Smithsonian.com

Image by Flickr user quinn.anya

Nature, for all of its free-wheeling weeds and lightning strikes, is also full of biological regularity: the rows of an alligator’s teeth, the stripes on a zebrafish, the spacing of a chicken’s feathers. How do these patterns arise?

Sixty years ago, with nothing but numbers, logic and some basic biological know-how, mathematician Alan Turing (best known for his pioneering work on artificial intelligence) came up with an explanation. He proposed that two chemicals—an “activator” and an “inhibitor”—work together, something like a pencil and eraser. The activator’s expression would do something—say, make a stripe—and the inhibitor would shut off the activator. This repeats, and voilà, stripe after stripe after stripe.

On Sunday, researchers reported the first experimental evidence that Turing’s theory is correct, by studying the eight evenly spaced ridges that form on the roof of a mouse’s mouth. (People, by the way, have four such ridges on each side, which help us feel and taste food.)

The scientists discovered that in mouse embryos, a molecule called FGF, or fibroblast growth factor, acts as a ridge activator, and SHH, or sonic hedgehog, acts as an inhibitor. When the researchers turned off FGF, the mice formed faint traces of the ridges that are normally made. Conversely, when they turned off SHH, the ridges morphed into one big mound. Changing the expression of one of these partners influenced the behavior of the other—just as Turing’s equations predicted.

Tragically, Turing would never know the importance of his contributions to developmental biology. The British government convicted him of homosexual acts in 1952 (for which it recently apologized), and punished him with chemical castration. Turing took his own life in 1954. This June is the 100th anniversary of his birth.

Studies show most football coaches make poor decisions on fourth down. Does Bill Belichick have a secret advantage? Photo by flickr user Keith Allison

Read our other posts about the history of football, what to bring to your Super Bowl party, the innovations of television advertising and much more.

This Super Bowl Sunday, as you watch grizzled coaches pace the sideline and bark at players, feel free to play armchair quarterback—or even head coach. Despite the hours they spend scouting players, analyzing game tape and drawing up complex tactical schemes, a pair of recent scientific studies indicates that many football coaches are no better at making some in-game decisions than you or I would be.

A 2006 paper by David Romer (pdf), a University of California at Berkeley economist, started things off by looking at a choice frequently encountered by coaches on fourth down: kick a field goal or try for a touchdown? Using data from more than 700 NFL games, Romer calculated the average chance of winning generated by each choice at different positions on the field. He then compared the data to the actual choices made by NFL coaches.

The conclusion: most avoid risk to an irrational extent, often opting to kick a field goal when going for a touchdown would provide a better chance of winning. Why would coaches—with their salaries and job security determined by on-field success—depart from the best possible choice? Romer speculates:

Perhaps the decision makers are systematically imperfect maximizers. Many skills are more important to running a football team than a command of mathematical and statistical tools…thus the decision makers may want to maximize their teams’ chances of winning, but rely on experience and intuition rather than formal analysis.

Another possible interpretation: for job security, coaches may prefer closer losses, coming after seemingly safe decision-making, to blowouts. A 23-0 loss may get a coach fired faster than a 23-6 score, which gives coaches incentive to kicking meaningless field goals rather than going for touchdowns.

Soon after the Romer study, Indiana University scientist Chuck Bower and partners from the business world went one step further. Using a similar dataset of actual NFL games, they built ZEUS: a powerful computer program that can analyze in-game situations on the fly and provide high-volume data analysis to coaches in real time. Bower said:

ZEUS is a valuable addition to a coaching staff’s tools, and one that can provide that elusive edge over the competition. The ZEUS engine is powerful enough to simulate the equivalent of every game played in the history of the NFL in less than a second. ZEUS can objectively assess crucial play-calling decisions with startling accuracy.

Comparing live data from the game with the historical track record of the NFL, ZEUS can indicate the choice that leads to a better chance of winning for a number of situations: not just what to do on fourth down, but whether to accept or decline penalties, attempt onside kicks, or try for two-point conversions.

In designing ZEUS, Bowers’ team drew upon many of the principles used in building computer models for other games—such as backgammon or chess—and applied them to football. “While the physical nature of the game is very different, the situational nature is strikingly similar. A football coach is constantly making decisions with respect to multiple variables: score, field position, down, yards to a first down, etc.,” said Bowers, an expert backgammon player.

NFL head coaches are a notoriously secretive bunch when it comes to strategy, so if anyone is currently using ZEUS, we’d likely not hear about it. But ZEUS’ own analysis indicates that one coach in particular might be using the cutting-edge program: New England Patriots coach Bill Belichick, set to coach in his 5th Super Bowl on Sunday.

The evidence? Belichick is famous for his unconventional decision-making, often opting to go for an aggressive play on fourth down when most coaches would punt or kick a field goal. The New York Times “Fifth Down” blog has used ZEUS to evaluate real-world decisions on a number of occasions. And when ZEUS was used to analyze a particularly controversial fourth down call made by Belichick—at the end of a crucial 2010 game against the Indianapolis Colts, he opted to go for it on his own 28-yard line, an unusually aggressive choice—ZEUS surprised many by saying he had, statistically, made the right call. The analysis indicated that, overall, it gave him team the best chance of winning.

Of course, statistical projections are not guarantees. In that case, the decision didn’t work out, and the Patriots lost the game. But if Belichick does have ZEUS on his sideline, it might give him that much better odds of being the winning coach on Sunday.

The first Death Star from Star Wars (via Wookieepedia)

Obi-Wan: That’s no moon. It’s a space station.

That space station was the Empire’s first Death Star in Star Wars: A New Hope. Obi-Wan and company had just bounced through a debris field, the remnants of the planet Alderaan. Such an act of destruction would seem impossible to us–it seemed so to many of the movie’s characters until it happened. But perhaps not, say three students at the University of Leicester in England who last year published a study on the subject in their university’s undergraduate physics and astronomy journal.

The study’s authors start off by making some simple assumptions: The planet being fired upon doesn’t have some sort of protection, like a shield generator. And it’s about the size of Earth but solid through and through (Earth isn’t solid, but the planet’s layers would have significantly complicated the math here). They then calculate the planet’s gravitational binding energy, which is the amount of energy required to pull apart an object. Using the mass and radius of the planet, they calculate that destruction of the object would require 2.25 x 10³² joules. (One joule is equal to the amount of energy required to lift an apple one meter. 10³² joules is a lot of apples.)

The energy output of the Death Star isn’t given directly in the movie, but the space station was said to have had a “hypermatter” reactor that had the energy output of several main-sequence stars. For an example of a main-sequence star, the authors look to the Sun, which puts out 3 x 10²⁶ joules per second, and they conclude that the Death Star could “easily afford to output [the energy required for an Earth-like planet's destruction] due to to its tremendous power source.”

It would be a different story, though, if the planet scheduled for destruction had been more like Jupiter than Earth. The gravitational binding energy of Jupiter is 1,000 times that of the Earth-like planet in the study. “To destroy a planet like Jupiter [the space station] would probably have to divert all remaining power from all essential systems and life support, which is not necessarily possible.”

Of course, that assumes that the Emperor wouldn’t be willing to sacrifice a space station full of people to wipe out his enemies. And considering that he was just fine with wiping out whole planets, I’m not sure I’d take that bet.

Sofia Kovalevskaya, Emmy Noether and Ada Lovelace are just three of the many famous female mathematicians you should know. Images courtesy of Wikicommons

If you haven’t yet read my story “Ten Historic Female Scientists You Should Know,” please check it out. It’s not a complete list, I know, but that’s what happens when you can pick only ten women to highlight—you start making arbitrary decisions (no living scientists, no mathematicians) and interesting stories get left out. To make up a bit for that, and in honor of Ada Lovelace Day, here are five more brilliant and dedicated women I left off the list:

Hypatia (ca. 350 or 370 – 415 or 416)

No one can know who was the first female mathematician, but Hypatia was certainly one of the earliest. She was the daughter of Theon, the last known member of the famed library of Alexandria, and followed his footsteps in the study of math and astronomy. She collaborated with her father on commentaries of classical mathematical works, translating them and incorporating explanatory notes, as well as creating commentaries of her own and teaching a succession of students from her home. Hypatia was also a philosopher, a follower of Neoplatonism, a belief system in which everything emanates from the One, and crowds listened to her public lectures about Plato and Aristotle. Her popularity was her downfall, however. She became a convenient scapegoat in a political battle between her friend Orestes, the governor of Alexandria, and the city’s archbishop, Cyril, and was killed by a mob of Christian zealots.

Sophie Germain (1776 – 1831)

When Paris exploded with revolution, young Sophie Germain retreated to her father’s study and began reading. After learning about the death of Archimedes, she began a lifelong study of mathematics and geometry, even teaching herself Latin and Greek so that she could read classic works. Unable to study at the École Polytechnique because she was female, Germain obtained lecture notes and submitted papers to Joseph Lagrange, a faculty member, under a false name. When he learned she was a woman, he became a mentor and Germain soon began corresponding with other prominent mathematicians at the time. Her work was hampered by her lack of formal training and access to resources that male mathematicians had at the time. But she became the first woman to win a prize from the French Academy of Sciences, for work on a theory of elasticity, and her proof of Fermat’s Last Theorem, though unsuccessful, was used as a foundation for work on the subject well into the twentieth century.

Ada Lovelace (1815 – 1852)

Augusta Ada Byron (later Countess of Lovelace) never knew her father, the poet Lord Byron, who left England due to a scandal shortly after her birth. Her overprotective mother, wanting to daughter to grown up as unemotional—and unlike her father—as possible, encouraged her study of science and mathematics. As an adult, Lovelace began to correspond with the inventor and mathematician Charles Babbage, who asked her to translate an Italian mathematician’s memoir analyzing his Analytical Engine (a machine that would perform simple mathematical calculations and be programmed with punchcards and is considered one of the first computers). Lovelace went beyond completing a simple translation, however, and wrote her own set of notes about the machine and even included a method for calculating a sequence of Bernoulli numbers; this is now acknowledged as the world’s first computer program.

Sofia Kovalevskaya (1850 – 1891)

Because Russian women could not attend university, Sofia Vasilyevna contracted a marriage with a young paleontologist, Vladimir Kovalevsky, and they moved to Germany. There she could not attend university lectures, but she was tutored privately and eventually received a doctorate after writing treatises on partial differential equations, Abelian integrals and Saturn’s rings. Following her husband’s death, Kovalevskaya was appointed lecturer in mathematics at the University of Stockholm and later became the first woman in that region of Europe to receive a full professorship. She continued to make great strides in mathematics, winning the Prix Bordin from the French Academy of Sciences in 1888 for an essay on the rotation of a solid body as well as a prize from the Swedish Academy of Sciences the next year.

Emmy Noether (1882 – 1935)

In 1935, Albert Einstein wrote a letter to the New York Times, lauding the recently deceased Emmy Noether as “the most significant creative mathematical genius thus far produced since the higher education of women began.” Noether had overcome many hurdles before she could collaborate with the famed physicist. She grew up in Germany and had her mathematics education delayed because of rules against women matriculating at universities. After she received her PhD, for a dissertation on a branch of abstract algebra, she was unable to obtain a university position for many years, eventually receiving the title of “unofficial associate professor” at the University of Göttingen, only to lose that in 1933 because she was Jewish. And so she moved to America and became a lecturer and researcher at Bryn Mawr College and the Institute for Advanced Study in Princeton, New Jersey. There she developed many of the mathematical foundations for Einstein’s general theory of relativity and made significant advances in the field of algebra.

A chocolate peanut butter pie for Pi Day (photo, and pie, by Sarah Zielinski)

Today is March 14, or 3.14, the day we celebrate the mathematical constant pi (π). Pi, the ratio of a circle’s circumference to its diameter, is an irrational number, meaning that it can’t be expressed as a simple fraction of two integers. It is also a transcendental number, which means it is not algebraic. The celebrated 3.14 is just the beginning of pi—it continues into infinity, and that may be one of the reasons people find it so fascinating. So in honor of Pi Day, here are some suggestions for how to celebrate:

1 ) Read about the long history of pi.

2 ) Memorize as many digits of pi as you can (here’s pi to a million digits). A Japanese man in 2005 memorized pi to 83,431 digits.

3 ) See how far you can calculate pi. Computer programmer Fabrice Bellard calculated pi to 2.7 trillion digits using his home computer.

4 ) Watch the movie Pi, a 1998 thriller about a paranoid mathematician.

5 ) Make a pi-themed pie (I went for chocolate peanut butter pie, but any flavor is appropriate).

6 ) Celebrate with music: Learn the song, “Pi, Pi, Mathematical Pi,” set to the tune of “American Pie”; listen to the Pi Rap; or sing Pi Day carols.