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Dana Mackenzie covers the solution to a problem that Martin Gardner posed 50 years ago in Scientific American. (Photo by Dana Mackenzie.)

 

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Ramanujan, Ono in Scientific American

"The Oracle" (subhead: "Mathematician Ken Ono has solved long-standing puzzles using insights hidden in the unpublished papers of Indian prodigy Srinivasa Ramanujan"), by Ariel Bleicher, appears in the Scientific American for May, 2014. Bleicher tells us how six equations buried in a manuscript of Ramanujan's came to Ono's attention in 1989, and changed the direction of his mathematical career. The equations "drew a parallel between modular forms [Ono's specialty] and ... partition numbers." (The $n$th partition number $p(n)$ "counts the combinations of positive integers that sum to a given integer $n$. For example, $~p(4)$ is 5: $ 1 + 1 + 1 + 1,~ 1 + 1 + 2,~ 2 + 2,~ 1 + 3$ and $~4$.")

Ramanujan had published in 1919 "one of the first real breakthroughs" in the study of the number-theoretical properties of $p(n)$. He and G. H. Hardy had commisioned the computation of the first $200$ values by a "calculation wizard named Percy Alexander MacMahon (aka Major MacMahon);" the results came arranged in a 5-column table, starting with $p(0)=1$. Ramanujan noticed that all the entries in the fifth column were divisible by $5$, and that starting with $p(5)$ every seventh entry was divisible by $7$; he proved that these patterns persist forever: $p(4+5k)\equiv 0 ~\mbox{mod}~5$ and $p(5+7k)\equiv 0 ~\mbox{mod}~7$; he also observed that $p(6+11k)\equiv 0 ~\mbox{mod}~11$, but remarked: "It appears that there are no equally simple properties for any moduli involving primes other than these."

1 1 2 3 5
7 11 15 22 30
42 56 77 101 135
176 231 297 385 490
627 792 1002 1255 1575
1958 2436 3010 3718 4565
5604 6842 8349 10143 12310

The first few rows of MacMahon's table of the partition numbers $p(n)$ for $n=0, 1, 2, \dots$, showing Ramanujan's congruences: ${\bf p(4+5k)\equiv 0 ~\mbox{mod}~5}$, $~\color{red}{ p(5+7k)\equiv 0 ~\mbox{mod}~7}$, $~\color{blue}{p(6+11k)\equiv 0 ~\mbox{mod}~11}$. Adapted, with two additional rows, from Scientific American 310 No. 5 (May 2014).

The six equations that were brought to Ono's attention in 1989 led him to understand that there was a very deep connection between the partition function and modular forms: "Ramanujan's six statements linked the two fields in a profound way that no one had anticipated." and in fact, "if he thought of the partition function as a modular form in disguise, he could show that they were true. Another thought immediately followed: ... with a few adjustments, the theories he had developed about modular forms could be powerful tools not just for verifying Ramanujan's genius but also for unearthing deeper secrets about the partition function. ... In this way, Ono was able to prove that partition congruences are not rare at all. Mathematicians had assumed there were few beyond 5, 7 and 11. But in fact, as Ono discovered, there are infinitely many."

After Ono's first round of discoveries, Bleicher tells us, "The problem of how to predict partition congruences lay dormant for five years, until postdoctoral fellow Zachary A. Kent arrived at Emory in the spring of 2010." Together, Ono and Kent, "little by little, ... built in their minds a labyrinthine superstructure into which the partition numbers could be neatly arranged." The congruences "obey a fractal structure --they repeat in near-identical patterns at different scales, like the branches of a snowflake." Working together with Amanda Folsom (Yale) the collaborators "were able to prove that partition congruences appear in a calculable manner. They exist for every prime and every prime power. Beyond 11, though, the patterns get much more complex, which is probably why Ramanujan never worked them out." Bleicher quotes George E. Andrews (Penn State): "It's a dramatic and surprising discovery. I don't think even Ramanujan could have dreamt it."

[This article exemplifies how hard it is to present mathematical research to a large audience. Scientific American avoids equations and when introducing an abstract concept like function feels obliged to explain: "describes a relation between two things: it takes a given input $x$ and spits out the corresponding output $f(x)$." Modular forms, which are mentioned 12 times, are only described as "abstract two-dimensional objects revered for their remarkable symmetry." Their deep connection to the partition function is stated, but cannot begin to be explained. This part of the story is vivid and compelling on the human side, but one could substitute "potato pancakes" for each occurrence of "modular forms" without diminishing the amount of mathematics that has been communicated. The article has links to the research papers; curious readers might also appreciate an expository work like Amanda Folsom's What is a Mock Modular Form? in the December 2010 Notices.-TP]

Sperner's Lemma helps split the rent

"To Divide the Rent, Start With a Triangle," by Albert Sun, appeared in the April 28 2014 New York Times. (The online version has excellent interactive graphics). "Last year, two friends and I moved into a small three-bedroom apartment in Manhattan. ... The bedrooms were different sizes, ranging from small to very small. Two faced north toward the street and had light; the third and smallest faced an alley. The largest had two windows; the midsize room opened onto the fire escape." How to split the rent? "The problem is that individuals evaluate a room differently. I care a lot about natural light, but not everyone does. Is it worth not having a closet? Or one might care more about the shape of the room, or its proximity to the bathroom." Sun discovered a mathematical solution to the problem in a 1999 Monthly article by Francis Edward Su of Harvey Mudd College. Su's algorithm is presented in the following three images and their captions, adapted from the Times article and Su's paper.

rental algorithm-a

a. Each point in the triangle represents three percentages adding up to 100%, telling how the occupants of rooms 1, 2, and 3 will contribute to the total rent. Along the bottom edge, for example, room 1 costs 0%, and the cost of room 2 varies from 0% at the far left to 100% at the right. At the midpoint of the triangle, each room costs $\frac{1}{3}$ of the total.

rental algorithm-b

b. The triangle is partitioned into a network of subtriangles fine enough so that the price for each room varies, say, by less than $1 between adjacent triangles, and in such a way that the vertices in the network can all be labeled with the initials A, B, C of the renters with every subtriangle getting one of each.

rental algorithm-c

c. At each of A's vertices, A chooses a preferred room at that particular price-point; similarly for B and C. Renters are supposed to be rational, so that for example at vertices where room 1 costs nothing, that room will be preferred.

A "trap-door" argument given by Su, guarantees that, no matter how the additional choices are made, somewhere in the network will be a triangle labeled 1, 2, 3. (Su explains how the diagram can be dualized so that the proof also follows from Sperner's Lemma). The combination of room-choices and prices encoded in that triangle should divide the total rent (plus or minus a couple of dollars) in a way agreeable to all three renters.

The online New York Times links to an updated version of a rent division calculator, due to Su and Elisha Peterson, which gives a dynamic implementation of the process.

Spontaneous hemihelices

"Things That Happen When Harvard Researchers Play With Rubber Bands" is the title for a video posted on the Slate website, May 1, 2014. The research in question is presented in a PLOS-one article, "Structural Transition from Helices to Hemihelices," by Jia Liu, Jiangshui Huang, Tianxiang Su, Katia Bertoldi, and David R. Clarke, all of Harvard. The article describes the fabrication of pre-stressed elastomer strips (two lengths are glued together, one relaxed and one stretched) and the behavior of the bi-strips when they are gradually allowed to take their "natural" shape.

strips

This image, downloaded from the PLOS-one article by Liu et al., shows how the pre-stressed strips are fabricated. The red strip, initially shorter and thicker, is stretched to the length of the blue one; then the two are clued together. When the bi-strip is allowed to contract, it curls. doi:10.1371/journal.pone.0093183.g002

The phenomenon the Harvard team discovered is what the title of their paper suggests. "[W]hen the bi-strips have a large aspect ratio, they spontaneously twist along their length to form a regular helix. ... [W]hen the aspect ratio is small, we observe the formation of periodic perversions, separating helical segments of alternating chiralities."

helices

"Illustration of a helix (top), a hemihelix with one perversion marked by an arrow (middle) and a hemihelix with multiple perversions (bottom). The scale bar is 5 cm, and is the same for each image. These different shapes were all produced in the same way as shown [above] with the same value of pre-strain $\chi=1.5$ but with decreasing values of the height-to-width ratio of the bi-strip's cross-section. $L=50cm, w=3mm, h=12,8,2.5mm$)." doi:10.1371/journal.pone.0093183.g001

The authors present a numerical model which replicates their experimental findings; they also attack the problem analytically, and derive equations with solutions which "clearly resemble the hemihelices observed in the experiments."

The authors remark that "it is highly probable that the reason hemihelices with multiple perversions have escaped notice in the past has been that most man-made materials, unlike elastomers, would fracture well before these strains could be achieved."

This research was widely picked up in the press: for example, the second image above was reproduced in the New York Times ("A Rubber Band With a Surprising Twist," April 29, 2014), in La Repubblica ("C'è una nuova forma geometrica: scoperta l'emielica," April 24) and in the Los Angeles Times ("With a little rubber, scientists make weird, twisted hemihelix," April 24).

Tony Phillips
Stony Brook University
tony at math.sunysb.edu

 

Math Digest Math Digest
On Media Coverage of Math

Math Digest includes posts throughout each month by Anna Haensch (2013 AMS Media Fellow) and Ben Polletta (Boston University). These early-career mathematicians provide their unique insights (and occasionally videos, interviews and podcasts) on math-related topics recently covered by the media.

Recently posted:

The All-Time Ultimate Greatest Ever, by Ben Polletta

If you could choose any eleven mathematicians from history to play on your World Cup team, which geometer would be your striker? If you were stuck on a desert island for an indeterminate amount of time, and you could only bring three mathematicians with you, which number theorist would go in your suitcase? If you had to eat only one mathematician for the rest of your life, which probabilist would be on your plate? We all have our own personal list of the top 100 mathematicians of all time (see Greatest Mathematicians of All Time), and we've all debated it with our friends and fought about it with our spouses.

archimedes

Kiersz's list isn't a top 13--he focuses on mathematicians whose research presaged modern innovations in mathematics and its applications--but he makes some thought-provoking choices. There are the few indisputable greats: Newton, Liebniz, Gauss, Euler, and Euclid are probably in everyone's top 10. Most people would include Pythagoras somewhere, if only for the ubiquitous Pythagorean theorem, and his followers' extreme intolerance for irrational numbers. Descartes for sure, because where would math be without the Cartesian plane? But what about Archimedes? Did you know that he used limit-like arguments to estimate the value of pi and the area under a parabolic curve almost two millenia before calculus was invented? Not to mention he was probably the first, and certainly the most famous, person to celebrate a breakthrough (his understanding of how solids displace liquids) by running through the streets naked shouting 'Eureka!' And it's a tragedy of cultural imperialism that Muhammed ibn Musa al-Khwarizmi, the namesake of the algorithm and father of algebra, mostly goes forgotten. John Napier, the little-known father of the controversial logarithm (it's an acquired taste), is another interesting pick.

Kiersz largely leaves out the modern era, but there are plenty of players, er, scholars, from the past three centuries worthy of attention. Bernhard Riemann, a star of the German team in the mid-1800s, made three great accomplishments, any one of which would put him on the list: formalizing what's now known as the Riemann integral; founding what's now known as Riemannian geometry; and, in a single short paper, introducing the world to the Riemann zeta function and the outstanding Riemann hypothesis. Or take Poincare, who E.T. Bell called "the last universalist": the father of chaos theory, the Poincare group, and the recently-proved eponymous conjecture, is surely one of the best. Of course, it's hard to narrow down any list of the all-time greats, because each mathematician has his own style (not to mention his own field). Analysts might include Cantor, Weierstrass, Cauchy, and Fourier; probabilists would surely argue for Pascal and Bayes, not to mention the Russian giant Kolmogorov; geometers and physicists might tout the accomplishments of Hamilton, Noether, Lie, Weyl, and Einstein; and number theorists would surely give Ramanujan a nod. The fathers of computer science--Babbitt, Lovelace, Turing and von Neumann--certainly deserve a place. And who could forget the prolific combinatorialist Paul Erdős? Then there are those working today, or very recently, whose accomplishments, in time, might be as impactful as those of their predecessors--mathematicians like Fermat-slayer Andrew Wiles, chaos-theorist and topologist Steven Smale, analyst Terrence Tao, probabilist Oded Schramm, and *cough* Benjamin Pittman-Polletta. They may not all be on your list, but there's definitely room for them on mine.

See "The 12 Mathematicians Who Set The Stage For The Modern World", by Andy Kiersz, Business Insider, 3 July 2014.

---Ben Polletta

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Citations for reviews of books, plays, movies and television shows that are related to mathematics (but are not aimed solely at the professional mathematician). The alphabetical list includes links to the sources of reviews posted online, and covers reviews published in magazines, science journals and newspapers since 1996.

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