MATH NEWS

How does metabolism scale with animal body size?
Peter Dodds' new paper in Physical Review Letters tosses out the
long-held belief in a 3/4 exponent in favor of the common sense 2/3.

Apparently, the mysterious "3/4 law of metabolism" -- proposed by Max Kleiber in 1932, printed in biology textbooks for decades, explained theoretically in Science in 1997 and described in a 2000 essay in Nature as "extended to all life forms" from bacteria to whales -- is just plain wrong.
"Actually, it's two-thirds," says University of Vermont mathematician Peter Dodds. His paper in the January 29 edition of Physical Review Letters helps overturn almost eighty years of near-mystical belief in a 3/4 exponent used to describe the relationship between the size of animals and their resting metabolism.

Two-thirds or three-quarters?

To understand the debate between 2/3 and 3/4, assume a spherical cow. "That's what a physicist would do," Dodds says, laughing. Basic geometry shows that the surface area of this difficult-to-milk creature would increase as the square of its radius while the volume would increase as the cube of the radius. In other words, the exponent that describes the ratio of surface area to volume is 2/3.
Next, assume a spherical mouse. OK, now compare the resting metabolic rates of these sorry animals. Since the point of resting metabolism is to keep a warm-blooded animal warm (and alive!) with the lowest necessary energy use, both geometry and common sense suggest that the cow would have a lower rate of metabolism per cell than the mouse: the mouse, with more surface area relative to its volume, would lose heat faster than our cartoon cow.
And what about in real animals? In 1883, a German physiologist named Max Rubner measured the heat output of some dogs ranging from a few pounds to nearly seventy. He plotted these numbers to show that the dog's metabolic rates were proportional to their mass with an exponent of 2/3 -- just like the geometry of an imaginary spherical beast would suggest.
But, in 1932, Swiss agricultural chemist Max Kleiber presented a paper with a now-famous graph. It plotted, on a logarithmic scale, the body weight of 13 mammals, ranging from rats to cows, against their resting metabolism. Strangely, the line traced through the data points did not conform to Rubner's observation nor common sense. Instead, it hewed to a line with a somewhat steeper slope of about .73. To make it easier for slide rule use, he rounded the exponent to a neat .75. Kleiber's 3/4-power law was born.
"Kleiber's original data is a mess, a complete mess," says Dodds, "but it became something everyone believed in. The idea of quarter-powers begins to take on this spooky, magical quality. Nobody can explain it, but it's a secret law of the universe. It's quarterology!"
Over the next decades, hundreds of animals' resting metabolisms were measured or estimated, from microbes to whales. The results in various groups of animals ranged from slopes of less than 2/3 to greater than 1. But as Vaclav Smil wrote in a sweeping "millennium essay" for Nature they were "close enough to the 0.75 line," and concluded that "the 3/4 slope is representative for all" animals.
"Some data seems to fit this 3/4 line -- if you're looking for it!" says Dodds. "It was pre-supposed to be true -- and became a universal overarching law that somebody needs to explain."
Instead of explaining it, in the 1960s a Scottish conference on energy metabolism simply voted, 29-0, to enshrine 3/4 as the official exponent. Then, in 1997, an elegant, though controversial, paper by Geoffrey West and colleagues was published in Science that claimed to derive 3/4 from first principles, drawing on ideas about fractals in networks and the growing length of tubes.
"The problem is their paper fell to pieces mathematically. It just didn't work. Unfortunately, I showed that and published a paper with my adviser and a fellow student in 2001," Dodds says. They also reanalyzed data from Kleiber and six other scientists and concluded that there is little empirical evidence for rejecting 2/3 in favor of 3/4. "But we didn't have a better theory," Dodds says, "or some way to clean it all up."

Network matters

Until now. Dodds's new paper explores the geometry of branching networks -- like blood supply -- to show how a material, like blood, can be most efficiently delivered. "If you're going to build organisms with a central source, like a heart, that places physical constraints that evolution has to run up against," he says. "These constraints won't let the ratio be too far away from 2/3."
"My new paper follows the argument that was put forth in 1997 -- that, somehow, networks give rise to the 3/4 law. They were right that supply networks are key to understanding the metabolic limitations of animals. Except my paper shows that networks give rise to the 2/3 law, actually," Dodds says, "If you do the math properly."
"What's good here is that the network supplying to the inside of this system matches with the 2/3 exponent you'd expect from surface area," he says. And recent statistical analyses continue to show that the 2/3 exponent fits well empirically with large data sets for both mammals and birds. It seems that Rubner got it right in the 1880's after all.
Of course, it may be that biologists -- who delight in detail, local mechanisms and exceptions -- will win out over statistical physicists -- who look for evidence of universal patterns in nature: there may be no single exponent to describe the scaling of metabolism. A line drawn through a confounding scatter of data about specific animals across orders of magnitude may be just a line, not a law. But a confluence of facts -- greater understanding about how a network best minimizes volume, as evolution would favor in the costly production of blood supply; surface area geometry; and re-analysis of Kleiber's and other data -- seem to be pummeling the once-beguiling 3/4 law into dust.
"Especially for smaller guys," Dodds says, "like birds, it's just absolutely, stone-cold 2/3."

Spherical Cows Help to Dump Metabolism Law

When Derren Brown ‘explained’ how he predicted the lottery, I suddenly experienced something psychologists must be familiar with. An impressive piece of television magic was followed by an expose episode full of pseudo-mathematical smoke-screen. Psychologists breathed a sigh of relief as this time Derren wasn’t using their subject as a decoy explanation for his impressive conjuring skills, while, as a mathematician, I was ready to throw a calculator at the TV screen. What has impressed me though is that now when I run mathematics lessons in various secondary schools, the students will suddenly get very excited when Derren Brown is mentioned. Discussions on how he actually did the illusion aside (such as split-screens and - my favourite - long-distance laser etching), the students want to talk about the mathematical red-herrings he threw out.
For the pedantic record: both the Wisdom of the Crowd and the Heads-Tails game are well understood bits of mathematics. Wisdom of the Crowd is a lovely example of how even though people make mistakes when trying to estimate a value, the errors follow a delightfully predictable distribution. If you average across everyone’s guess, the under-estimates compensate for the over-estimates and you get a fairly accurate answer; but only when people make consistent mistakes. Which is not the case when predicting a random future event, such as the lottery numbers. The Heads-Tails game is a probability trick to give yourself a seemingly-impossible advantage when predicting a coin flip. I’ve used it to win many a free drink, and if you want to learn how, the details are here.
So now I’ve ended up being extremely grateful to Derren Brown for continuing the age-old overlap between magic and mathematics. For generations, mathematicians have dabbled in magic and visa-versa (the magician S. Brent Morris completed a mathematics PhD based on techniques develop for his magic tricks). My colleague, and professor of computer science, Peter McOwan is also an amateur magician and uses magic in his lectures. He says “It’s no surprise that some mathematicians and computer scientists are also keen amateur magicians. Day-to-day we work on the maths that underpins our modern society, but from time-to-time it’s just good fun to use maths for entertainment too.”
This is also perfect for modern mathematics teaching. To teach maths in a secondary school, you need to engage the students in the subject and nothing does this better than showing them how the maths can be used to perform a magic trick. If a student knows he can use prime numbers to amaze his friends with a card trick, they will suddenly become much more engaged in the lesson. This is why I have worked with Professor McOwan to produce a Manual of Mathematical Magic. All copies of the book, each with a magic kit, are available free to schools. Any school nationwide can request a copy and this week we have posted a copy to every school in London. We hope that teachers will be able to use it in their class, or at the very least give it to an enthusiastic student. If you work in a secondary school, or send your child to one, please locate the copy we sent to the mathematics department and make sure it is being used.
Before we posted the Manual of Mathematical Magic to all the London schools, I ran an estimation-based competition to win one of five advance copies. Not entirely unexpectedly, one entry simply read “the average of all other guesses” and sure enough, they were one of the winners. If more maths teachers use magic to engage their students, we’ll end up with a more mathematically literate society and of course, more people winning free drinks in the pub.

Schools can get more information about the Manual of Mathematical Magic kit and order a free copy from: www.mathematicalmagic.com. Production and distribution of the kit is funded by the Higher Education Funding Council for England.

Matt Parker is based in the School of Mathematical Sciences at Queen Mary, University of London. He also gives talks about Mathematics to schools and wider audiences across the UK.

Posted by Hannah Devlin on February 4, 2010 in Mathematics , Young Scientists | Permalink | Post to Twitter

The mathematics of magic

- The Van Horn Eagle math, number sense and calculator team traveled to Canutillo last weekend to take part in the annual Canutillo Math Meet. The meet featured 22 schools and 20 of them were 4A or 5A schools.
The Eagle team, coached by B.O. Buchhorn took 2nd Place overall at the meet. Not bad for a 1A school!
Eagle scores from the meet are as follows:
Math: 9th Grade: Imisha Bhakta, 8th; and Savannah Corralez, 9th.
Calculator: 9th Grade - Imisha Bhakta, 5th; 10th Grade - Jackie Grado, 2nd; 12th Grade - Julio Baeza, 3rd; Rutesh Bhakta, 6th; and Rebecca Mendias, 8th. The Eagle calculator team took 2nd as well based on their 4 best scores. The four best scores were turned in by Julio Baeza, Rutesh Bhakta; Jackie Grado and Rebecca Mendias.
Taking 1st Place at the meet were the following schools: Number Sense - Canutillo; Math - Coronado; and Calculator - Jefferson/Silva.

Congratulations!

Van Horn team takes 2nd

Zipporah “Fagi” Levinson, the wife of the late Institute Professor Norman Levinson ’34, SM ’34, SCD ’35, died on Dec. 11, 2009, at the age of 93 after numerous strokes and pneumonia.
In the book Recountings: Conversations with MIT Mathematicians, editor Joel Segel described Levinson as “the unquestioned ‘den mother’” for decades to the likes of Norbert Weiner and John Nash,” and “a vibrant and irreplaceable part of the Mathematics Department’s institutional memory and perhaps the person best qualified to put a human face on the period when the department was coming into its own as a full-fledged research entity.”
Levinson was particularly effective as the moral and social historian of the department, and was frequently called upon to remind them of the Institute’s history, including its anti-Semitism and sexism, and how it had overcome them.
Levinson was born in 1916 in what she described as a tough Jewish ghetto in Brooklyn, the daughter of a dentist and dental technician. She was educated at City College of New York and was awarded her masters in education by Columbia. In 1938 she met Norman Levinson, who was then a young instructor in the MIT Department of Mathematics, and he proposed to her two days later. They married after knowing each other for only a week, and the department acknowledged the wedding by presenting her with a copy of Calculus Made Easy, signed by all members of the faculty.
Levinson often opened up her house to other mathematicians, providing a congenial and supportive environment, which was greatly appreciated by the rapidly expanding department. After her husband became head of the department and an Institute Professor, Fagi continued to act as the department’s social glue. After his death in 1975, she remained an active participant in the social life of several generations of the department, ultimately becoming the department’s oral historian.
She spent the last nine years of her life at Lasell Village in Newton. Until shortly before her death she stayed in touch with numerous friends from the mathematical world. She is survived by two daughters, Sylvia and Joan (Zorza) ’62, four grandchildren and five great-grandchildren.
Donations may be made to MIT noting they are in memory of Fagi Levinson (and/or Norman Levinson) and sent to Bonny Kellermann, MIT director of memorial gifts, 600 Memorial Drive, Room W98-516, Cambridge, MA 02139. Donations will fund opportunities for undergraduate research in applied mathematics.

Zipporah Levinson, Department of Mathematics’ ‘den mother,’ dies at age 93

A major hurdle in the ambitious quest to design and construct a radically new kind of quantum computer has been finding a way to manipulate the single electrons that very likely will constitute the new machines' processing components or "qubits."
Princeton University's Jason Petta has discovered how to do just that -- demonstrating a method that alters the properties of a lone electron without disturbing the trillions of electrons in its immediate surroundings. The feat is essential to the development of future varieties of superfast computers with near-limitless capacities for data.
Petta, an assistant professor of physics, has fashioned a new method of trapping one or two electrons in microscopic corrals created by applying voltages to minuscule electrodes. Writing in the Feb. 5 edition of Science, he describes how electrons trapped in these corrals form "spin qubits," quantum versions of classic computer information units known as bits. Other authors on the paper include Art Gossard and Hong Lu at the University of California-Santa Barbara.
Previous experiments used a technique in which electrons in a sample were exposed to microwave radiation. However, because it affected all the electrons uniformly, the technique could not be used to manipulate single electrons in spin qubits. It also was slow. Petta's method not only achieves control of single electrons, but it does so extremely rapidly -- in one-billionth of a second.
"If you can take a small enough object like a single electron and isolate it well enough from external perturbations, then it will behave quantum mechanically for a long period of time," said Petta. "All we want is for the electron to just sit there and do what we tell it to do. But the outside world is sort of poking at it, and that process of the outside world poking at it causes it to lose its quantum mechanical nature."
When the electrons in Petta's experiment are in what he calls their quantum state, they are "coherent," following rules that are radically different from the world seen by the naked eye. Living for fractions of a second in the realm of quantum physics before they are rattled by external forces, the electrons obey a unique set of physical laws that govern the behavior of ultra-small objects.
Scientists like Petta are working in a field known as quantum control where they are learning how to manipulate materials under the influence of quantum mechanics so they can exploit those properties to power advanced technologies like quantum computing. Quantum computers will be designed to take advantage of these characteristics to enrich their capacities in many ways.
In addition to electrical charge, electrons possess rotational properties. In the quantum world, objects can turn in ways that are at odds with common experience. The Austrian theoretical physicist Wolfgang Pauli, who won the Nobel Prize in Physics in 1945, proposed that an electron in a quantum state can assume one of two states -- "spin-up" or "spin-down." It can be imagined as behaving like a tiny bar magnet with spin-up corresponding to the north pole pointing up and spin-down corresponding to the north pole pointing down.
An electron in a quantum state can simultaneously be partially in the spin-up state and partially in the spin-down state or anywhere in between, a quantum mechanical property called "superposition of states." A qubit based on the spin of an electron could have nearly limitless potential because it can be neither strictly on nor strictly off.
New designs could take advantage of a rich set of possibilities offered by harnessing this property to enhance computing power. In the past decade, theorists and mathematicians have designed algorithms that exploit this mysterious superposition to perform intricate calculations at speeds unmatched by supercomputers today.
Petta's work is using electron spin to advantage.
"In the quest to build a quantum computer with electron spin qubits, nuclear spins are typically a nuisance," said Guido Burkard, a theoretical physicist at the University of Konstanz in Germany. "Petta and coworkers demonstrate a new method that utilizes the nuclear spins for performing fast quantum operations. For solid-state quantum computing, their result is a big step forward."
Petta's spin qubits, which he envisions as the core of future quantum logic elements, are cooled to temperatures near absolute zero and trapped in two tiny corrals known as quantum wells on the surface of a high-purity, gallium arsenide chip. The depth of each well is controlled by varying the voltage on tiny electrodes or gates. Like a juggler tossing two balls between his hands, Petta can move the electrons from one well to the other by selectively toggling the gate voltages. Prior to this experiment, it was not clear how experimenters could manipulate the spin of one electron without disturbing the spin of another in a closely packed space, according to Phuan Ong, the Eugene Higgins Professor of Physics at Princeton and director of the Princeton Center for Complex Materials.
Other experts agree.
"They have managed to create a very exotic transient condition, in which the spin state of a pair of electrons is in that moment entangled with an almost macroscopic degree of freedom," said David DiVincenzo, a research staff member at the IBM Thomas J. Watson Research Center in Yorktown Heights, N.Y.
Petta's research also is part of the fledgling field of "spintronics" in which scientists are studying how to use an electron's spin to create new types of electronic devices. Most electrical devices today operate on the basis of another key property of the electron -- its charge.
There are many more challenges to face, Petta said.
"Our approach is really to look at the building blocks of the system, to think deeply about what the limitations are and what we can do to overcome them," Petta said. "But we are still at the level of just manipulating one or two quantum bits, and you really need hundreds to do something useful."
As excited as he is about present progress, long-term applications are still years away. "It's a one-day-at-a-time approach," Petta said.

Research at Princeton was supported by the Sloan Foundation, the Packard Foundation and the National Science Foundation. Work at the University of California-Santa Barbara was supported by the Defense Advanced Research Projects Agency and the NSF.

Princeton scientist makes a leap in quantum computing