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A diagram showing partitions for the integers 1 through 8 (via wikimedia commons)

You’re familiar with partition numbers, even if you don’t recognize the term; even kindergartners know them. The partition of a number is all the ways that you can use integers to add up to that number. Start with 2. There is only one way to get there: 1 + 1. The number 3 has 2 partitions: 2 + 1 and 1 + 1 + 1. Four has 5 partitions: 3 + 1, 2 + 2, 2 + 1 + 1 and 1 + 1 + 1 + 1. And so forth. But partition numbers get unwieldy pretty quickly. By the time you get to 100, there are more than 190,000,000 partitions. We’re well beyond elementary school math.

Mathematicians have been searching for the past couple of centuries for an easy way to calculate partition values. In the 18th century, Leonhard Euler developed a method that worked for the first 200 partition numbers. Solutions proposed in the early 20th century for larger partition numbers proved to be inexact or impossible to use. And the search continued.

The most recent mathematician to tackle the problem was Ken Ono at Emory University, who had a eureka moment while on a walk through the north Georgia woods with his post-doc Zach Kent. “We were standing on some huge rocks, where we could see out over this valley and hear the falls, when we realized partition numbers are fractal,” Ono says. “We both just started laughing.”

Fractals are a kind of geometric shape that looks incredibly complex but is actually composed of repeating patterns. Fractals are common in nature—snowflakes, broccoli, blood vessels—and as a mathematical concept they’ve been hauled into use for everything from seismology to music.

Ono and his team realized that these repeating patterns can also be found in partition numbers. “The sequences are all eventually periodic, and they repeat themselves over and over at precise intervals,” Ono says. That realization led them to an equation (all math leads to equations, it sometimes seems) that lets them calculate the number of partitions for any number.

The results of their studies will soon be published; a more detailed analysis is available at The Language of Bad Physics.

The red and aquamarine highlights, and its giant size, set this crayfish apart (Photo courtesy of Carl Williams)

Crayfish, crawfish, crawdads. Call them what you will (tasty?), there are some 600 species found all over the world, and half of those in the United States and Canada. But if you’re looking for the real hotspot of crayfish diversity, head to Tennessee or Alabama. That said, scientists weren’t expecting to find a new species in Shoal Creek in Tennessee; aquatic biologists had been studying life in that little waterway for decades.

The tale starts in 2009, when Eastern Kentucky University biologist Guenter Schuster received some photos of a large crayfish found in Shoal Creek and shared them with Chris Taylor, an aquatic biologist at the University of Illinois. The crayfish had bearded antennae covered in bristly setae that enhance their sensory capabilities, and it looked a lot like Barbicambarus cornutus, a species that lives about 130 miles away from the creek in Kentucky and can grow as big as a lobster. Schuster and Taylor thought that perhaps a fisherman had brought the crayfish to Tennessee in a bait bucket. But when a colleague in Tennessee told them he’d found another giant crayfish in the creek, they had to check it out for themselves.

After a couple hours of wading through the water and upturning boulders, they struck the crayfish jackpot. Beneath a big, flat boulder under a bridge they found a male twice the size of any other crayfish they’d seen that day. And under an ever bigger rock they spotted a female. DNA analysis showed that these large Shoal Creek crayfish were their own distinct species, now named Barbicambarus simmonsi; a description of the new species appears in the Proceedings of the Biological Society of Washington.

The scientists aren’t sure why no one noticed the large crustacean before. “If you were an aquatic biologist and you had seen this thing, because of the size and the setae on the antennae, you would have recognized it as something really, really different and you would have saved it,” says Schuster. However, it appears that these crayfish are not common (only 5 have ever been caught) and their preference for living under large rocks in deep water may have made them easy to overlook, especially in times of high water.

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Should Wall Street be taking lessons in ecology? (courtesy of flickr user Craig S)

Can anyone explain the recent financial crisis? I’ve been listening to Planet Money and This American Life try to do so over the last couple of years, but it’s only driven home how complex everything is. Even simple questions like “what is money?” and “how much is there?” aren’t easy to answer. But metaphors are good. And the idea that ecosystems might be an appropriate analogue, as proposed in a Perspective in this week’s Nature, is intriguing.

Andrew Haldane, of the Bank of England, and Robert May, a biologist at Oxford University, draw analogies with food webs and infectious diseases in their attempt to describe the banking industry and find ways to better prepare it to prevent and weather future downturns. The biggest obstacle to their attempt is that we seem to have created a financial system far more complex than any natural ecosystem. But if you look at banks as nodes in a network, it’s easy to see the parallels with ecological concepts like food webs and epidemiological networks modeling disease spread. And perhaps, as in ecosystems, stability does not rise as the network gets beyond a certain size; at that point, problems that arise spread throughout the system, possible causing collapse.

There are lessons to be had from the world of ecology, say Haldane and May. We could be promoting and managing ecosystem resilience better by requiring banks to have a larger proportion of liquid assets on hand in case of some sort of shock to the system. Taking a lesson from epidemiology, we could focus on limiting the number of “super-spreaders” within the network; but instead of quarantining infected individuals we would somehow limit the number of “super-spreader institutions,” those banks more familiarly labeled as “too big to fail.”

Of course, the banking system is not an ecosystem, as a News and Views article that accompanies the piece cautions. But if the models the financial system has been using are part of what got us into this mess, perhaps they might be advised to look elsewhere for some help.

An 1849 photograph of Edgar Allan Poe (via wikimedia commons)

I’ve read my share of short stories by Edgar Allan Poe, but I was nonetheless intrigued by a caption in an article in the latest Smithsonian special issue, Mysteries of the Universe. It read: “The hollow Earth theory inspired authors from Edgar Rice Burroughs to Edgar Allan Poe.” I knew that Poe, like many writers, drew from the world around him. But it wasn’t until I started reading up on Poe’s scientific interests that I realized how far they went.

The Hollow Earth theory envisions the planet as something like a huge chocolate truffle with us living on its exterior surface. Inside, the theory states, there are continents and oceans floating on the interior of the outer shell surrounding a gooey, heavenly center. The idea was promulgated by Captain John Cleves Symmes, who toured the country in the 1820s, talking up his fantastical idea and trying to scrounge up funding for a trip to one of the poles where, he maintained, there were holes that would allow access to the center.

Poe used this theory in his sole novel, The Narrative of Arthur Gordon Pym of Nantucket, published in 1938, as well as the short stories “MS. Found in a Bottle” and “A Descent into the Maelstrom.” Each involves a sea journey, though none of the adventurers ever reaches that place where they could enter the center of the Earth.

But Poe’s work went beyond this early science fiction and into the world of science itself. He published a textbook on shell collecting, for example, during a time when these pretty beach finds were intriguing both scientists and obsessive collectors. But his biggest contribution is the prose-poem “Eureka,” published shortly before his death. “I design to speak of the Physical, Metaphysical and Mathematical — of the Material and Spiritual Universe:- of its Essence, its Origin, its Creation, its Present Condition and its Destiny,” he wrote before then pondering such things as Olbers’ Paradox, which argues that the night sky should be so full of stars as to appear as bright as day. It can be hard to read but is truly fascinating.

“No thinking being lives who, at some luminous point of his life of thought, has not felt himself lost amid the surges of futile efforts at understanding, or believing, that anything exists greater than his own soul,” Poe writes in “Eureka.” He was more than a bit of philosopher as well, it seems.

PS — Happy 202nd birthday, Mr. Poe!

The Romans build the Pont du Gard aqueduct during a time of climate stability (via wikimedia commons)

Even in our modern age, humans are incredibly vulnerable to changes in weather and climate. And earlier in human history, we were even more so. Even the Romans, who managed to build monuments, roads and aqueducts that still stand today, weren’t immune, according to a new study published last week by Science.

Scientists in Germany and Switzerland created a 2,500-year-long record of Central European summer precipitation and temperature variability from nearly 9,000 samples of larch, pine and oak tree rings. They found that the region experienced above average precipitation and little temperature fluctuation up until about A.D. 250, with a couple of colder periods around 350 B.C.—when the Celtic peoples began to expand across the continent—and 50 B.C., which was when the Romans were conquering Britain.

But around A.D. 250 began a 300-year period of extreme climate variability, when there were wild shifts in precipitation and temperature from one decade to the next. The Romans didn’t fare so well. The Roman Empire nearly fell during the Crisis of the Third Century and split into two in 285. In 387, the Gauls sacked Rome, followed by the Visigoths in 410 and the Vandals in 455. By 500, the western Roman Empire was gone.

“Relatively modest changes in European climate in the past have had profound implications for society,” Penn State University climate scientist Michael Mann told New Scientist.

Human history shows that we don’t deal well with times of climate upheaval. If things are good or bad, we can adapt if given enough time. But a small change in climate can have deadly consequences. The study also found that the period around 1300 saw wetter summers and colder temperatures; it was about that time that Europe experienced a famine and plague of such immense size that nearly half the population died.

“The provocative outcome,” of the study, University of Arkansas geoscientist David Stahle told ScienceNOW, “is that harsh climate conditions happen to be associated with upheavals in society, like the Black Death.”