September 23, 2013
Most people think of history as a series of stories—tales of one army unexpectedly defeating another, or a politician making a memorable speech, or an upstart overthrowing a sitting monarch.
Peter Turchin of the University of Connecticut sees things rather differently. Formally trained as a ecologist, he sees history as a series of equations. Specifically, he wants to bring the types of mathematical models used in fields such as wildlife ecology to explain population trends in a different species: humans.
In a paper published with colleagues today in the Proceedings of the National Academy of Sciences, he presents a mathematical model (shown on the left of the video above) that correlates well with historical data (shown on the right) on the development and spread of large-scale, complex societies (represented as red territories on the green area studied). The simulation runs from 1500 B.C.E. to 1500 C.E.—so it encompasses the growth of societies like Mesopotamia, ancient Egypt and the like—and replicates historical trends with 65 percent accuracy.
This might not sound like a perfect accounting of human history, but that’s not really the goal. Turchin simply wants to apply mathematical analysis to the field of history so that researchers can determine which factors are most influential in affecting the spread of human states and populations, just as ecologists have done when analyzing wildlife population dynamics. Essentially, he wants to answer a simple question: Why did complex societies develop and spread in some areas but not others?
In this study, Turchin’s team found that conflict between societies and the development of military technology as a result of war were the most important elements that predicted which states would develop and expand over the map—with those factors taken away, the model deteriorated, describing actual history with only 16 percent accuracy.
Turchin began thinking about applying math to history in general about 15 years ago. “I always enjoyed history, but I realized then that it was the last major discipline which was not mathematized,” he explains. “But mathematical approaches—modeling, statistics, etc.—are an inherent part of any real science.”
In bringing these sorts of tools into the arena of world history and developing a mathematical model, his team was inspired by a theory called cultural multilevel selection, which predicts that competition between different groups is the main driver of the evolution of large-scale, complex societies. To build that into the model, they divided all of Africa and Eurasia into gridded squares which were each categorized by a few environmental variables (the type of habitat, elevation, and whether it had agriculture in 1500 B.C.E.). They then “seeded” military technology in squares adjacent to the grasslands of central Asia, because the domestication of horses—the dominant military technology of the age—likely arose there initially.
Over time, the model allowed for domesticated horses to spread between adjacent squares. It also simulated conflict between various entities, allowing squares to take over nearby squares, determining victory based on the area each entity controlled, and thus growing the sizes of empires. After plugging in these variables, they let the model simulate 3,000 years of human history, then compared its results to actual data, gleaned from a variety of historical atlases.
Although it’s not perfect, the accuracy of their model—predicting the development and spread of empires in nearly all the right places—surprised even the researchers. “To tell the truth, the success of this enterprise exceeded my wildest expectations,” Turchin says. “Who would have thought that a simple model could explain 65% of variance in a large historical database?”
So why would conflict between societies prove to be such a crucial variable in predicting where empires would form? “To evolve to a large size, societies need special institutions that are necessary for holding them together,” Turchin proposes. “But such institutions have large internal costs, and without constant competition from other societies, they collapse. Only constant competition ensures that ultrasocial norms and institutions will persist and spread.”
The model shows that agriculture is a necessary but not sufficient precondition for a complex society, he says—these states can’t form without farming, but the persistent presence of competition and warfare is necessary to forge farming societies into durable, large-scale empires. Conventional analyses of history could come to this same conclusion, but they wouldn’t be able to demonstrate it in the same mathematically-based way. Using this approach, on the other hand, Turchin’s group could remove the influence of warfare and see the model’s accuracy in describing real historical data plummet.
Of course, there are limitations to viewing history through math—humans are more complicated than numbers. “Differences in culture, environmental factors and thousands of other variables not included in the model all have effect,” Turchin says. “A simple general model should not be able to capture actual history in all its glorious complexity.”
Still, the model is a unique and valuable tool. Going forward, Turchin’s team wants to develop it further—adding more nuance (such as including the quality of agricultural productivity, rather than merely toggling if farming exists in a given area or not) to improve on that 65 percent accuracy. Additionally, they’d like to expand the model, applying it to more recent world history and also pre-Columbian North America, if they can find relevant historical data.
Based on his experiences so far, Turchin thinks they’ll be successful in developing a model that better reflects the the rise and fall of civilizations. “It turns out that there is a lot of quantitative data in history,” he says, “you just have to be creative in looking for it.”
April 25, 2013
Google, as many researchers know well, is more than a search engine—it’s a remarkably comprehensive barometer of public opinion and the state of the world at any given time. By using Google Trends, which tracks the frequency particular search terms are entered into Google over time, scientists have found seasonal patterns, for example, in searches for information about mental illnesses and detected a link between searching behavior and a country’s GDP.
A number of people have also had the idea to use these trends to try achieving a more basic desire: making money. Several studies in recent years have looked at the number of times investors searched for particular stock names and symbols and created relatively successful investing strategies based on this data.
A new study published today in Scientific Reports by a team of British researchers, though, harnesses Google Trends data to produce investing strategies in a more nuanced way. Instead of looking at the frequency that the names of stocks or companies were searched, they analyzed a broad range of 98 commonly used words—everything from “unemployment” to “marriage” to “car” to “water”—and simulated investing strategies based on week-by-week changes in the frequencies of each of these words as search terms by American internet users.
The changes in the frequency of some of these words, it turns out, are very useful predictors of whether the market as a whole—in this case, the Dow Jones Industrial Average—will go down or up (the Dow is a broad index commonly considered a benchmark of the overall performance of the U.S. stock market).
The strategy was relatively straightforward: The system tracked whether a word such as “debt” increased in search frequency or decreased in search frequency from one week to the next. If the term was suddenly searched much less frequently, the investment simulation bought all the stocks of the Dow on the first Monday afterward, then sold all the stocks one week later, essentially betting that the overall market would rise in value.
If a term such as “debt” was suddenly searched much more frequently, the simulation did the opposite: It bought a “short” position in the Dow, selling all its stocks on the first Monday and then buying them all a week later. The concept of a “short” position like this might seem a bit confusing to some, but the basic thing to remember is that it’s the exact opposite of conventionally buying a stock—if you have a “short” position, you make money when the stock goes down in price, and lose money when it goes up. So for any given term, the system predicted that more frequent searches meant the market as a whole would decline, and less frequent searched meant it would rise.
During the period of time studied (2004-2011), making investment choices based on a few of these words in particular would have yielded overall profits several times higher than a conservative investment strategy of simply buying and holding the stocks of the Dow for the entire time. For example, basing a strategy solely on the search frequency of the word “debt,” which turned out to be the single most profitable term in the study, would have generated a profit of 326% over the seven years studied—compared to a profit of just 16% if you owned all the stocks of the Dow for the whole period.
So if you systematically bought a “short” position in the market every time the word “debt” suddenly started getting searched more often, you’d have made a ton of money over the seven years studied. But what about other words? The system simulated how this strategy would have performed for each of 98 words chosen, listed in the chart at right from most useful at predicting the movement of the markets (debt) to least useful (ring). As seen in the chart, for some of these terms the frequency that we type them into Google seems to serve as a very effective early-warning system for declines in the market.
Stock market declines typically reflect investors’ overall belief that, at any given time, it’s better to sell stock than buy it, and they often happen suddenly, when investors move in a herd to a new position—so the researchers speculate that rises in the terms’ frequencies in search convey a nascent feeling of concern about the market, before it’s expressed via actual transactions. All these searches might also reflect countless investors in an information-gathering phase, seeking to find out as much as they possibly can about an industry or a stock before selling it.
Even beyond the practical investment strategies that this type of analysis might generate, simply looking through the words provides a striking—and oftentimes confusing—window into the collective American psyche. It’s seemingly obvious why a sudden increase in the amount of people searching for the word “debt” might signal overall negative feelings about the market, and would likely precede a drop in stock values, and why “fun” might precede increases in the market. But why do searches for the words “color” and “restaurant” predict declines nearly as accurately as “debt”? Why do “labor” and “train” also predict stock market rises?
March 29, 2013
If the Easter Bunny comes to your house this weekend, you may find yourself with a plethora of marshmallows and Peeps. What to do with them all? Aside from simply eating them, cooking with them, or unleashing your artistic side by making dioramas, consider using them….for science!
Marshmallows, it turns out, are must-have pieces of equipment for at-home science experiments. Sure, you can use them test your kids’ self control through the the field of psychology’s notorious marshmallow test and its ever-more complex iterations. But if you’d rather not torture your kids by leaving tantalizingly in reach a marshmallow they’re ordered not to have, consider trying these easy science projects:
Marshmallows in a vacuum
No, not that kind of vacuum, despite the intriguing possibilities conjured by this phrase. You’ll need:
- A glass jar with a lid
- A mechanism to pump some of the air out of the jar
Place a few marshmallows in the jar, seal it, and then pump the air out:
What’s going on? Marshmallows are basically a foam spun out of sugar, water, air, and gelatin. The sugar makes them sweet, the water and sugar combo makes them sticky and the gelatin makes them stretchy. But the air–which actually makes up most of the confection’s volume–makes marshmallows the tastiest way to encapsulate a gas in a solid. As you pump air out of the jar, the air inside the marshmallow expands and the marshmallow puffs up. Release the seal, and the marshmallows return to their normal size.
Congratulations! You’ve just demonstrated Boyle’s Law, which states that when the temperature doesn’t change, that the relationship between pressure (which is decreased by pumping air out of the jar) and volume of any set amount of gas (the marshmallow) is inversely proportional. In other words, decreasing one necessitates an increase of the other.
If you can’t eat ‘em, nuke ‘em!
If you’ve ever roasted a marshmallow over a campfire, you’ll know where this next demonstration is going. You’ll need:
- A microwave
- A microwavable plate
- A standard-sized marshmallow (avoid minis or jumbos; the former will fry and the latter may make an enormous mess!)
Place the marshmallow on one of its flat sides in the center of a plate. Then microwave the marshmallow for, say, 45 seconds on high.
It’s alive! This time, rather than changing the pressure surrounding the marshmallow, you’re changing the temperature. As the microwave bakes the marshmallow, the water in the marshmallow heats up and warms the air. When air becomes hot, it expands, forcing the marshmallow to puff up. The confection’s water also softens the sugars, causing it to ooze, as seen in the video above (created by YouTube user bbbpwns).
The relationship between temperature and volume is representative of Charles’ Law, which holds that any set amount of gas will expand when heated–increasing the temperature of a gas necessitates an increase in the gas’ volume.
Trying this with Peeps makes for a slightly alarming outcome, showcased by YouTube user UBrocks:
If you flashed back to the Stay Puft Marshmallow Man, alas–the monster marshmallow you pulled from your microwave doesn’t last–it will cool and deflate into a glob of ooze. But before it cools completely, the ooze is quite malleable and can be sculpted into shapes. But careful! The marshmallow remnants are like naplam–they’ll stick to you and burn. After it cools a bit, brush some oil on your palms before you mold anything, else your sculpture will stay glued to your hands.
A gooey way to calculate the speed of light
For this demonstration you need a bit of background knowledge as you start out. The speed of a wave can be calculated by multiplying the wavelength (the distance from crest to crest) with the frequency (the number of crest-to-crest cycles that repeat in a stretch of time). Light is a wave, and its speed can be calculated the same way without fancy equipment. You’ll need:
- A microwave with the turntable removed
- A glass casserole dish or baking tray
- Mini marshmallows
- A ruler
- A calculator
Take the baking tray and pack one layer of marshmallows along the bottom, lined up like tiny puffy soldiers. Make sure the turntable is removed from the microwave–this allows microwaves to move through the glass and the marshmallows in a standing wave pattern. Cook for a few minutes on low, watching the marshmallows carefully. With the turntable removed, the microwave doesn’t heat evenly–you’ll notice melted patches forming in your marshmallow field.
As soon as you see a few such patches, remove the dish and measure the distance between two that form a line parallel to the microwave’s door–these mark the locations of highest amplitudes within the standing wave. Multiply this by two to get the full wavelength of the microwaves that passed through your marshmallows (if you look at the geometry of a standing wave, your initial measurement only gave you half the wavelength). Convert this into meters.
Multiplying this result by frequency of the microwave, found in the microwave’s manual or in a label inside the device, gives ~299,000,000 meters per second–roughly speed of light! Catch a video of this here.
March 14, 2013
March 14, when written as 3.14, is the first three numbers of pi (π). To commemorate the (completely artificial) confluence of the world’s most famous and never-ending mathematical constant with the way we can write the date, math enthusiasts around the country embrace their inner nerdiness by celebrating π, the ratio of the circumference of a circle and its diameter.
The date–which also happens to be Einstein’s birthday–inspires celebrations every year. Today. the Massachusetts Institute of Technology is posting password-protected decision letters on its admissions office site–would-be attendees can view whether they gained admittance at 6:28 pm (approximately equal to 2π, or the ratio of a circle’s circumference to its radius). Not to be outdone, Princeton’s celebrations of pi span an entire week, complete with a pie eating contest, an Einstein look-alike contest and a π-themed video contest (videos extolling pi and Einstein’s birthday must be less than 3.14 minutes; the winner will be announced at 3:14 today and will receive–you guessed it–$314.15).
Just why are people crazy about pi? The number–three followed by a ceaseless string of numbers after the decimal point, all randomly distributed–is the world’s most famous irrational number, meaning that it cannot be expressed as through the division of two whole numbers. In fact, it is a transcendental number, a term which boils down the idea that it isn’t the square root, cube root or nth root of any rational number. And this irrationality and transcendental nature of pi appeals, perhaps because pi’s continuous flow of numbers reflects the unending circle it helps to trace.
Pi has held an almost mystical quality to humans throughout time. Its unspoken presence can be felt in the circular ruins of Stonehenge, in the vaulted ceilings of domed Roman temples, in the celestial spheres of Plato and Ptolemy. It has inspired centuries of mathematical puzzles and some of humanity’s most iconic artwork. People spend years of their lives attempting to memorize its digits–they hold contests to see who knows the most numbers after the decimal, write poems–”piems,” if you will–where the number of letters in each word represents the next digit of pi, compose haikus (pikus)…the list goes on and on like pi itself.
Here are some notable moments in the history of pi:
1900-1650 BC: A Babylonian tablet gives a value of 3.125 for pi, which isn’t bad! In another document, the Rhind Mathematical Papyrus, an ancient Egyptian scribe writes, in 1650 BC “Cut off 1/9 of a diameter and construct a square upon the remainder; this has the same area as the circle” This implies that pi is 3.16049, “which is also fairly accurate,” according to David Wilson of Rutgers University’s math department.
800-200 BC: Passages in the Bible describe a ceremonial pool in the Temple of Solomon: “He made the Sea of cast metal, circular in shape, measuring ten cubits from rim to rim and five cubits high. It took a line of thirty cubits to measure around it” (I Kings 7:23-26). This puts pi at a mere 3.
250 BC: Archimedes of Syracuse approximates the area of a circle by using the Pythagorean Theorem to find the areas of two shapes–a 96-sided polygon inscribed within the circle and an equally faceted polygon within which the circle was circumscribed. The areas of the 96-sided shapes sandwiched the area of circle, giving Archimedes upper and lower bounds for the circle’s extent. Though he knew that he had not found the exact value of pi, he was able to approximate it to between 3 1/7 and 3 10/71.
Late 1300s: Indian mathematician and astronomer Madhava of Sangamagrama first posits the idea that pi could be represented as the sum of terms in an infinite sequence–for example, 4 – 4/3 + 4/5 – 4/7 + 4/9…His work helped inspire branch of mathematics that examines the results of mathematical operations performed over and over on a never-ending stretch of numbers.
1706: Welsh mathematician William Jones began to use π as a the symbol for the ratio of the circumference of a circle to its diameter. Famed Swiss mathematician Leonhard Euler adopted this usage in 1737, helping to popularize it through his works.
1873: Amateur English mathematician William Shanks calculates pi out to 707 digits–his number was written on the wall of a circular room–appropriately named the Pi Room–in the Palais de la Découverte, a French science museum. But his number was only correct to the 527th digit–in 1946, the error was finally caught, and in 1949, the number was corrected.
1897: Lawmakers in Indiana almost pass a bill that erroneously labels the value of pi to 3.2. Cajoled by an amateur mathematician Edwin Goodwin, the Indiana General Assembly introduced House Bill 246, which introduced “a new mathematical truth” for sole use by the state. The “truth” was an attempt to square the circle–a puzzle which requires that a circle and square of the same area be constructed using only a geometrical compass and a straightedge. The bill unanimously passed the house, but the senate and hence the state was spared from embarrassment by C.A. Waldo, a Purdue mathematics professor who coincidentally happened to be in the State House that day. “Shown the bill and offered an introduction to the genius whose theory it was, Waldo declined, saying he already knew enough crazy people,” Tony Long of Wired wrote. Waldo gave the senators a math lesson, and the bill died.
1988: Larry Shaw of San Francisco’s Exploratorium inaugurates the first Pi Day celebration. This year, even as it prepares for its grand re-opening in April, the museum holds its 25th annual Pi Day extravaganza.
2005: Chao Lu, then a graduate student in China, becomes the Guinness record holder for reciting digits of pi–he recited the number to 67,980 digits. The feat took him 24 hours and 4 minutes (contest rules required that no more than 15 seconds could pass between any two numbers).
2009: Pi Day becomes official! Democratic Congressman Bart Gordon of Tennessee’s 6th congressional district, along with 15 co-sponsors, introduced HR 224, which “supports the designation of a Pi Day and its celebration around the world, recognizes the continuing importance of National Science Foundation math and science education programs, and encourages schools and educators to observe the day with appropriate activities that teach students about Pi and engage them about the study of mathematics.” The resolution was approved by the House of Representatives on March 12 of that year, proving that a love of pi is non-partisan.
How are you celebrating Pi Day?
December 12, 2012
Today as you are slogging through the tasks marked on your calendar, you might notice the date: 12/12/12. This will be the last date with the same number for day, month and last two digits of the year until New Year’s Day, 2101 (01/01/01)–89 years from now.
Many are celebrating the date with weddings (the truly hard core are start their ceremonies at 12:00 pm, presumably so that they’d be mid-vow at at 12:12), concerts–such as this benefit for victims of Superstorm Sandy–even mass meditations. The Astronomical Society of the Pacific, based in San Francisco, has actually declared 12/12/12 “Anti-Doomsday Day,” the antidote to purported Mayan prognostications that the world will end on 12/21/12. Belgian monks have released the holy grail of beers–Westvleteren 12–for public sale today.
But even if you’re not doing something grand to commemorate the last such date in most of our lifetimes, you might find that a closer look at the date itself is intriguing from a mathematical point of view. As Aziz Inan, a professor of electrical engineering at the University of Portland whose hobby includes looking at number patterns in dates, describes (PDF) among other things:
- 12 = 3 x 4 (notice the numbers here are the consecutive counting numbers)
- 12 = 3 x 4, and 3 + 4 = 7; the date 12/12/12 happens to be the 347th day of 2012
On 12/12/12, there will be 12 days until Christmas. Twelve is also significant to society, the Astronomical Society of the Pacific reminds us. Aside from 12 inches in a foot, there are “contemporary calendars (12 months in the year), chronology (12 hours of day and night), traditional zodiac (12 astrological signs), Greek mythology (12 Olympic gods and goddesses), holiday folklore (12 days of Christmas), Shakespeare (Twelfth Night), and of course in our culinary world (dozen eggs, case of wine)…More importantly, in astronomy, Mars is 12 light minutes from the Sun, the average temperature of the Earth is 12 degrees Celsius, and Jupiter takes 12 years to orbit the Sun.”
The first 12 years of the next century will see 12 more dates with repeating numbers–01/01/01, 02/02/02, etc.–but other dates with numerical patterns are in our future. Here are a few categories:
Cheating but repeating: Every decade of this century will experience at least one date where all the numbers are the same–2/2/22, 3/3/33. 4/4/44, etc. The next decade will also have 2/22/22. Future dates out of reach for us–take 2/22/2222–may be truer representations of repetitive numbers in dates–imagine having that birthday!
Number palindromes: Palindromes–a number that reads the same forwards and backwards–are more common than repeats. This year hosted 2-10-2012. If you write dates in the “Gregorian little-endian” style of day/month/year, then 2012 had two: 21/02/2012 (in February) and 2/10/2012 (in October). The next palindrome date will be next year on 3/10/2013 (in March or October, depending on how you read the date). One-hundred and nine years from today, 12/12/2121 will also be a palindrome date. Inan has identified 75 palindrome dates this century–you can see the first 30 on a list he compiled. Of course, if you only use the last two digits of the year, then this past February (in the month/day/year way of noting dates) was full of them: 2/10/12, 2/11/12, 2/13/12, etc.
Perfect squares: Some dates, like March 3, 2009 (3/3/09) are unique in that their numbers form perfect squares and their roots (as in 3 x 3 = 9). Other such dates are 4/4/16, 5/5/25, etc. But in some cases, if you take out the punctuation separating the dates, the resulting number is a perfect square. Take April 1, 2009, written as 4/01/2009 or 4012009–the number is a perfect square, with a root of 2003 (2003 x 2003 = 4012009). Other dates, when written the same way are reverse perfect squares, as Inan coined, when written from right to left. One such date December 21, 2010–when reversed it is 01022121, which happens to be the perfect square of 1011. Only two more such dates will occur this century.
Still other categories abound. Dates that are the product of three consecutive prime numbers (PDF), such as July 26, 2011, are an example; the date, when written as 7262011, equals 191 x 193 x 197. One date that is a simple sequence of consecutive numbers–1/23/45–will pop up every century. And my personal favorite, pi date (3/14/15), is only about two years away!
What other mathematical patterns in dates tickle your fancy?