Math and Verbal Gymnastics

We are coming to the end of Math Awareness Month, whose theme this year was The Future of Predictions. A clever theme name, indeed. I do love when mathematics and verbal gymnastics come together. And on that theme of math and words, you should know that April is not only math awareness month, but it’s also National Poetry Month! In honor of this double-whammy, I thought we could take a moment to explore the intersection of math and poetry.

As it turns out, there are actually quite a few blogs dedicated to poetry and mathematics. If you, like me, are a mathematician who is new to mathematical poetry, a good place to start is with “Five Types of Mathematical Poetry” on the blog Mathematical Poetry.

If the strictly lexical type of mathematical poetry is what you prefer — that is, mathematical poems constructed from the written word and influenced by ideas in math — then I suggest JoAnne Growney’s blog, Intersections — Poetry with Mathematics. In the theme of math awareness month, Growney posted a beautiful poem by Joyce Nower, inspired by prediction, fate, and the tragic story of the mathematician Hypatia.

Another type of mathematical poetry melds the written word with mathematical symbols, not necessarily following any mathematical rules. The late Bob Grumman, a pioneer in visual mathematical poetry, described it as “poetry that does mathematics, rather than merely discusses mathematics.” In a post on the Scientific American Guest Blog, Grumman discusses the state of the art form and the work of fellow visual poet Karl Kempton.

Magic Square By BabelStone - Own work, CC BY-SA 3.0,

Magic Square By BabelStone – Own work, CC BY-SA 3.0,

Pure mathematics can also be seen as poetry. The patterns and repetition in numerical and symbolic mathematics do echo those in traditional lexical poetry. To the right you can see a magic square, an ancient example of pure mathematical poetry.

In the spirit of poetry and math awareness, let me close with a terrible haiku that I just wrote in honor of the close of the semester.

reaching the limit
harmonic series diverge
and so too do we

Share your #mathematicalhaikus with me on Twitter @extremefriday.

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Fold Your Way to Glory

Yesterday, I led a meeting of a Teachers’ Math Circle about the fold and cut theorem. This theorem says any region with a polygonal boundary can be folded and cut from a sheet of paper using only one cut. I learned about the theorem last year when Numberphile posted this excellent video featuring a virtuoso performance from Katie Steckles, who folds and cuts every letter of the alphabet from memory.

I was a little nervous about leading the program because I had prepared almost nothing to say. Everything I thought about saying was boring, so I decided the best way to approach the activity was to just get people started on it. Luckily for me, the group was ready to jump right in. I dumped a bunch of paper into the middle of the table, and people started folding.

I encouraged people to try the most symmetric shapes first, but other than that, I didn’t have to give them many suggestions. I was prepared for some frustration when they started trying the scalene triangle because it’s a big step up in difficulty, but several of them got the scalene pretty quickly, and no one seemed to give up. In general, the strange shapes people got when you mess up were amusing rather than frustrating.

My favorite fold-and-cut mistake. I was trying to make a rectangle with a smaller rectangle inside it. Image: Evelyn Lamb.

My favorite fold-and-cut mistake. I was trying to make a rectangle with a smaller rectangle inside it. Image: Evelyn Lamb.

Participants almost immediately started asking mathematical questions and trying to extend the activity: do we have an existence theorem? Must we always fold along every angle bisector? Is there a general theory of folding? I liked Anna Weltman’s suggestion of trying to make things without drawing on the paper, and I spent some time trying to fold stars without drawing them, but the teachers didn’t really bite on that. Instead, some of them started thinking about minimal folding numbers for different shapes, and some of them worked on developing a folding algorithm.

Erik Demaine is one of the pioneers of fold-and-cut theory and the mathematics of paper folding in general. His page about folding and cutting has links to all the gory mathematical details as well as some templates. I ended up bringing copies of his swan to the teachers’ circle. They are beautiful, but I had mixed feelings about bringing them because they have the fold lines marked on them already. I didn’t hand them out until one group had started talking about how to use angle bisectors and perpendiculars in their folding algorithm, and I thought the swan template might give them some ideas. Because I gave them only the template, not any explanation of how it was made, I think it didn’t take away too much of their fun.

My fold-and-cut swan now enjoys pride of place on my new hexagonal shelf. Image: Evelyn Lamb.

My fold-and-cut swan now enjoys pride of place on my new hexagonal shelf. Image: Evelyn Lamb. Swan template: Erik Demaine.

In addition to Demaine’s swan, I brought templates for lots of different shapes from Patrick Honner and Joel David Hamkins, who uses hole punching symmetry activities as a warm-up for cutting. I also got ideas from Mike Lawler, who has done fold and cut activities with kids, and Kate Owens, who ran a fold-and-cut workshop for teachers.

I’ve done a little bit of origami, but I’ve never gotten good enough to feel like I had geometric intuition for doing it. I’m still at the level where I follow directions and get what the book says I should. Making these fold-and-cut shapes, though, is an easy way to start thinking about paper folding mathematically and creatively. Thanks to the resources I mentioned above, you too can easily introduce people to the joys of mathematical paper folding.

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Packing Heat: The World Of Sphere Packings Is On Fire

Suppose somebody hands you a bunch of oranges and asks you to stack them on a shelf, I’ll bet I can guess how you would do it. You’d build a pyramid by laying down a base layer and then fill in the upper levels by placing oranges in each of the divots provided by the layer below. If you’ve done this before, you might have noticed that the oranges in the base layer create a repeating hexagon pattern. In case you don’t have a crate of oranges next to you right now to try this out, check out the photo below.

A hexagonal packing of oranges creates the base layer on the left, and then the pyramid continues to rise using the hexagonal close packing.  Image via Jeremy Jenum Flickr Creative Commons.

A hexagonal packing of oranges creates the base layer on the left, and then the pyramid continues to rise using the hexagonal close packing. Image via Jeremy Jenum Flickr Creative Commons.

This is called a hexagonal circle packing, and it’s the densest way to pack a bunch of circles together. By densest, I mean that any other way you pack together circles is going to have much more empty space left over. When you place the subsequent layers on top by filling in the divots, what you’re doing is creating a well-studied arrangement called the hexagonal close packing of spheres. Just like the hexagonal packing in 2-dimensions, the hexagonal close packing is the densest way you can pack 3-dimensional spheres together. This was a result proved by Thomas Hales in 1998.

These both belong to the broader family of n-dimensional sphere packings, and it’s been a long standing open problem to find the densest sphere packings in each dimension. While we have the nice orange stacking analogy to help us visualize dimensions 2 and 3, in higher dimensions we can’t visualize things in the same way. But here is the essence of the problem. In any dimensions, a sphere is just a set of points that are equidistant from some center point, and a dense sphere packing is just an arrangement of non-overlapping spheres that fills up as much ambient space as possible.

Maryna Viazovska unlocked the solution in 8 dimensions, paving the way for the 24-dimensional solution. Photo Courtesy of Oberwolfach photo archives.

Maryna Viazovska unlocked the solution in 8 dimensions, paving the way for the 24-dimensional solution. Photo Courtesy of Oberwolfach photo archives.

A few weeks ago Maryna Viazovska, currently a post-doc at the Berlin Mathematical School and the Humboldt University of Berlin, solved the sphere packing problem in 8-dimensions. Erica Klarreich, a math journalist for Quanta Magazine gives details on how Viazovska arrived at her solution, and some stories about the people she met along the way.

And then not a week went by before she and her coauthors, Henry Cohn, Abhinav Kumar, Stephen D. Miller, and Danylo Radchenko generalized her methods to solve the sphere packing problem in a 24-dimensions. On his blog, mathematician Gil Kalai gives some historical background for the 8- and 24-dimensional sphere packing problems.

In a video posted by the Institute for Advanced Study, Stephen Miller gets into the details of the proof, he says “there’s something very special about 8 and 24, we can’t expect every week to keep proceeding like this.” Although the dimensions 8 and 24 might seem totally random, the reason these solutions came so close on each other’s heels is that these sphere packings — unlike those in other dimensions, as discussed on the n-category cafe — are related to two special lattices, E8 and the Leech lattice. Having this connection to lattices, which, full disclosure, I’m obsessed with, means that there is a world of machinery in the realm of modular forms for dealing with the packings. In a very broad sense, solving the packing problem came down to finding some suitable modular function that satisfied an appropriate list of properties that are derived from methods in harmonic analysis.

Sphere packing problems, of course, have many interesting applications, but the one that has always fascinated me is the link between dense sphere packings and error correcting codes. Trying to pack n-dimensional spheres as close to each other as possible is like trying to find points (namely, the center point of the sphere) that are as close to each other as possible, while maintaining some prescribed amount of distance between them (namely, the buffer created by the sphere around each center point). This acts just like an error correcting code, in the sense that we want to find code words that are similar enough that we can build a language out of them, but far enough apart that they can be transmitted over noisy channels and not be totally degraded by interference.

Like all good problems, sphere packings touch on many branches of mathematics: number theory, geometry, analysis. The fact that this problem has so many approaches and that its solutions are simultaneously so diverse in flavor, John Baez points out so perfectly in his blog post, “hints at the unity of mathematics.”

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How to Celebrate Square Root Day

Apparently today, 4/4/16, is Square Root Day. (I supposed we could also have celebrated 4/2 to have a long Square Root Weekend.) How should a math enthusiast celebrate this holiday, which won’t come again until May 2025?

8 squares arranged in a curve of pursuit. Image: Evelyn Lamb.

8 squares arranged in a curve of pursuit. Image: Evelyn Lamb.

Of course, one option is to be a curmudgeon. As a curmudgeon myself, I heartily support you in this endeavor, and Michael Lemonick of Scientific American does as well. If, however, you are not as cold and dead inside as he and I are, I’ve got some lovely square facts and activities for you.

I’ve got to lead off with one of the coolest things I learned last month, courtesy of Matt Baker.

Let n be a positive integer.  It is easy to see that a square can be dissected into n triangles of equal area if n is even (Exercise).  What if n is odd?  If you play with the question for a bit, you probably won’t be surprised to learn that in this case it’s impossible.  But you may be surprised to learn that this result was not proved until 1970, that the proof involved p-adic numbers, and that no proof is known which does not make use of p-adic numbers!

I find it shocking and delightful that a fairly simple question about plane geometry requires p-adics to solve. If, like me, you’re a bit uncomfortable with p-adics, cut-the-knot math has a p-adic page where you can learn about these strange completions of the rationals.

A more traditional way of celebrating might be to ponder the wonderful Pythagorean theorem. Cut-the-knot has more proofs of it than you can shake a stick at, and I’m still enchanted by Albert Einstein’s elegant proof, which Steven Strogatz wrote about last November.

Arts and crafts have many opportunities for square-making. One of my favorite square-centric designs is a curve of pursuit: out of tilted squares, curves seem to appear. The excellent mathematical knitting site Woolly Thoughts has some information about how to knit curves of pursuit. A few years ago, my grandparents celebrated their 82-th anniversary, so I made them a tablecloth with 8 squares arranged in a curve of pursuit. You can see it in the picture at the top of this post or read about it here.

Last year, Math Munch shared a project called SquareRoots by John Sims. Inspired by the quilts of Gee’s Bend, he made mathematical quilts based on the base 3 digits of pi. And for more mathematical quilts, check out the gorgeous ones on Elaine Ellison’s website.

If you want to take your square explorations up a dimension or two (because who has time to wait until 4/3/64 or 4/4/(2)256?), Numberphile has a lovely video about higher-dimensional Platonic solids, including the hypercube, and Mike Lawler has been making hypercubes with his kids.

It’s baseball opening day today as well (√ √ √ for the home team!), and Patrick Vennebush of Math Jokes for Mathy Folks has combined the two topics for a guess-the-graph game.

How will you be square today?

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A conversation with The Accidental Mathematician

Izabella Laba is a fantastic blogger. She is also a professor at the University of British Columbia. She is widely published in harmonic analysis, geoemtric measure theory and additive combinatorics. And most recently, she is one of the founding editors of the new ArXiv overlay journal, Discrete Analysis. But her blog, The Accidental Mathematician, was how I first became familiar with Laba, and it was the reason that I called her for a conversation last week. I was curious how she found her niche in the math blogosphere, and to hear her thoughts on the state of the art.

photo courtesy of Izabella Laba.

photo courtesy of Izabella Laba.

One of the pillars of Laba’s blog is the issue of gender imbalance in mathematics. I spoke with Laba about tackling this topic, “It’s never just gender inequality by itself. It’s not separated from everything else that happens to us. Gender inequality does not exist in a vacuum. It manifests itself in specific ways. And both depends on the rest of who we are and what we do.” It’s important to put things in a rich context, “if I were to just write a post about ‘men think that women are worse at math but they really aren’t,’ then there’s like one sentence I can write about that. And that’s where I have to stop,” Laba says, “it’s not a terribly interesting thing to either read or write.”

The context tells the story, and accordingly, the post of Laba’s that has stuck with me the most is “Gender, conferences, confrontations, and conversations.” For anyone considering organizing a conference, attending a conference, existing as a woman in math, existing as a man in math, seeking equality, recongnizing inequality in all its shadowy forms, or just generally getting it, Laba takes you there.

As a full professor at a top research university, Laba is able to scope out the gender terrain from a unique vantage point. Although it’s 2016, she says, “people get the impression that this progress is happening really quickly and this problem is going to be all fixed in a few years, and I don’t think that’s going to happen.” Laba cites the high numbers of women who fall through the so-called leaky pipeline, an idea that is confirmed by the AMS Annual Survey of the Mathematical Sciences. While 32% (which has held relatively stead over the past 10 years) of the mathematics PhD recipients in 2014 were female, when you compare the number who eventually get tenure track jobs at large research universities, that number is much smaller.

Sometimes it can be hard to see the big picture when we are so focused on our home institutions, Laba says, “blogging is important that it allows people to make those connections, by reading a lot of blogs and communicating with people you get a bigger picture than you would have on your own.”

Blogging, Laba says, is an important tool to shed light on all aspects of the profession, particularly for those who exist outside of the narrow confines of academia. “I don’t really think that I write to humanize mathematicians, maybe it has that effect to some people, but that’s not something that I aim for,” rather, Laba says, “if I decide to write a post it’s about a specific issue that I’ve been thinking about or discussing with people.” Laba has written about the duties of an academic mathematician, the way we choose speak as mathematicians, and the pervasive kookification of mathematicians in the media.

It took Laba some time to find her voice as a blogger, always an outspoken person, she says “I’m actually really embarrassed to look at some of my earlier posts. I’m sometimes tempted to delete all of that. Leaving that there for other people, especially other women who think that they might want to start blogging but they don’t write so well. Ok. Look at what I did!”

But the medium of blogging, Laba claims, is a good and important one. “It was a long time ago I came across personal blogs, political blogs, academic blogs, and it was just amazing how much people could do with that form,” she says, “you could speak your own voice, you could speak for yourself.” Even better, she says, “you could develop your voice gradually, you did not have to write a book, or do something big right away. You could start with small posts and work towards something bigger.”

Posted in people in math, women in math | Tagged | 1 Comment

All the P-values Fit to Print

A bell curve. Image: Cris, via Flickr.

A bell curve. Image: Cris, via Flickr.

I feel like I’ve seen news stories or blog posts about p-values every day this month. First, Andrew Gelman reported that the editor of the journal Psychological Science, famous to some for publishing dubious findings on the strength of p<0.05, will be getting serious about the replicability crisis. (The editorial he referenced came out last November, but Gelman tends to write posts a few months in advance.) Then the American Statistical Association released a statement about p-values, and a few days later, the reproducibility crisis in psychology led to some back-and-forthing between groups of researchers with different perspectives on the issue.

At the heart of much of the controversy is that much-maligned, often misunderstood p-value. The fact that the ASA’s statement exists at all shows how big an issue understanding and using the p-value is. The statement reads, “this was not a lightly taken step. The ASA has not previously taken positions on specific matters of statistical practice.” Retraction Watch has an interview with Ron Wasserstein, one of the people behind the ASA’s statement.

At 538, Christie Aschwanden tries to find an easy definition of p-value. Unfortunately, no such definition seems to exist. “You can get it right, or you can make it intuitive, but it’s all but impossible to do both,” she writes. Deborah Mayo, “frequentist in exile,” has two interesting posts about how exactly p-values should be interpreted and whether the “p-value police” always get it right. Mayo and Gelman were also two of the twenty people who contributed supplementary material for the ASA statement on statistics.

Misuse and misinterpretation of p-values are part and parcel of the ongoing reproducibility crisis in psychology. (Though some say it isn’t a crisis at all.) Once again, Retraction Watch is on it with a response to a rebuttal of a response (once removed?) about replication studies. The post goes into some depth about a study that failed to replicate, and I found it fascinating to see how the replicating authors decided to try to adjust the original study, which was done in Israel, to make it relevant for the Virginians who were their test subjects. Gelman also has three posts about the replication crisis that I found helpful.

One of the underlying issues with replication is something a bit unfamiliar to me as a mathematician: inaccessible data. Not all research is published on the arXiv before showing up in a journal somewhere, so there are still paywalls around some articles. More troubling, though, is the fact that a lot of data never makes it out of the lab where it was gathered. This makes it hard for other researchers to verify computations, and it means a lot of negative results never see the light of day, leading to publication bias. The Neuroskeptic blog reports on a lab that has committed to sharing all its data, good bad and ugly.

So what’s the bottom line? It’s easy to be pessimistic, but in the end, I agree with another post by Aschwanden: science isn’t broken. We can’t expect one experiment or one number to give us a complete picture of scientific truth. She writes,

The uncertainty inherent in science doesn’t mean that we can’t use it to make important policies or decisions. It just means that we should remain cautious and adopt a mindset that’s open to changing courses if new data arises. We should make the best decisions we can with the current evidence and take care not to lose sight of its strength and degree of certainty. It’s no accident that every good paper includes the phrase ‘more study is needed’ — there is always more to learn.

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Algebra: It’s More Than Just Parabolas

Our old pal Andrew Hacker is back at it again. With the publication of his new book and a spate of recent media appearances, he is a man on a mission. A professor emeritus in the Department of Political Science at Queens College, Hacker first rose to fame notoriety when he called for an end to high school algebra requirements in his 2012 op-ed, “Is Algebra Necessary?” This year, he’s doubled down on that demand in his book “The Math Myth and other STEM delusions.”

Naturally, mathematicians and educators have had a lot to say about this. Hacker’s main thesis is that we need to get away from this idea of teaching the arcane rules and formulas of algebra, and instead, replace it with something more intuitive and relatable which he calls “numeracy.” In an excerpt from his book, Hacker says, “Calculus and higher math have a place, of course, but it’s not in most people’s everyday lives. What citizens do need is to be comfortable reading graphs and charts and adept at calculating simple figures in their heads.” Numeracy, he explains, entails a broader sense of quantitative literacy and ability to interpret numerical information without all that rote memorization and jargon.

Hacker is right, cramming rules and pointless seeming equations into the brains of young people is painful for everyone and it totally misses the point of math, which contains so much beauty, utility, and historical context — or ideally at least two of those three. Instead, says Simon Jenkins in The Guardian, “the prominence of maths in the curriculum is education’s version of Orwell’s imaginary boot, ‘stamping on your face … forever’.” Sounds pretty grim. In a post on Math With Bad Drawings, Ben Orlin claims, “Andrew Hacker has a coherent and lovely vision for how to teach mathematics. But to treat his work as a blueprint for all of mathematics education is to make a category error.” Orlin points out that there’s a looming backstory to why we teach the math that we do, competing interests from teachers, future employers, governing bodies, administrators, you name it. To malign the subject of algebra is to ignore the fact that the way we teach those rules and equations is actually more relevant than the rules and equations themselves.

And this is why I will now point out that Hacker is wrong. And the basis of his wrongness, as Keith Devlin breaks down in a post on Devlin’s Angle, seems to stem from the fact that he doesn’t really know what algebra is. Algebra, according to the Khan Academy, is simply “the language through which we describe patterns…Once you achieve an understanding of algebra, the higher-level math subjects become accessible to you. Without it, it’s impossible to move forward.” Algebra is not, as Hacker tries to claim, the inane study of parametric equations, polynomial functions, and vectorial angles. Algebra is the way that we learn to wrangle not just numbers, but concepts and unknowns. It is an ancient art form that allows us to frame questions about numerical things we don’t totally understand and march slowly towards an answer using a systematized approach. For example, suppose that a movie studio earned $15 million with 2 million total transactions. Part of that coming from $6 video rentals, and the other part from $15 video sales. How could you find out how many videos were rented versus sold? Algebra! It is incredibly powerful and as far as gaining a sense of numeracy and quantitative literacy, it can’t be beat.

Personally, I think the part that really sticks in my craw is that Algebra, when taught properly, is just not that hard. I am totally on board with toppling calculus from its place of prominence in the high school math echelon. Because it’s true, not all people need calculus. In fact, most people don’t. But Algebra? I think it’s ok to suffer through a year of Algebra I just to be aware of the fact that we can talk about the general behavior of math using equations and unknowns. To me, that’s the mathematical equivalent of learning how to read a chapter book. If we were willing to accept an equivalently low bar for literacy, then most of our nation’s high schoolers wouldn’t even be able to read Hacker’s book. Or any grown-up book for that matter. And what a scary world that would be.

Posted in K-12 Mathematics, Math Education | Tagged , , , , | 1 Comment

The Creativity of Approximation

As a mathematician, I am frequently frustrated with the world’s stubborn refusal to mirror mathematical perfection. No “circle” made of atoms actually has a circumference-to-diameter ratio of π; no population’s growth is exactly an exponential function. The overwhelming approximate-ness of the world generally distresses me, but a recent post on Craig Kaplan’s blog  has me looking for creative possibilities in the messiness of the real world.

A solid that doesn't quite exist. Image: Craig Kaplan.

A solid that doesn’t quite exist. Image: Craig Kaplan.

I met Kaplan, a computer scientist at the University of Waterloo, last year at the Bridges math+art conference, but I didn’t know he had a blog until a friend shared his delightful post about a solid he built. It appears to be 4 dodecagons, 10 decagons, and 28 equilateral triangles, but as he writes, “Unfortunately, there’s a small problem with this polyhedron: it doesn’t exist. Mathematically, you can prove that if you want all the faces to be regular polygons, there’s no way that these shapes will close up into a perfect solid.”

Instead, the solid only appears to exist because of the messy real world: “the real, mathematical error inherent in the solid is comparable to the practical error that comes from working with real-world materials and your imperfect hands.” Kaplan writes beautifully about the serendipity of finding near misses in geometry and closes by asking, “Where else in mathematics or beyond it might we find near misses, once we adopt this mindset?”

I’ve been reading a lot about temperament and tuning recently, so my mind turned to tuning systems. As I have written in the past, no piano can be perfectly in tune because (3/2)12 (twelve perfect fifths) is close, but not quite equal, to 27 (seven octaves). All tuning systems tweak various near misses, especially that one, to create as many intervals as possible that are as close to perfect as possible. Our ears are approximate enough that we can tolerate the little deviations from perfection that make pianos possible.

Recently, I exploited the near miss idea to make bias tape for a sewing project in a new toroidal way. True bias tape has a slope of 1, but I figured 9/8 was close enough and made my bias tape using a closed geodesic on a flat torus. You can read more about my method on Roots of Unity.

I’ll echo Kaplan now and ask you: Where have you found near misses in mathematics? Have you ever used near misses to unlock a new creative possibility in your art?

Posted in Mathematics and the Arts | Tagged , , | 1 Comment

Opening The Cryptographic Backdoor

By Kārlis Dambrāns from Latvia - Apple iPhone 6, CC BY 2.0 courtesy of Wikimedia Commons

By Kārlis Dambrāns from Latvia – Apple iPhone 6, CC BY 2.0 courtesy of Wikimedia Commons

Unless you’ve been living off the grid somewhere in an igloo build out of old discarded iPhones, you’ve probably heard about the recent standoff between Apple and the US government. The short story, is that the US Government has demanded that Apple build a tool to access the data stored on one particular iPhone. Essentially, the government is asking Tim Cook to find some sort of backdoor entry to the securely encrypted interior of the iPhone, and Tim Cook definitely doesn’t want to do that. Computationally, this is sort of difficult — but doable — but the big problem rests in the legal precedent set by handing over this sort of unfettered access.

We’ve talked about encryption here before, and since this is such an interesting story, with so many broad implications, I thought it would be good to give a rundown of the actual cryptography at play here. If you are curious about the legal and ethical implications, well, maybe you can find a blog on legal blogs somewhere, or you can read this quick explainer of the legal pieces of the puzzle.

All of Apple’s security schemes are described in tediously technical detail in the iOS Security Guide, and what we can get from that is that the iPhone’s security system is multi-tiered and really complicated, but there are two important numbers to keep in mind:

  1. PIN: This is some number of several digits that you, the user, picked out the first time you booted up your phone. This PIN number resides in your brain and nowhere else.
  2. UID: This is some 256-bit key that’s unique to your phone and baked into the hardware during manufacturing. This resides only on your personal hardware and nowhere else. Even Apple itself doesn’t keep a copy of this.

The first time you boot up and enter your PIN, your phone will use these two numbers to generate a third number called the passcode key — this process takes about 80 millisecond and it’s personalized to your phone. The UID and the precise recipe for tangling up the UID and PIN are both stored in the so-called secure enclave of your phone’s hardware. On his cryptographic engineering blog, Matthew Green gives a bunch of this information and more in a nice plain(isn) language primer on iPhone security at least up until iOS8.

Now, we can definitely give a hat-tip to Apple and say that cryptographically, this sucker is pretty airtight. There’s no way to open up the enclave or find some brute force method to extract the 256-bit UID encryption key, it’s just way too hard and it would take forever. Strictly speaking, there’s also no way to extract the PIN since it’s in the secure enclave of your brain. But the fortress of the iPhone is not impregnable, because of one thing: there are only a finite number of PIN choices, in fact 10,000 if we restrict to 4 digit numbers. So provided you can guess the PIN, the rest of the system will fall into place for you. But that, as they say, is where they get ya.

Apple is so clever that they have build a second barrier of security, namely you have to wait some number of minutes between each attempted PIN, and you can only try out 10 different PINs before you get locked out and the data on the phone gets wiped forever. This “some number of minutes” amounts to several days once you’ve made more than 4 attempts. So what the FBI is really asking is for Apple to write a tool to disable these lockout features. This seems like a feasible project, but due to special signature recognitions in the software, such a program can only be written in-house at Apple.

More than half of people think that Apple's in the wrong, and that the phone should be unlocked.

More than half of people think that Apple’s in the wrong, and that the phone should be unlocked.

What’s not totally obvious is to me is how dangerous it would be to birth a tool like this into existence. Yes, in some sense it does open up a backdoor to the iPhone security system, but in another sense it may be ad hoc to the point of being useless outside the context of this one particular iPhone in question. I suppose the larger question is whether it’s appropriate for the FBI to make these sorts of demands on industry. A recent PEW Research poll shows that 51% of American say yes. It suffices to say, it’s a complicated issue and a worthwhile reminder that mathematical cryptography is only as strong as the humans who hold the keys.

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Beyond Banneker: Resources for Learning about Black Mathematicians

Benjamin Banneker, one of the first African American mathematicians. Image: Public domain, via Wikimedia Commons.

Benjamin Banneker, one of the first African American mathematicians. Image: Public domain, via Wikimedia Commons.

Part of the reason Erica Walker wrote Beyond Banneker: Black Mathematicians and the Paths to Excellence was that she was tired of hearing the response “Are there any?” when she talked with people about her research on Black mathematicians. On my blog Roots of Unity, I just published a Q&A with Dr. Walker about the book. She helped me compile some resources about Black mathematicians so you can spare yourself the embarrassment of asking the question “Are there any?” the next time you’re talking with Dr. Walker! These resources include information about Black mathematicians throughout history as well as sites and organizations with opportunities for Black mathematicians and students today.

Probably the first place that comes to mind if you’ve ever gone looking for information about Black mathematicians is the Mathematicians of the African Diaspora page maintained by Scott Williams. In addition to profiles of hundreds of Black mathematicians, the site has several articles about the history of African and African American contributions to mathematics. In a similar vein, the Mathematical Association of America SUMMA (strengthening underrepresented minority mathematics achievement) program maintains an archive of biographies of mathematicians from underrepresented groups.

There are Black mathematicians sprinkled throughout other general math history sites. I particularly find the MacTutor archive and Agnes Scott College collection of biographies about women in mathematics useful. Grandma Got STEM, a blog about older women in STEM fields, has several entries about Black women, most recently Della Bell, a mathematician at Texas Southern University.

Of course, I must recommend Walker’s book Beyond Banneker: Black Mathematicians and the Paths to Excellence as a resource for learning about the experiences of Black mathematicians (If you don’t have the book yet, you can listen to her talk about her work on YouTube. If you’re a cheapskate like me, check out your local library. I have an n=1 study that shows a 100% success rate in convincing your library to buy it if you ask for it.) She suggested some of the references she used as she was writing her book, highlighting Pat Kenschaft’s work, James Donaldson’s chapter in A Century of Mathematics in America, and the recent paper “African-American Mathematicians and the Mathematical Association of America” (pdf) by Asamoah Nkwanta and Janet Barber.

I’ve written about Black mathematicians a few times on my blog. I wrote about Evelyn Boyd Granville twice (because mathematicians named Evelyn are great), and I’ve published interviews with Sudanese computer scientists Rasha Osman and African American mathematician Trachette Jackson. I recently saw another interview with Jackson for the Society for Mathematical Biology newsletter (pdf). Last year, number theorist Piper Harron’s thesis hit my math social media network like a flower-sprouting seed bomb, and I wrote about how it contrasts with other number theorists’ work. My co-blogger Anna mentioned it here as well.

Presh Talwalker wrote a post earlier this month about David Blackwell, the first Black tenured professor at Berkeley, and some of the work he did in game theory. I’ve also enjoyed recent press in the AMS Notices (pdf) and elsewhere about John Urschel, the Baltimore Ravens lineman and math nerd who is busting stereotypes about athletes and math. He just started graduate school at MIT in the offseason. And I’m ridiculously excited that there is a movie coming out in 2017 about Katherine Johnson, an African American mathematician who worked for NASA and helped get John Glenn into space.

The National Association of Mathematicians (NAM) and Conference for African American Researchers in Mathematical Sciences (CAARMS) are two good places to learn about opportunities for African American mathematicians and students. NAM often shares information about African American mathematicians on their facebook page.

Many educational organizations aim to encourage mathematics students from underrepresented groups. One that is near and dear to my heart because I used to live in Houston is the Cougars and Houston Area Math Program (CHAMP) at the University of Houston. Director Mark Tomforde has a guest post about it today at

As it is Black History Month, I have also seen resources about African Americans in other STEM fields. D.N. Lee, a biologist who writes on the Scientific American blog network, wrote a post about decolonizing STEM earlier this month. I also follow the @BlackandSTEM Twitter account and enjoy perusing the History Makers website, which has an excellent collection of video interviews with prominent African Americans in a wide variety of fields.

Do you know of other resources that should be added to this list? Have you written about Black mathematicians on your blog? Please share in the comments!

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