Sesame Street is not simply a play on the word. Sand is deeply associated with numbers, particularly large or extremely large ones, and counting sand is associated with the challenge of handling such numbers. This challenge can, of course, be looked at as a harmless and absorbing way of whiling away some time, hence the tradition in some eastern European cultures of putting sand in a coffin to offer the dead an opportunity for a peaceful preoccupation. It can also be looked at, not surprisingly, as a waste of time: “counting sand” is a pejorative Buddhist expression used to describe those who merely study the details of doctrine rather than putting effort into the grander scheme of things. The ability to count sand is traditionally regarded as beyond mere mortals, a divine or supernatural skill. According to Herodotus, Croesus, the fabulously wealthy king in ancient Greece, decided to arrange a performance appraisal of the available oracles. The crucial test was the ability to predict, remotely, what the king would do on a defined day. The oracle at Delphi, who won the contest, prefaced her accurate prediction by declaring her credentials: “the number of sand I know, and the measure of drops in the ocean.” Although it is true that counting sand grains is occasionally the task of geologists (or, in all likelihood, geology students), it is an essentially impossible, and generally futile, activity. Imagine, for example, that you poured out merely a liter or a quart of sand as your loogaroo deterrent and you decided to check its efficacy by counting the grains yourself. At the rate of one grain per second, twenty-four hours per day, the pile would occupy you for many months (even assuming that you could say a number such as 9,702,337 in a second).
It is worth noting, however, that counting in sand is part of the origins of mathematics. The abacus evolved from a flat stone covered with sand as a renewable medium for counting and calculating, using symbols drawn into the sand or small stones placed on top. The word may derive from the Phoenician word for sand, abak, or the Hebrew word for dust, avak (or abhaq). An ephemeral system, its geographic and cultural origins are disputed. Equally debated is the relationship between the abacus and the origins of Arabic (or, strictly, Hindu-Arabic) numerals. The story goes that eighth-century scholars in Baghdad were inspired by the work of Indian mathematicians and began to use a new system of numerals that they called huruf al-ghubar, “letters of sand”—or “dust.” It seems quite likely that this derives from writing on a sand table or abacus. The system was introduced into western Europe by Arabic mathematicians in the eleventh century and their symbol for zero by Fibonacci two hundred years later. Does our 0 represent the depression left in the sand when a pebble is removed?
ZILLIONS AND ZILLIONS
Even small quantities of sand rapidly start amounting to significant numbers. Draw a 1-centimeter (½ in) square on a piece of paper and spread a layer of sand one grain thick so as to fill the square. Take a magnifying glass and, testing how it feels to be a loogaroo, count the grains: there will be, typically, several hundred of them. A handful of sand, before it runs through your fingers, will contain tens or hundreds of thousands of grains, depending on their size. This was unfortunate for the Sybil, gatekeeper of the underworld in Greek mythology: having been granted her wish that she should live as many years as the grains of sand in her hand, she deeply regretted her failure to request youth as well. In Knutsford, Cheshire, the first Saturday in May is Royal May Day, when sand drawings and modest sand castles are constructed outside the houses of brides. The “Royal” in Royal May Day refers to King Knut, who is said to have emptied his shoe of sand on encountering a wedding party, wishing the happy couple as many children as there were grains of sand. If, on this May Day, he was returning from his experiment at the beach—another popular legend has him demonstrating that even the king of England is powerless to turn back the tides—the couple would have had a crowded house.
Sand as a symbol of fertility crops up around the world. The Hindu goddess Parvati, the consort of Shiva, takes on a number of divine forms, friendly and terrible, but always powerful. She is sometimes represented by five piles of sand by a river, and these are worshipped by girls. As Gauri, her maternal form, she sculpted a son from sand and commanded him to guard the entrance to her cave. Shiva, not much of a family man, was prevented from entering and so he beheaded the boy; to make up for this, he attached an elephant’s head, and so created Ganesh, ironically the god of good luck. Parvati, in her various guises, is a popular subject in Indian sand-sculpting festivals.
In Cambodia and Laos and other countries of Southeast Asia, New Year ceremonies require piles of sand to be made, in varying scales of size and sophistication. The numbers of sand grains are associated with the days of a happy and prosperous life or with the number of sins for which forgiveness is being asked. In other parts of Asia, sand symbolizes rain, and rituals involve participants dressed as cows and farmers hurling sand at each other.
Although the number of grains in a small pile may seem considerable as a representation of quantities of children, sins, or days of happiness, it is, nevertheless, still a trivial number in the world of sand. It is in the realm of really, really large numbers, which stretch the imagination to and beyond its limits, that sand provides the most powerful imagery. And that imagery is long and deeply established in the consciousness of many cultures. In Shakespeare’s Richard II, the challenge of the military task faced by the Duke of York is described as “numbering sands and drinking oceans dry.” Mrs. Alving, in Ibsen’s play Ghosts, laments that “this country is haunted by ghosts—countless as grains of sand.” Something that is incredibly big is often referred to as being of “biblical proportions,” and challengingly large numbers are invariably evoked in the Bible through reference to sand, particularly “the sand which is by the sea shore”: the number of people that the Philistines gathered to fight with Israel, the extent of Solomon’s wisdom, the quantity of the seed of Abraham and David, and the number of feathered fowls rained down on the children of Ephraim, all of these are as the sand of the sea. Indeed, “the number of the children of Israel shall be as the sand of the sea, which cannot be measured nor numbered.”
The imagery of unimaginable and all-encompassing quantities occurs also in the Koran: “There is not even a single grain of sand in the obscure depths of the Earth, nor any plant, blooming or withering, that is not recorded in the Transcendent Koran.” In Buddhism, the quantities of sand evoked are those transported by one of the world’s great rivers, the Ganges. Traditionally, a bodhisattva goes through forty stages of practice before reaching the forty-first stage of Buddha-hood. The first thirty stages require worshipping Buddhas “as many as the sand grains of five Ganges Rivers,” the focus then escalating until, at the fortieth stage, the number of sand grains is equal to those of eight Ganges Rivers. Buddha thought past ages “were more in number than the grains of sand from the mouth of the Ganges.”
The number of sand grains cascading across the Ganges delta, the world’s largest, is a measure that successfully evokes the immeasurable, but reason requires at least an attempt to quantify such yardsticks. The fact remains that, if science and mathematics are to handle the unimaginably large and inconceivably small, then some system of calculation is necessary. Archimedes recognized this need, close to two millennia before we recognized atoms, quarks, or pulsars.
As the introduction to a manuscript known as The Sand Reckoner, Archimedes, the son of an astronomer, wrote the following to Gelon, king of Syracuse:
There are some, king Gelon, who think that the number of the sand is infinite in multitude; and I mean by the sand not only that which exists about Syracuse and the rest of Sicily but also that which is found in every region whether inhabited or uninhabited. Again there are some who, without regarding it as infinite, yet think that no number has been named which is great enough to exceed its magnitude. And it is clear that they who hold this view, if they imagined a mass made up of sand in other respects as large as the mass of the Earth, including in it all the seas and the hollows of the Earth filled up to a height equal to that of the highest of the mountains, would be many times further still from recognizing that any number could be expressed which exceeded the multitude of the sand so taken.
Archimedes took up the challenge of handling very large numbers. It was the third century B.C.: our system of place-based numerals would only be introduced to Europe by Fibonacci 1,500 years later, and the existing Greek mathematical notation used a complex series of symbols that were cumbersome to work with. But Archimedes overcame this handicap in his extraordinary development of a means of calculating very large numbers.
Using measures of finger-breadths and grains of sand, Archimedes devised a basic large number, ten thousand, which he called a myriad, derived from the Greek word myrios, “uncountable.” (The Greeks had another phrase for a countless multitude: psammakosioi, “sand-hundred,” expanded by the satirical dramatist Aristophanes to psammakosiogargaroi, “sand-hundred-and-plenty,” to describe a character’s pains versus his pleasures—of which he had precisely four.) Multiplying myriads by myriads gave very large numbers indeed, each multiplication creating a new order. For example, in modern terms, Archimedes’ system for the third order included numbers up to 1024. But he continued to pile order on order up to the 108 order, at which point he began periods, with the first order of the second period and so on—he had probably lost the good King Gelon long before this. Archimedes built this system up to a number that we would represent as 1080,000,000,000,000,000. Archimedes went on to estimate the diameter of the universe (the sphere of fixed stars) from a ratio involving the diameter of the sphere of the Earth’s orbit around the Sun (yes, the ancient Greeks knew this was the way it worked) and the diameter of the Earth. His result was only the diameter of Mars’s orbit, but Archimedes was moving in the right direction. From there, it was easy to calculate the number of grains of sand in the universe—ten million units of the eighth order, or 1063. He told the king that, while this would appear incredible to the great majority of people, for anyone conversant with mathematics, this would be a convincing proof.
How the king might have taken all this, or applied the conclusion, is lost in the mists and the sands of time. Various modern comparisons have been made between Archimedes’ extraordinary number and modern estimates of the numbers of various particles in various volumes of space, and occasionally one of these calculations will be “eerily close” to that of Archimedes. That such conclusions are essentially meaningless should not detract from the ingenuity and accomplishment of Archimedes. (Sand would, in the end, contribute to Archimedes’ downfall: though he had helped defend Syracuse against the Romans with various ingenious inventions, the Romans prevailed, and one of their soldiers is said to have killed him when the mathematician berated him for disturbing his drawings in the sand.)
Centuries ahead of his time, Archimedes was the first to develop a means of representing very large numbers, but our current system of powers of ten—a million, 106, a billion, 109—is more convenient. A rough calculation of the number of “grains of sand from the mouth of the Ganges,” assuming a typical annual delivery, gives 25 × 1016, or 250 million billion, a successful evocation of an unimaginably large number. The human mind seems to have some capacity limit for grasping large numbers, after which adding zeros and giving such things names (octillions, zillions, and so on) becomes merely a game in a virtual arena. And the endgame, infinity, can never be reached merely by adding more zeros or Ganges together—however many years you spend counting the sand grains pouring into the Bay of Bengal, you will never reach infinity.
In 1980, Carl Sagan, the enthusiastic popularizer of all things astronomical, kicked off one of the most enduring, entertaining, but quantitatively pointless debates about large numbers. He declared, in his television series Cosmos, that “the total number of stars in the universe is larger than all the grains of sand on all the beaches of the planet Earth.” The calculations are ongoing and the debate rumbles on, particularly in the ethereal realms of the internet, and there are, predictably, two schools of thought. While estimates are always increasing, the number of stars is the easier number to calculate: anywhere between 1020 and 1022. As for the grains of sand—well, it depends. What are the assumptions in terms of grain size and, indeed, what counts as a beach? Only the areas of sand above high tide, or areas underwater as well? Depending on how you choose to do the calculation, you can derive a number that is larger or smaller than 1022. And if all the sand grains of the Earth are included, not just those on beaches, then it’s again a different matter. Or, invoking the biblical description of the number of the seed of Abraham, “as the stars of the heaven, and as the sand which is upon the seashore,” we can call it a tie. At the end of the day, it doesn’t matter: Sagan successfully evoked an extremely large number. He set up a powerful comparison between two images of vastly different scale—a comparison drawn also by the science fiction writers Brian Aldiss, in his Galaxies Like Grains of Sand (1959), and Samuel R. Delany, in Stars in My Pocket Like Grains of Sand (1984).
Sagan also commented on the finite size of the universe, asserting that if you wrote out a googolplex in full on a strip of paper, the paper would be bigger than the universe. (He demonstrated this by starting, and abandoning, the process in one of his television programs.) The terms googol and googolplex are the names for numbers substantially greater than the number of sand grains on Coney Island. In 1938, Edward Kasner, the prominent Columbia University mathematician, was talking with a group of kids, pursuing his hunch that the young have no trouble visualizing very large numbers. The children generally agreed that the number of raindrops falling on New York on a rainy day would be similar to the number of grains of sand on Coney Island, and that the numeral one followed by twenty zeros was probably a good expression of that number. Kasner suggested thinking about a larger number, one followed by a hundred zeros, and asked what it might be called. It was apparently his nine-year-old nephew Milton who came up with the name googol—and then googolplex, a one followed by, according to Milton, as many zeros as you could write before your hand became too tired. A googol (10100) and a googolplex (10googol) were formalized by Kasner and James Newman in their 1940 book, Mathematics and the Imagination. While an enjoyable term, the googol is so large a number that it is not represented by anything in the known universe, whether it be sand grains, stars, or atoms; it is not widely used, but it is reputed to have been the origin for the name of a certain internet search engine.
GEOLOGIC TIME AND ETERNITY
Unimaginably large numbers of particular importance include measures of geologic time. The Earth is 4.6 billion years old; the dinosaurs died out 70 million years ago. By the human measure of three score years and ten, these descriptions of time are impossible to grasp. The term deep time is sometimes used to refer to the geologist’s clock, as if these expanses are domains to which only geologists, through some arcane perspective, are privy. The fact is that these measures of time are as difficult for a geologist to come to terms with as for anyone else, but, of necessity, we do. Perhaps the trick is looking closely at natural processes, how slowly they work, yet what dramatic changes they bring about. Entire mountain ranges have risen, yet are destined to be worn away until only their roots remain. Entire oceans have been created and destroyed, great thicknesses of sediment have been built up from the steady work of rivers. Looking at the rates at which these processes are happening today, and assuming that they have always happened at roughly the same rate, the only conclusion is that the development of our planet as we see it today took a vast, almost unimaginable length of time. But imagine it we must—and we shall return to this in chapter 7.
Any number of ingenious ways of visualizing and illustrating the scale of geologic time have been devised. Condense the Earth’s life span into twenty-four hours or a year and note how utterly trivial is our time on Earth: if the Earth is a year old, we appeared at seventeen minutes to midnight on December 31. If one sheet of paper represents one year of the Earth’s history, the stack would be 450 kilometers (280 mi) high. A particularly vivid method, if you have a long enough corridor, is to roll out toilet paper. However, if every sheet were to represent a century, the corridor would have to stretch from London to New York and the history of Homo sapiens would cover the length of a Manhattan city block.
But since this book is about sand, and you are now intimately familiar with both heaps and individual grains of the stuff, perhaps a sand pile would be an appropriate illustration of geologic time. You can illustrate this yourself with a good quantity of medium-grained sand (I have simply used builder’s sand). Put a single grain on the floor—this grain represents one hundred years of the Earth’s history. A century is a length of time we can grasp, an aspirational but realistic life span. A cup of sand contains around three million grains, so now add three heaping cups of sand to make a pile (Figure 17, top). The Earth, as represented by our pile of sand, is now one billion years old, and, finally, the first primitive single-celled life forms emerge. Add another eleven cups of sand, and the Earth has reached the time when simple animals first occupied the land, just over 400 million years ago (Figure 17, middle). A final addition of one and a third cups brings the pile to the present day (Figure 17, bottom)—you have to look very carefully to see the difference. Remove a good pinch and the pile represents the first appearance of modern humans; take away ten teaspoons to show when the dinosaurs were wiped out.
Very large numbers present the same problem for us as a landscape without trees—we need some yardstick, some analogue to grasp to enable us to appreciate scale. We have seen in chapter 1 how our imagination can use an individual grain of sand as a measure of something minuscule, often to help us comprehend something vast. An accumulation of sand grains can similarly provide an image that resonates with large numbers, immense lengths of time—and a means of approaching the concepts of infinity and eternity.
FIGURE 17. A sandpile history of the Earth. (Photos by author)
A googolplex is no closer to infinity than it is to the number one. In 1714, Jonathan Swift, the Irish satirist, churchman, and author of Gulliver’s Travels, wrote to the Spectator:
Supposing the whole body of the earth were a great ball or mass of the finest sand, and that a single grain or particle of sand should be annihilated every thousand years. Supposing then that you had it in your choice to be happy all the while this prodigious mass of sand was consuming by this slow method, until there was not a grain of it left, on condition you would be miserable ever after; or supposing that you might be happy ever after, on condition you would be miserable until the whole mass of sand were thus annihilated at the rate of one sand in a thousand years: which of these two cases would you make your choice?
A difficult question that prompts a vague idea of eternity. The same approach was later taken by another Irishman, James Joyce, in A Portrait of the Artist as a Young Man, contemplating the eternity of hell—“Eternity! What mind of man can understand it?” To assist in the task, Joyce has his sermonizing priest use as his yardstick a child’s small handful of tiny grains of sand. He then asks that we imagine “a mountain of that sand, a million miles high, reaching from the earth to the farthest heavens, and a million miles broad, extending to remotest space.” Multiply this mountain by the number of drops of water in the ocean, and then imagine that, every million years, a bird arrives and carries off a single grain of sand. By the time the bird had removed all the sand, “not even one instant of eternity could be said to have ended.” An effective calibration. And read Jorge Luis Borges’s “The Book of Sand” (1975), the haunting story of a terrible, infinite book, so called “because neither the book nor the sand has any beginning or end.”
For millennia, sand has had a place in our imagination to represent numbers too large to grasp, the passage of time, and eternity; at the same time, it is a symbol of things almost too small to see, the microscopic. The power of this imagery derives in part from the familiarity of sand, its ubiquity as an everyday material. It is one of our planet’s most fundamental materials, shaping the surface of the Earth and our environment. The following chapters will look at those diverse habitats and the dynamics of the sands that sculpt them.