In prehistoric times, as human beings began to spread through the world in several waves of expansion, communities isolated from each other were appearing. Thus, it is not strange that also many different languages had appeared and had developed, languages that in a number of cases have virtually nothing to do with each other. It seems that the ability to invent languages is innate in "Homo sapiens", and this is evident when we ponder that even though there are only about 200 countries, there are between 3000 and 6000 languages on planet Earth. Part of this development of language has been dedicated, not surprisingly, to quantify things. From the most simple, for example how many animals I caught today, or how many fruits I collected, how many children I have, how many are we or how many are you, to the most complicated such as what elements are to be taken into account to calculate the trajectory of a rocket launching humans to the Moon.
But of course, as each culture developed words in different ways, they also developed numbers differently. For example, in sub-Saharan Africa there is an ethnic group, that for their purposes, are content to just say "one" and "two". In their environment, it's probably difficult to find situations where they need to count a thousand objects or a thousand people, or a hundred, or even tens. When this ethnic group needs to explain that there are more than two people, or more than two animals hunted, or have more than two children, they just say "many". "One", "two" and "many" are the numbers in this culture. So simple, and so they are happy.
Other ethnic groups gave a little more individualism to their many children, however. For example, the Tabajaras, a Brazilian Amazon tribe from which the virtuoso classical-guitar duo "Los Indios Tabajaras" came. These famous brothers arrived to the erudite stages of New York leaving behind their "names" Muçaperê and Erundi, "fifteen" and "sixteen" in their native language, or better understood, in the native language of their parents.
And do not think that this is a custom that is not widespread among the most civilized. In the United States of America, the black activist Malcolm X used the surname "X" because he did not know the true name of his ancestor who was brought from Africa as a slave, but there is one person whose full name, first and last name, carries an X at the end because he belongs to the tenth consecutive generation to carry the same name. His father was "IX", his grandfather "the Eighth", his great grandfather "the Seventh", his great great grandfather's surname was "VI", etc., etc., up centuries ago to "Jr." and finally to the proud patriarch, the origin of the closely guarded and reproduced names and surnames in the lineage of this family.
If we think about it, other cultures devised ingenious ways to designate plurals. The Guarani designed the "ore" and the "ñande", which have no direct equivalent in Spanish, Portuguese or English. "Ore" for "us but not you" and "ñande" for "us including you". Or the "kuéra", shorthand way of transforming sentences from singular to plural. Continuing on with the interesting Guarani language, we see that this South American ethnicity did need to describe a wider cosmos and introduced the "peteĩ", "mokoi", "mbohapy", "irundy" for "one", "two", "three", "four", up to a revelation: "po". "Po" means "hand", but is also used to designate the number five. And if we look closely, "potei", "pokoi", "poapy", "porundy" which may be neologisms but nevertheless indicate that this ethnic group used the fingers of one hand to count, until using them all and then restart the cycle. It is a base-five mathematical system. Upon completion of the fingers of both hands comes the word "pa", which means "complete", but can also be used to say "ten". And so going through the word for "twenty", the word for "thirty", the word for "forty", etc., always starting a new round every five units.
The Maya, admittedly the most-advanced pre-Columbians in Math, did not content with five nor ten, but with twenty: apparently they did not only used the fingers of one hand or two, but also the toes of both feet. Thus, only after every 20 units they passed to a higher level, what would be something like "the Mayan tens". Each 20 "Mayan tens" they passed to another level, what we may call "the Mayan hundreds". Each 20 "Mayan hundreds" they went to the level of the "Mayan thousands". And so on.
In Europe, and obviously completely independently, in ancient times a similar system was developed no less than in France and other countries with Flemish influence. It survives today in the language of Molière with the "quatre-vingts", a common way of saying "80", as it is four times 20. Notable. And it does not stop there: we have the "quatre-vingt-un" (81), the "quatre-vingt-deux" (82), "quatre-vingt-trois" (83), passing by the "quatre-vingt-dix", four times 20, plus 10, that is, 90, etc., etc., until the "quatre-vingt-dix-neuf", four times 20, plus 10, plus 9, viz 99. And so this country's aeronautical engineers used their CATIA binary digital computers for designing the Airbus "trois cent quatre-vingts" (380), the largest passenger plane in the world.
To do math, some numbers seem better than others, especially if they are easy to multiply or divide. "Per dozen is cheaper" because apparently 12 can be divided not only by a pair (two things), but by 3, 4 and 6, and of course all these factors can be multiplied to re-sell the complete dozen. Good for quick business in a busy market.
The dozen is apparently a legacy of the Babylonians, who also imagined twelve constellations in the sky, found that in one year there are about 12 full moons, and then put the twelve everywhere: twelve hours in a day, and why not 360 days in a year: 12 full moons * 30 days (the approximate synodic period of the Moon), thus obtaining the approximate length of the solar year. And apparently even twelve spokes for the wheels of their wagons. Speaking of "cycles" or "circles" (remember the bicycles, tricycles, and "quads", short for quadricycles), the number 360 can be divided into so many parts that it is very useful to measure a circle: 360 degrees.
The Babylonians also gave us the magical seven, for the seven "planets" or "wandering stars": the Sun, the Moon, Mercury, Venus, Mars, Jupiter and Saturn, which were what they could see millennia before the invention of the telescope. They considered them sacred, and not only the Babylonians, many of the cultures of the Middle and Near East as well. From there come the days dedicated to honor the Moon, Mars, Mercury, Jupiter, Venus in Spanish (Lunes, Martes, Miércoles, Jueves, Viernes) and for Anglo-Saxons Saturday and Sunday (days of "Saturn" and of the "Sun", respectively).
Returning to the XXIst Century and leaving behind superstitions: 7 is the independent variable we use on the two-dimensional matrix computer that we hung on the wall to know when we have to go to work to produce capital, and when we can relax and go out to spend our dividends at the malls.
The 13 is unlucky for some (there is no car numbered 13 in Formula 1) but it seems to bring good luck to others, and also the number 17: both are prime numbers, that is, they are not multiples of other numbers except 1, which puts them in a place apart. So aside that cicadas "Magicicada sp." apparently survive because they reproduce at those frequencies: it is rare that their potential predators live that many years, and so when these cicadas are born they are out of synchronization with the reproduction of their possible enemies. This is understandable because other "Animalia" usually have cycles of 2, 3, 4, 5, 6, 7, 8 or 10 years, which has as a consequence that when the cicadas "Magicicada sp." are born every 13 or 17 years they do so abruptly that their number exceeds overwhelmingly, literally, everyone who wants to eat them.
These mathematical "tricks" are not isolated: Biology is full of numbers. For example, the monk Gregor Mendel discovered in 1865, by statistical analysis of more than 20 000 isolated plants, the genes and the proportions of inheritance, in which on average one quarter of the progeny develops recessive traits, while the other three inherit the dominant traits.
Other mathematical cases lead to death and destruction: if the population of a species becomes unbalanced, their numbers can grow in geometric progression (i.e., an increase in which each number is a multiple of the former, like 1, 2, 4, 8, 16, 32, etc.). This process is called exponential growth. The population of "Homo sapiens" is doubling every forty years, which is disastrous for the planet's ecosystem. The first scholar who warned us of this danger was precisely another priest, Thomas Malthus, way back in the year 1798.
Of course, geometric progressions are also used in positive ways, such as to calculate the exponential increase in the capital earned by each person, which is what, with great sacrifice, take us out of poverty.
Incidentally, proportions and averages are to be well understood, otherwise it might happen to us what happened to a highly respected President of a great country of the West, who in the last century desperately ordered that something be done to solve the "problem" that half of his countrymen had below-average IQ.
Mathematics is present in other places that may seem unusual. For example in Music: the note A in the C-major scale is defined, according to the ISO 16:1975 standard, as 440 cycles or vibrations per second, 440 Hertz in scientific terms. For musicians of European (a.k.a. Western) artistic ancestry, this is the note used to tune all the rest. So in keyboard or guitar music, each note above or below this A follows a logarithmic scale in which the difference between a certain musical note and the following one (no intermediate frequencies between adjacent keys) is a multiple, in Hz, of the twelfth root of 2, or 2 to the power of 1/12, approximately the number 1,059 463 094 36... . This is called "equal temperament tuning". When we run "do-re-mi-fa-so-la-ti" on a keyboard or guitar, and then to the next "do", we have run an octave, a word derived from "a group of eight" (seven, plus one, musical notes). But there are actually 12 frequency intervals in an octave: the steps between "do-re", "re-mi", "fa-so", "so-la" and "la-ti" are twice as large as the steps between "mi-fa" and "ti-do". The larger steps are called tones, and the smaller steps are called semitones. Two semitones make a tone. So the mathematical underlying of an octave is actually 2 + 2 + 1 + 2 + 2 + 2 + 1 = 12. When we calculate the frequencies with the ratio "2 to the power of 1/12", given above, we see that after running these 12 semitones, that is, 1 octave, we have doubled the frequency of the note we are playing. The explanation is that 2 to the power of 1/12 becomes 2 to the power of 12/12, that is, 2 to the power of 1, or just 2. Thus each octave is twice the previous octave. We have started at "do" and we have ended at "do" again, but one octave higher, that is, between one "do" and the other "do" the frequency doubles. And a similar mathematical phenomenon can happen with the plethora of instruments of a classical orchestra or with a choral ensemble. Again, the note A in the C-major scale (440 Hz) is used to tune the instruments of the classical orchestra or the voices of the choral, but now "do-re-mi-fa-so-la-ti" and then to the next "do" follow a simplified series of ratios, namely "1", "9/8", "5/4", "4/3", "3/2", "5/3", "15/8" (say, by changing the length of a string), and "2", respectively. This is called "just or harmonic tuning", and it was known to Pythagoras way back in the Sixth Century B.C.E., who correctly regarded it as a mathematical manifestation of Nature, a major discovery.
Furthermore, when a man and a woman sing together, the pitch of the voices is usually higher for the female voice, but the duo may tune the frequencies of their voices to a ratio of 2 to 1, or 3 to 1, or 4 to 1. That is a reason why they can sing the familial "Happy Birthday" in harmony: the harmonics are full integer multiples of a frequency, so their voices fit together easily, as if they were singing the same note even though one person is actually singing at a much higher (or lower) frequency than the other person. Thus, although the distances between notes in classical music scales are slightly different between themselves (purists read more perfect) than the distances between notes in keyboard and guitar music scales, once again each scale ("do-re-mi-fa-so-la-ti-do") also begins at 1 time the frequency and ends at 2 times the frequency, and the following "do-re-mi-fa-so-la-ti-do" at 3 times the frequency, and the following "do-re-mi-fa-so-la-ti-do" at 4 times the frequency. Just perfect for the harmonics of Händel's "Halleluja". Amazing.
Although there are other pairs of notes that do not come as unpleasant when sounding together, like the so-called just fifths, and to a lesser extent, the major and minor thirds (or even in many instances, for dramatic effects, the opposite is sought: pairs of notes that intentionally sound unpleasant or stressful), only the octave generates perfect harmony, due to its underlying mathematics of being multiplied by integers.
The physical explanation is that with perfect ratios each wave rides at a constant distance from the preceding and the following one, be it over the others, be it in between, or at any other position, but without slipping out of synchrony. You can help a child ride a swing by pushing her or him each time her or his back comes to you, or every 2 times, or every 3 times, or even every 4 times, but you cannot help her or him if your efforts fall out of synchrony. You can even achieve it by pushing when the child is still rising or when the child is already receding from you: granted, the swing will not be as large, but as long as you pay attention to her or his frequency, you will generally succeed in keeping the child swinging. You and the child become a single motion entity, working in "harmony", and the child will smile. If you pay attention to your fellow musical performers' frequencies, you will complement each other and the classical orchestra or the choral becomes a single waves-generating entity, keeping everything in harmonic motion, consonant but not dissonant. By paying attention to this underlying mathematics, what could potentially be a haphazard, incongruous, ugly "noise" becomes beautiful "sound", and the audience will smile.
In the first universities, which appeared in the Middle Ages, the curriculum began with the "Trivium": Grammar, Logic and Rhetoric, before moving on to more advanced studies, the "Quadrivium": Arithmetic, Geometry, Astronomy and Music. Note that these two phases correspond to Language and Mathematics, respectively, the two main measurements that are taken today from candidates who want to enter the most prestigious universities. Of course, university curricula are much broader now, but at the same time more specialized. No wonder then that Wernher von Braun, the engineer who created the rocket that carried human beings to the Moon, the Saturn V (5 because it had five Rocketdyne F-1 engines), advised his young followers to study Science and Mathematics in depth, but at the same time reminded them that in his youth he also had played piano and cello, and at home he had a painting by Rubens. Forgetting the Arts and Humanities results in a stunted personality development, he said.
As it is known, the Romans wrote numbers with the same signs they used for Literature: I, V, X, L, C, D, M. Incidentally, 4 can be written IIII, and not IV, and the horizontal bars above may indicate multiplication per thousand.
THE DECIMAL SYSTEM
The signs we use for our one, two, three, four, five, six, seven, eight and nine derive from the Arabic notation: 1, 2, 3, 4, 5, 6, 7, 8, 9. It is much easier to solve "3456 * 789" than "three thousand four hundred fifty-six multiplied seven hundred eighty-nine times". The Arabic zero apparently derives from India, a civilization with whom the Middle East had nautical communication. Probably the "0" actually began as a period (.), placed discreetly by calculists when there was no other sign to place, that is, there was an empty or null value in the operation.
The Mayans also had their zero and it seems that they realized more quickly about its importance because they drew it as a stylized seashell.
In ancient times the simplest multiplications and divisions were by two or by three. So, in the English System of Measurement submultiples of an inch (in, or ") are 1/2, 1/4, 1/8, 1/16, 1/32, 1/64 of an inch. The inch in turn was the size of three grains of barley, three inches is a palm, four palms are one foot and three feet make a yard. The mile derives from the Romans: "milia passum", one thousand (double-step) paces of a legion of soldiers, or 1609 m. The league is what a person can walk in an hour, or about 5 km (poorly standardized: for example, the league of Paraguay was in the Eighteenth Century 4180 m, according to the cartographer and naturalist Félix de Azara, with the warning of his colleague Francisco de Aguirre that it could vary by about 3%). The maritime league was other, different one: five leagues of Paraguay were four maritime leagues.
The nautical mile is more modern and much more mathematical: it was designed with the idea that each boat could calculate its travel distance by measuring the difference of latitude achieved. A maritime, marine or nautical mile equals one minute (') of latitude. As the Earth is not a perfect sphere, measurements at the equator yield differences from measurements near the poles. Therefore, it was standardized halfway between zero degrees and ninety degrees latitude: at 45 degrees latitude. The standard value is therefore 1852 m.
The knot as a measure of speed (both for boats and for aircraft) comes from an archaic speedometer from the era of Columbus' ships, or even earlier: the "log's line" or "log and line" method, consisting in a bucket or in a flat piece of wood (log) tied to a rope, that was thrown into the sea leaving it "anchored" in the water while the boat continued its march. The rope, or line, had knots at well calculated intervals, thus by counting the number of knots that ran in a given time while the rope uncoiled, the speed could be estimated, in "knots" (1 knot = 1 kt = 1 nautical mile per hour; this gives 1,852 km/h according to its modern value). By multiplying the speed by the travel time the distance covered by the ship was estimated.
To avoid dangerous confusions, the International Civil Aviation Organization (ICAO) is using nautical miles (NM) for distances, knots (kt) for the velocities and feet (ft) for aircraft heights, due to the enormous amount of U.S.-made equipment. In other applications, the International System is recommended . Worth mentioning that the symbol for meter, "m", is a symbol and not an abbreviation, so it is wrong to write "m.", "mts.", "M", or anything else. The same goes for km (kilometers), kg (kilograms), V (volts), A (amperes), N (newtons, a measure of force), K (kelvins, a measure of temperature), etc.. Incidentally, the right thing to say is "kelvin" and not "kelvin degree". "C" degrees is not "Centigrade degree" but "Celsius degrees", for its inventor. However, this last unit should be avoided in scientific notation in preference of kelvins.
The Fahrenheit degrees are very interesting. They were designed to measure ambient temperature: as temperature extremes in Anglo-Saxon countries varies between the freezing point of the sea and the temperature of the human body, these were the values used to calibrate the scale. In keeping with the custom of dividing and multiplying in the simplest possible way, it ranges from zero (0) to 96 degrees, not 100. The scale of a Daniel Fahrenheit's thermometer could be divided into 2, 3, 4, 6, 8, 16, 24, 32, 48 or 96 equal parts. It makes sense, like the inches (the size of the distal phalanx, or last bone of the thumb of an adult man), the feet (of an adult male, according to tradition this was no less than the French Emperor Charlemagne), the yard (the distance reached by the arms outstretched to the sides of the body, counting from the nose to the tip of the thumb of an adult male, according to tradition this was no less than King Henry I of England). They were easy to use without a measure tape and easy to teach to children. An English aeronautical writer tells how during the Cold War he had the rare opportunity to be next to a Soviet military plane and could rapidly estimate its capacity to wage war. Its maximum speed can be inferred by paying attention to its shape. By walking around it he could measure its overall size; by using his hands to measure its tires he calculated its maximum weight, and with these data he estimated its fuel capacity and its maximum range. By subtracting the fuel capacity he could estimate the armament load it could carry to different distances and at different speeds. All in a few minutes without the Soviet officials having realized the amount of data that this Englishman, with his "obsolete" system of measurement, was obtaining.
But despite this, the decimal system is clearly superior to perform complex calculations, because it is only a matter of just adding or removing zeros, or to run a comma. A note on this: the International System of Units ("Système international d'unités", whose symbol is SI in any language, and not IS, S.I.U., S.I., Metric System, MKS or whatever) recognizes a particular symbol for decimal separation but not for anything else: pi is approximately equal to 3,141 592 653 59 ... . Note the spaces left every three digits. A million is written 1 000 000 and not 1,000,000. A thousand is written 1000 and not 1,000. The idea is to avoid problems when translating from one language to another, where a point can be used as comma: 6.346 or 6,346 may be the same value or a thousand times different. The exception is when it comes to money: obviously no empty space is to be left when typing a monetary value on a check.
So, the decimal system is so good that inadvertently we use it in the wallet. The people of the United Kingdom were well aware of this in 1971, when sterling pounds ceased to be 20 shillings and shillings stopped to be 12 pences. Now one sterling pound still has pences but they are 100, on par with the rest of the world. And by the way, an inch is no longer three grains of barley but it is now defined as being exactly 25,4 mm.
The SI gained great momentum after the French Revolution, to the point that circumferences of one hundred degrees were proposed and even watches of 10 hours a day, with one hundred minutes and a hundred seconds. Today only the slope in percentage survives (a slope of 45 degrees is a slope of 100 %, because we climb one hundred meters vertical as we advance one hundred meters horizontal), while degrees are measured in radians (the radius superimposed on the circumference, so that a semicircle is 3,14... radians in aperture and a full circle has a circumference of 6,28... radians). Believe it or not, radians, microrradians, and milirradians are extremely useful in calculations, much more so than talking about 57 degrees, 17 minutes and 45 seconds and go you find the multiples and submultiples.
The seconds (s) are the standard marking of time, no matter how many millions are at stake. Spacecraft computers work this way. Hours and minutes do not matter: that's what the powers of 10 are for: 10 to the power of 3, 10 to the power of 6, 10 to the power of 9, etc.. Or kilo (k), mega (M), giga (G), etc..
This way of writing the multiples is very useful when talking about billions. In the U.S., 1 000 000 000 is one billion, while in the rest of the world they are only one thousand millions. A billion would be a million millions, or 1 000 000 000 000. Although etymology tells us that they are not so wrong: the Italians began using million to say "huge thousand", or a thousand times a thousand, therefore a billion (from obsolete French) would be "a thousand twice huge", or a thousand times a thousand times a thousand, that is, 9 zeros, not 12. To that, people in the U.S. still cling.
Returning to the modern French, a meter in SI is measured by measuring the speed of light. By convention, we say that light travels in a vacuum exactly 299 792 458 m within 1 s. What we need to measure is therefore the second, and to that end atomic clocks increasingly accurate are being built. After obtaining the datum, the arithmetic division is performed and the value of the meter is obtained. No more "the platinum-iridium bar kept in Paris", etc.. It may seem unnecessarily complicated, but it's better than "the length of the sword of Napoleon when dismounted after the battle of Waterloo" or something like that. What is still not solved is the issue of the standard kg, "kept in the offices of the Bureau International des Pois et Mesures, etc..". We still have to go to Paris to calibrate the balances, but the day will come ... .
MATHEMATICS: THE UNIVERSAL LANGUAGE
What remind us about something: no matter the notation, the system of measurements, be it decimal, binary, Mayan, Guarani, etc.: numbers are universal. So universal that it would probably be the perfect language to try to communicate with extraterrestrial beings. We have already applied this idea (still without success, as we have "not" learned): since 1960, E.T. signals have been sought with radio telescopes and laser detectors. One of the few projects, SETI @ home, uses hundreds of thousands of volunteers' computers, through the Internet, to analyze data from the Arecibo radio telescope, the largest in the world. In 1972 and 1973 the Pioneer 10 and 11 probes left the Earth with plaques identifying, in scientific language, their planet and time of origin. They were followed by the Voyager 1 and 2 in 1977, with LP disc records containing images and sounds. In 1974, the Arecibo radio telescope sent a powerful message to a distant cluster of stars 25 000 light-years away.
The universality of Mathematics also carries with it intriguing things. For example, there are numbers that are considered fundamental in the Universe in which we live, such as the Universal Gravitational Constant in Isaac Newton's system (6,673 84 * 10 to the power of -11 cubic m / kg * square s), the Speed of Light in Einstein's Relativity (299 792 458 m/s), the Charge of the Electron sought by Millikan (1,602 176 565 * 10 to the power of -19 C), Planck's Constant in Quantum Mechanics (6,626 069 57 * 10 to the power of -34 J * s), etc.. If these numbers were quite different, planets would end up lost in space, stars (like the Sun) would shine for too short a time, molecules could not form, atoms would not exist, etc..
We can think of other universes in which these constants are different. Maybe those universes never get to evolve to harbor intelligent life. Though out of an infinite number of universes, one of them should evolve to become the one we live in. There is a postulate that says that an infinite number of monkeys typing on an infinite number of typewriters for an infinite amount of time would write the complete works of Shakespeare. Chance becomes certainty when it comes to endless possibilities.
However, it is useless to seek other universes. If we ever get to enter in communication with one of them, it would simply mean that it is a continuation of our own universe. And if it really is a parallel universe, it would have no communication or passage whatsoever with ours and therefore we would never know of its existence. So for practical purposes, the Universe is all there is, all there was and all that will be, and therefore it is one only: the one we inhabit.
Wondering why the Universe has these Constants, why it works how it works, why it allows Life, why we are here, is the same as the Queen of England wondering why she is the Queen of England and not someone else. "Someone" has to be the Queen of England, and if not her, someone else would be wondering that. And if there is no Queen of England, "no one" would be wondering that. We reflect about the Universe simply because we are here to do it. If we were not here, we would not be thinking, for starters. Therefore, it is useless to ask why we are here; we can only be grateful that we are here. If we're not alive, we won't have much opportunity to have fun.
Way back in time, four centuries ago, Galileo Galilei discovered that in order to understand Nature we have to understand the language it speaks. Nature has a language, and that language is the language of Mathematics. The most important book in History was published by Isaac Newton, its first edition in 1687: "Mathematical Principles of Natural Philosophy". In it it was first demonstrated that Nature goes up to Heaven, and that, to the best of our knowledge, everything that happens here happens there, and that everything that happens there happens here. The Earth is not a thing that is here and the Universe a separate thing that is out there, but the Earth and all that it contains, including ourselves, are part of the Universe. We are the thinking part of the Universe. That's what we have achieved, the hydrogen atoms produced in the Big Bang, after 13 700 million years of evolution.
As Richard Feynman said, it is unfortunate that many people have trouble understanding Mathematics. It is not a language invented by us, but it is the language that comes to us from outside of us, and perhaps that is why it is so difficult to learn. But we can not appreciate the beauty of the Universe if we do not understand its language. One can try to explain orally to a blind person the beauty of a landscape, but if he or she does not see it by him or herself, unfortunately he or she will not have the same experience that a person with the ability of sight has. The latter has a much greater chance to marvel at it. So, although it may sound strange to most of the dear readers, talking about the world around us without Mathematics is just like taking a look through a window without ever walking out through the door to really enjoy the world.
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Photograph: A 17-year periodical cicada "Magicicada sp.", brood XIII, 2007. Photographed in Lisle, Illinois. Shown with a Kentucky Coffee tree, "Gymnocladus dioicus". Photo credit: Bruce Marlin (original license, of the photo only, obtained at: http://creativecommons.org/licenses/by/3.0/deed.en). © Bruce Marlin, http://www.cirrusimage.com, via Wikimedia Commons.