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Elisabetta Matsumoto investigates the mathematics and mechanics of knitting. The New York Times covered her work and Rachel Crowell followed up with a Q & A posted here on Math in the Media.(Image courtesy of Elisabetta Matsumoto)
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A monthly survey of math news
This month's topics:
"Math Teachers Should Be More Like Football Coaches"
Goro Shimura obit in the New York Times
Archimedean protein nano-cages
Math and the candidate
is the title of an Op-Ed piece (New York Times, May 11, 2019) by John Urschel, who played three years for the Baltimore Ravens and is now a math Ph.D. student at MIT. Urschel tells us how in high school his dream was to play college football even though "I weighed 'only' 220 pounds — about 80 pounds less than a big-time college tackle." and "I was an above-average athlete, but not a freak of nature." But that "didn't stop my coaches from encouraging me to believe I could reach my goal, and preparing and pushing me to work for it. When they told me I had potential but would have to work hard, I listened." He goes on: "Football coaches can be easy to caricature: all that intensity, all those pep talks, all those promises to build character.... But I wish math teachers were more like football coaches." Here's what he means: "A growing body of research shows that students are affected by more than just the quality of a lesson plan. They also respond to the passion of their teachers and the engagement of their peers, and they seek a sense of purpose. They benefit from specific instructions, constant feedback and a culture of learning that encourages resilience in the face of failure — not unlike a football practice." At Penn State, where he played football and majored in mathematics, Urschel met the teacher/coach we all need. "Until I got to college, I didn't really know what mathematics was.... Then one day, one of my professors summoned me to his office, handed me a book and suggested that I think about a particular problem. Understanding it, I realized, required reading other, more elementary books.... It wasn't easy, but it was fascinating. My professor kept giving me problems, and I kept pursuing them, even though I couldn't always solve them immediately. Before long, he was introducing me to problems that had never been solved before, and urging me to find new techniques to help crack them. The mathematical research I was doing had little in common with what I did in my high school classrooms. Instead, it was closer to the math and logic puzzles I did on my own as a boy. It gave me that same sense of wonder and curiosity, and it rewarded creativity."
"I still feel that childlike excitement every time I complete a proof. I wish I'd known this was possible when I was a kid."
Kenneth Chang wrote the obituary (New York Times, May 13, 2019) for the Princeton mathematician Goro Shimura, who died on May 3. For the general public, the most recognizable element of Shimura's work is its connection to Fermat's Last Theorem. Chang reminds us what that is: "a seemingly simple statement made by the French mathematician Pierre de Fermat in 1637: Equations of the form $a^n + b^n = c^n$ do not have solutions when $n$ is an integer greater than 2 and $a, b$ and $c$ are positive integers.... In his writings, Fermat claimed that he had figured out a proof but that he did not have enough room to write it down. For centuries, mathematicians sought unsuccessfully to figure out what Fermat was referring to." Fermat's Last Theorem turned out to be linked to the Taniyama-Shimura Conjecture, a statement about elliptic curves formulated in 1955 by Shimura and Yakuta Taniyama, but the connection took a while to establish. In 1985 Gerhard Frey (Saarland) showed that a solution of Fermat's equation led to an anomalous elliptic curve that was almost a counterexample to the Taniyama-Shimura Conjecture. That "almost" was eliminated the next year by Kenneth Ribet (Berkeley). Back to Chang: "Thus, a proof of a form of the Taniyama-Shimura conjecture would also prove Fermat's Last Theorem. In the 1990s, Andrew Wiles, then also at Princeton, figured out how to do just that, and Fermat's Last Theorem had finally been proved true."
Chang interviewed Peter Sarnak, who had been a colleague of Shimura's at Princeton. "Dr. Sarnak ... recalled visiting Dr. Shimura's house and seeing two desks in his office. In the morning, Dr. Shimura would work at one, exploring new ideas. In the afternoon, he would work at the second, polishing papers for publication. Once he made a breakthrough and finished a draft of a paper at the morning desk, he would place it in a drawer in the second desk and not return to it for about a year. 'A very meticulous and unusual way of working,' Dr. Sarnak said. 'By himself, almost always.'"
The May 16, 2019 issue of Nature includes a "Letter" ("An ultra-stable gold-coordinated protein cage displaying reversible assembly") from an international team of 21 led by Jonathan Heddle (RIKEN, Jagiellonian University) together with a "news and views" assessment ("Protein assembles into Archimedean geometry") by Todd Yeates (UCLA). As Heddle et al. explain in their Abstract, "Here we demonstrate an ultra-stable artificial protein cage, the assembly and disassembly of which can be controlled by metal coordination at the protein–protein interfaces... The geometry of these structures is based on the Archimedean snub cube and is, to our knowledge, unprecedented... The cage shows extreme chemical and thermal stability, yet it readily disassembles upon exposure to reducing agents."
The building blocks of the cage were derived from TRAP (trp RNA-binding attenuation protein)—"a bacterial ring-shaped protein amenable to genetic modification;" the authors were able to engineer a double-mutant TRAP with "11 equally spaced thiol groups along the outer ... border." They observed that in the presence of gold ions resulted within minutes in the self-assembly of spheres ("TRAP-cages") around 22 nm in diameter.
The construction exploits a mathematical near-coincidence. In the snub cube (illustrated below), each vertex is shared by four equilateral triangles and one square and therefore has total surface angle $4\times 60 + 90=330$ degrees, a number evenly divisible by 11, so a ring-shaped molecule with 11-fold symmetry can fit at each vertex compatibly with the geometry.
A molecule with 11-fold rotational symmetry (an undecamer) fits naturally at one of the 24 vertices of the snub cube. The molecules themselves lie in the faces of the polyhedron dual to the snub cube, the pentagonal icositetrahedron. Image from Nature 569 341 (Yeates), used with permission.
The pentagonal icositetrahedron. This polyhedron, like the snub cube, exists in two distinct mirror-image configurations. Image from Nature 569 438-442 (Heddle et al.), used with permission.
The near-coincidence that allows the 24 rings to lock stably together is that the angle between next-to-nearest-neighbor edges in an 11-gon is $114.54...^{\circ}$, whereas the obtuse angles of the pentagonal faces measure $114.81...^{\circ}$. This is close enough for 10 of the thiol groups on each of the 24 rings to bond (bonding mediated by gold ions) with thiol groups on adjacent rings. The 11th group sits in the acute angle of the face, too far from neighboring rings for the gold-mediated interaction.
The regular 11-gon fits almost exactly into a face of the pentagonal icositetrahedron: the angle $B = 114.54...^{\circ}$ between non-consecutive edges of the 11-gon is close enough for practical purposes to $A = 114.81...^{\circ}$, one of the obtuse angles of the pentagonal face. Image from Nature 569 438-442, used with permission.
Cryo-electron microscopy density maps of the left-handed and right-handed forms of the TRAP-cage constructed by Heddle et al., seen from the top. The inner diameter of the cage is 16nm. Note that of the 11 groups forming each ring, only 10 make contact with adjacent rings. The other ones dangle, four by four, in the gaps corresponding to the square faces of the snub cube. Image from Nature 569 438-442, used with permission.
At the end of the Letter occurs this remark: "The architecture described by TRAP-cage has, to our knowledge, never been observed in nature, although a similar arrangement of 11-pointed stars appears in Islamic art," with a reference to "The Hendecagonal Stars in the Alhambra" by Antonia Redondo and Dirk Huylebrouck.
Andrew Yang, a candidate for the Democratic presidential nomination, has made math a "rallying cry" for his campaign, as Matt Stevens reports in the New York Times (May 23, 2019; headline: "'Nerdiest Presidential Campaign in History'"). Stevens attended a Yang rally in Washington Square Park: "And when the 2,500 rain-soaked supporters of Andrew Yang realized he was about to drop his biggest applause line, they screamed the words to help him finish his New York rally with a bang. 'The opposite of Donald Trump,' Mr. Yang yelled, pausing to let his fans join in, 'is an Asian man who likes math!'" That's the last mention of math in the article, which concentrates on Yang's ethnicity in the context of national politics.
Photos accompanying the article show Yang and some of his supporters wearing dark blue baseball caps with MATH in large letters across the front. These hats are available (currently back-ordered) at the campaign merchandise shop where you can read: "The MATH hat is an essential item for any true Andrew Yang supporter. Celebrating numbers and facts an essential part of our campaign."
tony at math.sunysb.edu
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