Carnival of Mathematics #170

I’m hosting issue number 170 because I have a thing for the number’s largest prime factor, but it turns out there’s a reason for a Martin Gardner fan like me to appreciate the number itself: 170 is the number of steps¹ needed to solve a classic mechanical puzzle called The Brain invented by computer scientist Marvin H. Allison, Jr., described by Martin Gardner in his Scientific American essay “The Binary Gray Code”, and still available from Amazon.

The Brain, aka The Brain Puzzle, aka The Brain Puzzler.

Here’s what Gardner says about The Brain:

It consists of a tower of eight transparent plastic disks that rotate horizontally around their centers. The disks are slotted, with eight upright rods going through the slots. The rods can be moved to two positions, in or out, and the task is to rotate the disks to positions that permit all the rods to be moved out. The Gray code supplies a solution in 170 moves.

To find out about the Gray code, you can read Joseph Malkevitch’s September 2008 article “Gray codes“.

It’s interesting to compare speed-solving The Brain with speed-solving Rubik’s cube (check out the new Wired Magazine video about the latter!). The two solving processes might seem similar to an uninformed observer who only sees flying fingers and hears clacking plastic, but in terms of the mental processes involved, the tasks couldn’t be more different. At every stage in the process of solving a Rubik’s cube there are a dozen moves available, so if you’re hoping to make progress you need to have a very clear idea of what you’re doing and what you’ll do next. In contrast, The Brain only permits two moves at each stage along the way, so you can’t help making progress toward your goal as long as you don’t get confused and undo the move you just made. In that respect, solving The Brain is less like exploring a maze and more like walking through a labyrinth with a single non-branching path, such as one sees in a finite approximation to the infinitely twisty Hilbert curve. Thing-maker Santiago Ortiz has devised a Hilbert curve labyrinth that’s so easy even a marble can solve it. One could build a scaled-up version of this marble run that would accommodate a human rather than a marble, but I predict that anyone going down such a slide would feel quite ill by the time they reached the bottom.

Hilbert curve marble run.

ESSAYS

There were a number of good essays this month. Evelyn Lamb’s meta-parody I Can Has Numberz? was one of the oddest and (I have to say this) one of the cutest. First the Internet begat “Aren’t cats cute?” posts; then there were parodic “Aren’t humans cute?” posts; and now Lamb gives us an “Aren’t humans cute when they do math?” post as the first entry in a new genre that so far has but a single exemplar. Here’s hoping she does a sequel on the philosophy of mathematics. I mean, don’t you love it when a species that hasn’t encountered even one civilization from another planet pontificates about the timeless, universal features of its mathematics? It’s like when toddlers proudly announce what they’re going to be when they grow up. It’s sooooo adorable.

John Urschel wrote an op-ed that appeared in the New York Times, on Why Math Teachers Should Be More Like Football Coaches; it made me wonder if I’ve been sufficiently encouraging to students of mine who could have used encouragement.

Mark Dominus wrote an interesting piece called Math Jargon Failures about ways in which standard mathematical terminology makes things harder, rather than easier, for people who are trying to get a handle on the underlying concepts.

I liked many things about Robbert Dijkgraaf’s The Subtle Art of the Mathematical Conjecture. My favorite bit was this passage:

In fact, the metaphor of scaling a summit does not adequately capture the full impact of a proof. Once the conjecture is proved, it is not so much the endpoint of an arduous journey but rather the starting point of an even greater adventure. A much more accurate image is that of a mountain pass, the saddle point that allows one to traverse from one valley into another.

NEWS

Michael Griffin, Ken Ono, Larry Rolen and Don Zagier have given new life to an old idea about the Riemann Hypothesis; their work may or may not help our planet’s mathematical culture get an actual proof anytime soon, but it sheds new light on the Hypothesis. See the press release Mathematicians revive abandoned approach to Riemann Hypothesis, Enrico Bombieri’s summary of the paper, or the paper itself.

This year’s Abel Prize went to geometer Karen Uhlenbeck. Now that the Abel Prize and the Fields Medal (recently awarded to Maryam Mirzakhani) and the Salem Prize (awarded in 2006 to Stefanie Petermichl and more recently to Maryna Viazovska) have gone to women, it doesn’t seem unreasonable to hope that I’ll live to see the last “first woman to win …” mathematical news story! (I’m looking at you, Wolf Foundation.)

Speaking of Viazovska and her groundbreaking work in 2016 (applying modular forms to the study of higher-dimensional sphere-packing), there’s been a fabulous follow-up result, wonderfully described by Erica Klarreich. For those just now learning about this saga-in-progress, here’s a synopsis of the action so far: Cohn and Elkies showed in 2001 that if a certain “magic” function existed, it would have implications for sphere-packing in 8- and 24-dimensional space; Viazovska showed in 2016 that the function existed; and now Cohn and Viazovska, joined by Kumar, Miller, and Radchenko, have extended those results to other sorts of geometric problems about sticking together things that can’t (or “don’t want to”) get too close to each other. ²

The 2001 article contained a near-miss construction of the required magic function, foreshadowing Viazovska’s eventual exact construction. But some near-misses can’t be fixed. For instance, the infamous “Horgan surface“, which some computer simulations seem to approximate, doesn’t actually exist, and Evelyn Lamb wrote an article a couple of years ago about polyhedra that don’t actually exist either, even though one can build paper models of them. (The catch is that the models aren’t — can’t be — mathematically perfect, but the imperfections are hard to spot with the naked eye.) Lamb’s article centered on Craig Kaplan, who has developed a fondness for these impossible shapes and their nearly-accurate real-world models. Now a cabal of chemists, with Kaplan’s help, have actually built such near-miss objects at molecular scale.

For those more interested in stacking 3-dimensional spheres than 8- or 24-dimensional ones, there’s something new to report in that department as well. If you want to make a stable stack of tennis balls, you can make ridiculously big ones by getting gravity and friction to work to your advantage.

A still from a video made by Andria Rogava. Guess what happens when the top ball gets removed?

The crucial idea is that a tennis ball at the top of a stack can serve as a kind of keystone, frictionally locking the balls below it into their proper places. Andria Rogava, who has been exploring this world of novel structures, writes “I can find no mention of such structures online and am sure they would have interested Martin Gardner — that great fan of recreational science — were he still alive.” I agree! I’m glad to see that Rogava’s work got the attention of the Daily Mail. Rogava has created a Facebook page for people interested in knowing more about these structures.

The butterfly attractor, discovered by Edward Lorenz with vital technical assistance provided by Ellen Fetter and Margaret Hamilton.

Here’s some new news about some old news. The old news is Edward Lorenz’s work on the weather, and its implications for the role of chaos in physical systems; the phrase “butterfly effect” has even become a part of popular culture. (The original metaphor involved seagulls, not butterflies, flapping their wings, but I think “butterfly effect” is a better name, not only because of the internal alliteration, and because butterflies are smaller than seagulls, but also because the Lorenz attractor, one of the first rigorously analyzed examples of a chaotic system, has a phase plot resembling a butterfly.) What’s recently come to light is that two women played an important role in Lorenz’s early work on chaos theory, as Lorenz himself acknowledged in his pioneering article but other people did not. Find out about Ellen Fetter and Margaret Hamilton in Joshua Sokol’s article Hidden Heroines of Chaos.

Lastly, there’s been some work in social psychology that uses mathematics as a testing ground for theories about when people claim knowledge they don’t have. The researchers asked subjects to assess their state of knowledge about such bogus concepts as “proper numbers” and “declarative fractions”. Who do you think was more likely to claim they knew the meanings of these meaningless terms: boys or girls? kids from wealthy households or kids from poor households? (Spoiler alert: You probably won’t be surprised by the answers.)

ODDS AND ENDS

I learned about many of the above articles through Twitter. If you aren’t on Twitter and you want to see what’s available, I suggest the list at TrueSciPhi as a good place to start. Some people whose tweets I especially enjoyed in May were Robert Fathauer (@RobFathauerArt), Roice Nelson (@TilingBot), Vincent Pantaloni (@panlepan), Catriona Shearer (@Cshearer41), and Dave Whyte (@beesandbombs).

There have been some good new videos this month. Check out Chalk of Champions and Numberphile: Peaceable Queens. There are also two new Math Encounters videos, courtesy of the National Museum of Mathematics (where the videos were made) and the Simons Foundation (which paid for production costs): Doug McKenna’s Golden textures: the art of dissecting golden geometries (January 2019) and Robbert Dijkgraaaf’s Space, time and the fourth dimension (March 2019). There are lots of older Math Encounters videos worth checking out. Since the word “ergodic” was a spelling word in this year’s national spelling bee³, I recommend Bryna Kra’s talk Patterns and disorder: how random can random be? (February 2014). There’s a short film about Karen Uhlenbeck, courtesy of the Abel Prize Institute. And lastly, if you missed the National Math Festival in May, you can still watch a Numberphile video showing some highlights from the festival.

There’s also some good math content in audio form on the internet (though most mathematical content creators understandably gravitate toward media that make it easier to incorporate visuals). There was nothing new from Samuel Hansen’s Relatively Prime podcast this month, but there was a new episode of Kevin Knudson and Evelyn Lamb’s My Favorite Theorem podcast,  featuring Moon Duchin. You can also learn more about Duchin’s work from an interview with her that appeared in Science News.

Finally, there’s David Eppstein’s article Playing with model trains and calling it graph theory describing work he’s done with with Demaine, Hesterberg, Jain, Lubiw, Uehara, and Uno; the authors showed that certain sorts of routing problems (think Rush Hour, but with trains) are PSPACE-complete. Results like that are really encouraging for folks like me (and Martin Gardner) who like puzzles but aren’t actually that good at solving them, because what these theorems say to us is: “You’re not stupid; these puzzles are genuinely hard!”

Next time (June 17): Mathematical Flimflam.

ACKNOWLEDGMENTS

Thanks to Tom Duff, Sandi Gubin, Brian Hayes, Michael Joseph, Michael Kleber, Evelyn Lamb, Andy Latto, Joseph Malkevitch, Evan Romer, and Katie Steckles.

ENDNOTES

#1. Why 170? The number 170 belongs to the exclusive club of numbers whose binary expansion alternates between 1 and 0 (170_ten equals 10101010_two). This set of positive integers occurs as entry A000975 in the Handbook of Integer Sequences (the most fantastically useful book for mathematicians in the galaxy, now available as an online resource).

#2. The history of the new results in sphere-packing is a little bit more complicated than was described above.

Viazovska found the magic function for 8-dimensional sphere-packing on her own; she, Cohn, Kumar, Miller and Radchenko found the (different) magic function for 24-dimensional space a week later. This solved the sphere-packing problem in dimension 8 and dimension 24.

Now the five have extended their earlier work to apply to problems involving mutually-repelling points in 8-dimensional space and 24-dimensional space. The connection between sphere-packing and mutually repelling points comes from looking at the centers of the spheres in a packing. If the spheres are of radius r, the centers can never be closer than distance 2r from one another.  One could instead imagine a physical system in which points can be close but don’t “want” to be, in the sense that having points be close together requires expending energy to overcome a repulsive force, and the system tries to find equilibrium configurations that minimize total energy. This is the setting in which the new work takes place. Part of what makes the new result so amazing is its generality; you might expect that details about the force-law would play a role, but the new result is remarkably general.

#3. If you check out the painful-to-watch video of elite 8th grade speller Aanson Cook misspelling “ergodic” (starting at 0:30), you’ll notice that he went to some trouble to try to determine whether the third syllable had a “d” or a “t”. In the end he incorrectly guessed that the word was “ergotic” and that the announcer was doing that thing wherein native English speakers pronounce a “t” like a “d”. (As linguists say, “Phones aren’t the same as phonemes“.) Or maybe Mr. Cook isn’t good at decoding the high-frequency part of a vocal waveform that determines the difference between “d” and “t”.

REFERENCES

Enrico Bombieri, New progress on the zeta function: From old conjectures to a major breakthrough, Proceedings of the National Academy of Sciences, June 2019.

Henry Cohn, Abhinav Kumar, Stephen D. Miller, Danylo Radchenko, and Maryna Viazovska, Universal optimality of the E₈ and Leech lattices and interpolation formulas (preprint), February 2019.

Martin Gardner, “The Binary Gray Code”, available as Chapter 2 in Gardner’s book Knotted Doughnuts and Other Mathematical Entertainments.

Michael Griffin, Ken Ono, Larry Rolen, and Don Zagier, Jensen polynomials for the Riemann zeta functions and other sequences, Proceedings of the National Academy of Sciences, June 2019.

Evelyn Lamb, The Impossible Mathematics of the Real World, Nautilus, June 2017.

Joseph Malkevitch, “Gray codes”, AMS Features column, September 2008.