Regression, Twitter, and #Ferguson

Emma Pierson's analysis of hashtags that have appeared in tweets about Ferguson, Missouri. Click for the image on her website. Image: Emma Pierson.

Emma Pierson’s analysis of hashtags that have appeared in tweets about Ferguson, Missouri. Click for more information on her website. Image: Emma Pierson.

Like many people, I have been following news about the events in Ferguson, Missouri with shock and sorrow for almost two weeks. I have been following these events as a human, not as a mathematician. But there’s a mathematical side to this story, too. I’m not just talking about the statistics on how many people are killed by the police each year (which we don’t even know for sure) and the racial composition of the Ferguson police force versus the people they stop and arrest, although those are both important. I’m talking about Twitter. It’s been a crucial part of how the Ferguson story has become international news, but it’s also a useful source of data about how people are responding to the tragedy.

Emma Pierson is a computational biologist currently working for 23andMe, and her blog, Obsession with Regression, focuses on data analysis, often with Twitter’s data. She writes,

“I am very excited about Twitter because it combines two qualities.

“1. People actually use it. Famous people — it’s become standard for celebrities to say “Follow me on Twitter!” — and more importantly, lots of people.

“2. It makes massive amounts of data available in a way you can process with a computer. 500,000,000 tweets are sent every day and Twitter will give you up to 1% of those. And if I know what 1% I want — for example, only Tweets containing the word “Spock” — it will give me all of them, which means I can actually hear everything that’s being said on a topic by millions of people worldwide. And not just what’s being said, but who’s saying it — how they describe themselves, where they live, who their friends are, and the last few thousand things they said.”

She has been blogging about Twitter data since December 2013, when 23andMe was ordered to stop providing disease risk information to their customers. She wrote a post about who was reacting to the news on Twitter and how they felt about it. Of course, being an employee of the company represents an obvious potential source of bias, so she also included a link to the tweets she analyzed so others could study them. She’s done several other interesting data analyses as well. Earlier this summer, she wrote an interesting analysis of tweets about LeBron James’ most recent career move, and of personal interest to me is her post about gender in the symphony. (Her analysis seems to match my experience. In my four years in the orchestra in college, I think we only had two men in the viola section.)

On Tuesday, Pierson wrote a post about using Twitter to study people’s reactions to current events, focusing on Ferguson. She mined a few hundred thousand tweets about Ferguson and analyzed the diferent hashtags that appeared in tweets with #Ferguson. (Part of the visualization she made is at the top of this post.) She also put her mineTweets program up on Github so others can use it to collect tweets about any topic in real time. She has some ideas for further analysis, particularly about whether the day/night-peace/violence pattern is apparent in tweets, and she’s invited others to contribute either ideas or analyses of their own.

The events in Ferguson have also highlighted the difference between the way Twitter and Facebook work. I’m not the only one whose Twitter feed has been saturated with #Ferguson, while Facebook has been nearly silent on the topic. In a Medium article, Zeynep Tufekci explains how Facebook’s algorithm for deciding what to show us caused this discrepancy and wonders what would have happened to Ferguson without Twitter. “It’s a clear example why net neutrality is a human rights issue; a free speech issue; and an issue of the voiceless being heard, on their own terms,” she writes. “Algorithms have consequences.”

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Medaling Mathematicians

You may consider the Fields Medal a boon to the mathematical community as it showcases amazing young mathematicians and brings math into the limelight. Or you may view the Fields Medal as an unfortunate reinforcement of the notion that mathematics is the work of lone geniuses. Whichever the case, you can blog about it, and you’ll be in good company.

From left to right :, Subhash, Khot, Martin Hairer, Manjul Bhargava,  the South Korean President Park Geun-Hye , Maryam Mirzakhani , the President of the International Mathematical Union (IMU) Ingrid Daubechies, and Artur Avila.

From left to right: Dr. Martin Hairer, Dr. Manjul Bhargava, the South Korean President Park Geun-Hye , Dr. Maryam Mirzakhani , the President of the International Mathematical Union (IMU) Dr. Ingrid Daubechies, and Dr. Artur Avila. Photos Courtesy of Dr. Alina Bucur.

The four winners of the Fields medal are pictured above at the International Congress of Mathematics (ICM).  Hands down, the best summaries of these winners’ contributions as well as the contributions of the Nevalina Prize winner were written by Erica Klarreich and Natalie Wolchover at Quanta Magazine. The accompanying videos provided at the Simon’s Center site are great for showing in a classroom. It’s particularly inspiring to me to see Maryam Mirzakhani drawing diagrams of manifolds on giant pieces of paper in her living room and talking about how her three-year-old daughter probably thinks she’s an artist. Also very interesting is the non-academic background of Artur Avila, who is Brazilian, and likes to think about math problems while he walks down the beach. Short videos like these will likely prove quite inspirational to young kids.

Lucky for you and me, I happen to subscribe to the email list of Women In Number Theory, on which Alina Bucur, a Number Theorist from UCSD, posted some great photos of the ceremony that she agreed to share. The beaming smiles of the recipients are much more lively to me than the official pictures making the rounds. In particular, I like these of Mirkazhani and Bhargava with Ingrid Daubechies, the inventor of wavelets and president of the IMU.


Because of the Fields Medal ceremonies, there were a few unexpected corners of the web where math showed up this week that actually described the mathematical accomplishments in some detail. Some of the basic ideas behind winner Maryam Mirzakhani’s research were covered in a Business Insider Article and an inforgraphic at a tutoring website (matific).

It was particularly interesting to read the comments on the BI article, in which a large portion of the discussion was based around whether a practical application of her work existed, or whether it mattered.

The Aperiodical blog’s great round-up pointed out that even popular outlets like Elle and Jezebel covered the medal this year. Four posts that I’d like to highlight that have recently come out are:

Jordan Ellenberg’s Slate piece, Math is Getting Dynamic, details the rise of Dynamics, the field of two of the young winners – Maryam Mirkhazani and Avila. He does a great job of making this field relatable and approachable.

Cathy O’Neil at mathbabe expresses her dismay at the tendency of awards such as the Field’s medal to paint a picture of mathematics as non-collaborative by awarding individuals rather than groups.

Izabella Laba’s short post expresses her feelings about attending the awards ceremony.

Mike Lawler’s reflection on how Mirzakhani’s high school experience made him reflect on his own great teachers and mentors.

In closing, it’s disappointing to all of those suave mathematicians out there I’m sure, but Nobel didn’t leave math out of his prizes because his wife slept with a mathematician.  As Evelyn points out in her recent post about how to talk about the Fields Medal at cocktail parties, the inventor of dynamite wasn’t even married.


Posted in Events, Math Education, Mathematics and the Arts, Number Theory, people in math, Theoretical Mathematics, Uncategorized, women in math | Tagged , , , | Leave a comment

Alias, Schmalias

While the great line from Romeo and Juliet: “a rose by any other name would smell as sweet” rings true, would a digital rose smell as sweet?  We often think of the digital world as a mere “renaming” of the real world.  But some interesting effects emerge from digitizing, and one of them is commonly known as aliasing.

If you’ve danced in a club with strobe lights, you’ve laughed at the slow motion effect that results from your eyes interpolating between positions. Sampling data can be thought of in much the same way. As a young child I was in love with both dance and mathematics, and I recall my uncle describing to me a David Parson’s piece involving a strobe light. The piece, Caught, involves a man appearing to fly through the air as a strobe light flashes on at just the right moments in a progression of jumps.

While I certainly didn’t think of it this way at the time, the choreographer Parsons was sampling his dancing at the same rate that he jumped. In this way he could appear to be at the same height above the ground (i.e. floating/flying) at every moment. If we think of Parsons (who it doesn’t hurt to think of since he was pretty gorgeous) moving up and down over a sine wave as he jumps, then we are simply sampling at a rate of exactly once per period and essentially leaving out the information that he ever touches the ground. So what made me think of all this again?

Mathalicious’s blog! While Mathalicious’s lessons require a subscription to view, the blog (and a few sample lessons) are free, and the most recent lesson entitled “Spinning your Wheels” is about aliasing, the same phenomenon mentioned above, in which we sample less frequently than is necessary to faithfully reproduce the information (movement) that is occurring. Mathalicious explains the “Wagon Wheel effect” which results in a car’s wheels appearing to stop moving or to spin backwards even as the car (or wagon) is moving forwards.

Chris Lusto's illustration of how a wheel that is spinning can appear still if it's movement is captured at a rate that coincides with it's turning through an angle of rotational symmetry.

Chris Lusto’s illustration of how a wheel that is spinning at a rate of 72 degrees per frame may appear still if consecutive frames are as shown above (where the green spoke would not be colored green in the film).  Any object that undergoes a symmetry in the time between successive frames will appear stationary.

A nice point made at the beginning of the post is that this effect is due completely to the nature of digital media in which the image is sampled. In the same way that we interpolate position when strobe lights are flashing, we interpolate the movement of the wheel as pixels on a screen change color. While a car wheel doesn’t have spokes, its hub caps often have the typical five-fold symmetry described in the blog post, and Chris (the blogger) has put in some great animation widgets to help the reader understand this phenomenon without ever mentioning Nyquist, sampling rate, or the word aliasing. While I applaud his explanation, I also think it would be worth plotting the sine wave traced out by one point on the wheel.  Then the typical definition of aliasing would be more directly connected to this wonderful example. Another great post of Chris’s on the Mathalcious blog that is in the same Digital Signal Processing vein is the Siren Song which illuminates the Doppler Effect and even addresses what happens if the siren travels at more than the speed of light!

Likewise, there are spatial examples of aliasing such as the Moire patterns that emerge from digital photographs of objects that have periodic patterns (like brick walls and textured clothes).

Taken from a 2012 gizmag post on super-resolution. Photo by C. Burnett.

Taken from a 2012 gizmag post on super-resolution. Photo by C. Burnett.

Also aliasing manifests in music as what we commonly call distortion when higher pitches are aliased down to lower ones. Any analysis of a cyclic phenomenon can be colored by the affects of aliasing. I happened upon the arxiv paper “Could sampling make Hares eat Lynxes?” which discusses the potential of aliasing as an explanation for misinterpretations of cyclic behaviors of populations in the context of the Lotke-Volterra predator-Prey model. What examples of aliasing have you experienced or found interesting?

Hares high-fiving after eating a lynx? :)

Hares high-fiving after eating a lynx? :)


Posted in Applied Math, K-12 Mathematics, Math Education, Mathematics and Computing, Recreational Mathematics | Tagged , , , , , , , , , , | 1 Comment

The Funny Pages

Ah, summer! Sleeping in, reading fiction, traveling, and, of course, preparing for fall classes. I’ll be teaching a math history class, which will be fun but is entirely new to me. As I cling to the last few weeks of freedom before the semester starts, when I have the luxury of prepping at a nice leisurely pace, I sometimes find my browser wandering from the Convergence website, with its useful articles about teaching mathematics using history, to a couple of my favorite funny math blogs.

Math Prof 4 Life illustrates the glamorous life of a math professor with the finest animated gifs that imgur can provide. From students who don’t understand how exams work to students who nail a proof at the board, from Joint Meetings exhibit hall candy to interminable committee meetings to those pesky problems you just can’t quit, the author has an appropriate gif for all sorts of awkward and awesome academic occasions.

By the way, undergraduate abstract algebra professor, I’m sorry for that time I thought all finite groups were abelian. Now I know that this is how you felt inside.

Math Professor Quotes is a dangerous blog to visit if you teach math and think you’re funny. (Guilty as charged.) Because you might be disappointed when you refresh it after class and find that the students surreptitiously using their phones under the desk weren’t submitting your wit to the blog for posterity. But maybe you’ll find a few quotes that will spice up your lectures. (Because recycled jokes never feel forced!) Here are a few of my favorites from the site.

“Using the chain rule is like peeling an onion: you have to deal with each layer at a time, and if it is too big you will start crying.”
“Differentiating is like squeezing toothpaste out of the tube. Integrating is like putting the toothpaste back into the tube.”
“There are precisely as many numbers between zero and one as there are between zero and two. #thefaultinourℝ”
And especially for my math 3220 students from last year: “Heine-Borel is the kind of theorem that is essential for your life. I mean, you can handle doing grocery shopping without Bolzano-Weierstrass, but you would never succeed without Heine-Borel.”

Do you have any favorite funny math blogs?

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Alan Turing on Stage and Screen

A statue of Alan Turing at Bletchley Park, by artist Stephan Kettle. Photograph: Sjoerd Ferwerda, via Wikimedia Commons.

A statue of Alan Turing at Bletchley Park, by artist Stephan Kettle. Photograph: Sjoerd Ferwerda, via Wikimedia Commons.

It was a big week for Alan Turing dramatizations. On Monday, the UK and US trailers for the Turing biopic The Imitation Game, starring Benedict Cumberbatch, were released, and on Wednesday, the 102nd anniversary of Turing’s birth, the Pet Shop Boys premiered A Man from the Future, an opera they wrote about Alan Turing, at the BBC Proms. Both works are based on Andrew Hodges’ biography Alan Turing: The Enigma.

I heard about the movie first from Christian Perfect at the Aperiodical. It seems to focus on Turing’s contributions to cracking the enigma code. Here’s the US trailer.

And here’s the UK trailer.

This morning, James Grime, a Turing and Enigma machine expert, wrote about the trailers, also for the Aperiodical. Grime has made several videos about Turing and the Enigma machine himself (see here, here, and here). His post is quite thorough, and it will help me go into the movie with a little better idea of what is factual and what is embellished. Grime’s review is largely positive. He writes, “Sure there are inaccuracies, but I think that is forgivable in a dramatisation of events. I think the film will actually inspire people to find out more about Turing, Enigma, and the work at Bletchley Park.”

Jeffrey Bloomer wrote on Slate’s Outward blog that the trailer was “disgraceful” in the way it failed to portray Turing’s sexuality, saying it, “frames the movie as a wartime epic and romance between Turing and his contemporary, Joan Clarke (Keira Knightley). For viewers in-the-know about Turing’s sexuality, there are some coy allusions to what’s really going on (‘What if I don’t fancy her in that way?’). But it’s not long before we’re back to tender scenes of the photogenic couple in duress.”

I felt like this criticism was overblown. First, I think Turing’s homosexuality is better-known than Bloomer realizes, making the allusion not so coy to a large number of viewers, particularly in the UK. (I could be wrong about that, of course.) Second, while I hope the movie doesn’t “straightwash” Turing, I also hope his accomplishments, not his sexuality and persecution, are the focus of the film. In a two-minute trailer, I thought the amount of time spent alluding to his relationships sexuality was about right. And while his relationship with Clarke is probably overstated, the two were engaged for a short time, and we can hardly expect a major movie not to exploit that fact a bit. I’m curious how much the script has changed since Hodges criticized it on this matter last year.

I’m more concerned with another inaccuracy the film, pointed out by Grime: “The film seems to be setting up [Commander] Denniston as an antagonist to Turing, which is probably a great disservice to Denniston, who by all accounts understood the difficultly of the work, deliberately recruiting the professor type, and was proud of their achievements.” Although Grime says that others in the military may have been less supportive of the Bletchley Park work than Denniston, this seems like a fairly big problem. It changes the way I will watch the film more than the embellished romance with Clarke.

A Man from the Future, on the other hand, focuses as much on Turing’s sexuality as it does on his science. The opera is not nearly as plot-oriented as a movie. It features snippets of narration drawn largely from Hodges’ biography interwoven with choral songs. Bits of Morse code bubble up here and there in the music. When work on the opera began, the Queen had not yet pardoned Turing for his so-called “gross indecency.” At the time, Neil Tennant, half of the Pet Shop Boys, was quoted as saying, “Of course the reason they won’t pardon Alan Turing is because they’d have to pardon all those homosexual men.” Now Turing has been pardoned, but the others have not been. The end of the opera notes, “an exception was made. The convictions for gross indecency of tens of thousands of other men, dead, and alive, remain unpardoned.”

You can listen to the BBC Proms concert online for the next 29 days. A Man from the Future starts around the 37 minute mark. The Imitation Game will be showing up at some film festivals early this fall, but most of us will have to wait until November to see it. I’m looking forward to it!

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Mathematician Presents Flawed Proof – in a work of fiction

Following Evelyn’s last post about the new Breakthrough Prize in Mathematics, I will now discuss the opposite of wild mathematical success.

Depending on how excited you are about public speaking, the moments before giving a talk at a math conference may be full of anticipation or anxiety.  So what happens when the speaker says something incorrect? At best, it’s embarrassing – like messing up in the middle of a recital in front of other musicians who know that music. What if the speaker presents an argument that is somehow fundamentally flawed? We talk a lot about how to handle students’ mistakes and how valuable they are – see Evelyn’s February blog post. But what about colleagues’ mistakes? How can we take it ins stride when we make mistakes in front of peers?

That’s one topic of the entertaining short story “The Penultimate Conjecture” by the late celebrated writer Leonard Michaels. One of a series of seven stories featuring fictional mathematician named Nachman, this story is read by Rebecca Curtis at the New Yorker’s monthly podcast posted on the first of this month. In just 45 minutes listeners can soak up some literature with a mathematical flavor and a dark sense of humor. According to Alex Kasman, who maintains the site Mathematical Fiction, this is the most mathematical of all of Michaels’ stories about Nachman, who, after attending fellow mathematician Lindquist’s much-anticipated presentation of the proof of the long-outstanding Penultimate Conjecture, realizes that the presenter’s proof is flawed. Should he say something to Lindquist? It’s not clear what he will do, and we share his uneasiness at being the messenger of bad news. Kasman files all the Nachman stories under the “anti-social mathematician” banner, but unlike the clichés in other stories about mathematicians, this one seems more true-to-life to me. Like the New Yorker’s fiction editor who chats with Ms. Curtis at the end of the reading, I am interested to know what the Ultimate Conjecture might be, and I tend to agree with this blogger that it is probably meant to be that of Nachman. However, one thing that isn’t discussed during the New Yorker podcast is the remaining possibility that Lindquist’s work will be fruitful in other ways besides what he aimed. My thoughts turn to the Math Overflow post concerning the Most Interesting Mistakes in Mathematics , in which many fascinating examples of mistakes made by preeminent mathematicians led to innovations. The most recent example mentioned concerns the Perko Pair, a pair of knots once thought to be non-isomorphic (due to a theorem that was later disproven), and later shown to be isomorphic (by a lawyer named Perko).

Most of us lack preeminence, and for the pessimists among us, the quality of famous mathematicians’ mistakes might just be a reason to save the contents of every Field’s medalist’s wastebasket. But for the optimists among us, it’s also encouraging to think that exercising ones intellect is bound to be fruitful even if it’s not in the manner intended. An interesting side note – in an effort to find a discussion about how mistakes during talks should be handled, I found pretty much nothing, which reinforces my belief that mathematicians are a very polite (or perhaps just confrontation-averse) bunch. I did, however, find a post on Math Overflow about how to correct mistakes in published work. Of course, step one is always to email the author. I also ran into this recent nice post by Orr Shalit at his blog Noncommutative Analysis, in which Shalit discusses how he handled the discovery of a mistake in a 16-year-old paper.  What are your thoughts on mistakes in our field? On mathematical literature?

Posted in History of Mathematics, Mathematics and the Arts, people in math, Publishing in Math, Recreational Mathematics, Theoretical Mathematics | Tagged , , , , , | 4 Comments

The Inaugural Breakthrough Prizes in Mathematics

Image: Julie Rybarczyk, via flickr.

Image: Julie Rybarczyk, via flickr.

Last month, the inaugural Breakthrough Prizes in mathematics, founded and partially funded by internet billionaires Yuri Milner and Mark Zuckerberg, were awarded to five people: Simon Donaldson, Maxim Kontsevich, Jacob Lurie, Terence Tao, and Richard Taylor. The prize is $3 million per person, and the first five winners will be on the committee for the selection of future winners. (In the future, there will be only one prize awarded per year.)

I was a bit surprised that there hasn’t been much talk on blogs about the prizes, but there has been a bit. Peter Woit wrote about the prize on Not Even Wrong, and the comments to his post are interesting. “Shecky Riemann” also has a post on Math-Frolic.

I must admit that I am somewhat cynical about the prize. (Now might be a good time to reiterate the disclaimer that appears on the sidebar of this blog: my opinions do not necessarily reflect the opinions of the AMS.) The five winners are all productive, brilliant mathematicians who have enhanced their fields immensely, and they deserve to be recognized. But $3 million is just so much money! It’s hard for me to see how concentrating that much money in the hands of so few people is an efficient way to support mathematics.

Woit’s post voices some similar concerns. He writes,

“…it’s still debatable whether this is a good way to encourage mathematics research. The people chosen are already among the most highly rewarded in the subject, with all of them having very well-paid positions with few responsibilities beyond their research, as well as access to funding of research expenses. The argument for the prize is mainly that these sums of money will help make great mathematicians celebrities, and encourage the young to want to be like them. I can see this argument and why some people find it compelling. Personally though, I think our society in general and academia in particular is already suffering a great deal as it becomes more and more of a winner-take-all, celebrity-obsessed culture, with ever greater disparities in wealth, and this sort of prize just makes that worse. It’s encouraging to see that most of the prize winners have already announced intentions to redirect some of the prize moneys for a wider benefit to others and the rest of the field.”

In fact, the New York Times reports that Tao, one of the winners, has similar feelings:

“Dr. Tao tried to talk Mr. Milner out of it, and suggested that more prizes of smaller amounts might be more effective in supporting mathematics. ‘The size of the award, I think it’s ridiculous,’ he said. ‘I didn’t feel I was the most qualified for this prize.’

“But Dr. Tao added: ‘It’s his money. He can do whatever he wants with it.’

“Dr. Tao said he might use some of the prize money to help set up open-access mathematics journals, which would be available free to anyone, or for large-scale collaborative online efforts to solve important problems.

As a young academic who has seen postdoc positions seem to dry up since the beginning of the financial crisis, I can’t help but do a little arithmetic. $50,000 is a nice round salary for a postdoc. Before benefits are factored in, that makes each $3 million prize the equivalent of 60 postdoc years. Even if we add another $50,000 a year for health insurance, travel, and other research expenses, that money could fund 30 postdocs a year, or create 10 three-year postdoc positions each year.

But 30 postdocs a year wouldn’t make a good press release. The New York Times wouldn’t write an article about their multimillion dollar minds. And the funders of the Breakthrough Prize want to encourage mathematical celebrity, which supposedly will lead to public awareness, not to fund worthwhile math research in the most efficient way possible. In a Scientific American article about the prize, Ben Fogelson writes,

“Milner’s goal, however, is to increase the popularity of science by celebrating the scientists. ‘Dividing [money] in small pieces and distributing it widely has been tried before and it works,’ Milner says. ‘I think the idea behind this initiative is to really focus on raising public awareness.’”

A commenter on Woit’s post suggested that each year, the prize money could be used to endow a research position at a university, noting that at MIT, you can endow a professorship for $3 million. Would that be high-profile enough? I think you could still write a press release about it!

I had some interesting discussions about the prize on Twitter after the prizewinners were announced, mainly focused on the utility of mathematical celebrity. Those discussions helped me frame a few questions about celebrity and public awareness. I’ve tried to figure out some analogous questions about movies and the Oscars because the Breakthrough Prizes have been described as the Oscars of science.

  • Will people think mathematics is more valuable because a few people can earn giant prizes from it? (Do people think filmmaking is more valuable because the Oscars exist?)
  • Will people want to become mathematicians because they think they could earn a big prize from it? (Do people become actors or filmmakers because they think they could win an Oscar?)
  • If the ultimate goal of the prize is to raise public awareness of math, what is a more effective way to do that: tell them about a successful mathematician, or tell them about an idea in math? (If someone doesn’t know much about cinema, would it be more effective to tell them about an Oscar-winning actor or show them a movie?)
  • Are these even the right questions and analogies?

This post might sound like I’m saying, “I don’t like this new prize because I’m never going to get it, but I would like it if it funded people more like me.” But I don’t think I’m quite there either. I have reservations about the suggested alternate uses of the prize money as well, thanks largely to two posts by Cathy O’Neil about billionaire money in mathematics and in academia in general.

On a lighter note, if you are a mathematician who is a bit embarrassed about a recent windfall, Persiflage suggests that “a bottle of Chateau d’Yquem 1967 does wonders to wash away any last remaining vestiges of embarrassment…”

Posted in people in math | Tagged , , , , , , , , , , , , | 6 Comments

Visualize Your Algorithms

As a college student in the ‘90’s with a penchant for “visual learning” I was never drawn to computer science. My one computer science class focused mostly on syntax and basic logic. Had shuffling and sorting been presented as eye-catching animations and colorful braids, I might have taken more CS classes. Mike Bostock presents such eye-catching images, specifically concerning four different goals: sampling of images, shuffling, sorting, and maze generation. For each goal, Bostock chooses several algorithms and explains how they each work visually. Even a brief look at the visualizations, some of which are animated and some static, gives a sense of the differences between the algorithms. As Bostock points out, visualizing algorithms is entertaining, makes understanding the algorithm more intuitive to others, helps in debugging your algorithm, and deepens your own personal understanding. In short, Bostock wants you to visualize not just data, but the algorithms used to analyze the data.  From a mathematical perspective, this is similar to insisting that we find ways to visualize the proof mechanism itself and not just the concept that it justifies.  In other words, we can draw a right triangle with appropriate labeling and call that a visualization of the pythagorean theorem, but it’s not nearly as instructive as this short animation or this geogebra applet.

Sampling is as ubiquitous in digital computing as the Pythagorean theorem is in geometry.  And while it is easy to understand as a concept, sampling is difficult to execute well. Bostock takes as an example an impressionistic painting. How can we choose samples in a way that will best digitally reproduce the original image? It turns out that we can learn form our own biology; the eye’s photoreceptors “sample” the world around us and are distributed in an irregular but uniformly dense manner. Such a sampling helps combat aliasing, a phenomenon whereby regularly spaced samples distort data with high levels of periodicity (Imagine a picture of a striped sweater that is sampled at a resolution lower than the spacing between stripes). This is where Poisson Disk sampling comes in. It is computationally more complex than other options, but produces a uniformly dense yet random sampling. As Bridson’s algorithm for Poisson Disk sampling plays, you immediately notice how the samples emanate from an initial seed rather than populating the entire sample space over time. This is one example of how visualization can helps us distinguish between algorithms.

Bostock works for the New York Times, and has many wonderful examples and posts on his site. Being a delinquent topologist, I was drawn to “How To Infer Topology”. Bostock is very interested in Cartography, and this leads to his writing about how to think about the data in a map more efficiently using a topological perspective. Several animations break down how one might encode a map by recording a sequence of arcs and special intersection points.

After a bit of perusal I realized that I had seen some of the work of one of Bostock’s collaborators, Jason Davies on Math Munch’s page in the past. Looking over Davies’ site, I can’t help but recommend that you take a look at his simple but beautiful graphic of a tiling of the Poincare disc. Zoom in and you can see the upper half plane model of the hyperbolic plane!

Posted in Applied Math, Math Education, Mathematics and Computing, Mathematics and the Arts | Tagged , , , , , | 1 Comment

The Human Side of Computer Science

Charles Dick-Kens, the inspiration for Dick Lipton and Ken Regan's joint alter ego, Pip. Image: public domain, via Wikimedia commons.

Charles Dickens, the inspiration for Dick Lipton and Ken Regan’s joint alter ego, Pip. Image: public domain, via Wikimedia commons.

Dick Lipton is a computer science professor at Georgia Tech who thinks P=NP, and Ken Regan is a computer science professor at the University of Buffalo who thinks P≠NP. Together, they are “Pip,” a Dick-Kens character.

Today I want to tell you about their blog, Gödel’s Lost Letter and P=NP.

Lipton and Regan write mainly about computer science, mathematics, and the history of those subjects, but they always put people first. Most posts start with a brief bio sketch of the person or people whose research they discuss, followed by a little background on the question they’re writing about and the technical content. In the “About” page, Lipton writes, “my real reason for making “who” central is to explain what researchers do when they work on hard open problems. They make mistakes, they go down dead-ends, they make progress. Also students may not know, but even experts sometimes forget, that ideas that we now see as easy were once unknown.”

The blog’s archives are extensive, and I’ve fallen down quite a few rabbit holes browsing through them. Rather than try to give any kind of a comprehensive overview, here’s an assorted (I dare not say “random”) collection of posts you might enjoy if you like creative, personal writing about some pretty technical topics.

I’m writing about this blog now because of how much I’ve enjoyed several of the most recent posts:
Avoiding Monsters and Non-Monsters, on Weierstrass’s “monsters” (continuous, non-differentiable functions) and some of their monstrous and non-monstrous friends
How to Avoid Leaving Clues, on how to have your memory and use it too
Is This a Proof? On the Weak Goldbach Conjecture and computer-aided proofs.
Proof Envy, on the theorems they wish they could use. “Call the phenomenon theorem envy: There are theorems that I have long known, but have never been able to use in any actual paper, in any proof of my own. I find this curiously unsatisfying, to know a cool result and yet be unable to use it in one of my papers.”
The Problem of Catching Chess Cheaters, on Regan’s recent work on the topic. Another good post on chess cheating is the Crown Game Affair.
In Praise of P=NP Proofs, on why reading P=NP proofs is a good idea, even if (so far) they’re all wrong.
I also stumbled on an entertaining post from 2010 about how well mathematicians have done at guessing whether conjectures are true.

Lipton and Regan suggested a few other posts to me, including some that are more computer-focused:
Perpetual Motion of the 21st Century, a double guest post from Gil Kalai, who does not think large-scale quantum computers are possible, and Aram Harrow, who does.
Quantum Chocolate Boxes, on using linear extensions to figure out whether your chocolate box is full of pralines or not.
Hedy Lamarr the Inventor, on amateurs in science and technology. In a similar vein, it takes guts to do research.
Finally, check out their Gödelinterviewseries, which gets more and more imaginative as it goes on.

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When Life Hands You Lemons, Make Fibonacci Lemonade

I’m so glad I found Andrea Hawksley’s blog earlier this year! Hawksley is a software developer, mathematical artist, co-founder of the Octahedral Group, an organization of Bay Area mathematical artists. She works on the eleVR project, where she helps make 3-D VR videos for viewing on a spherical viewer. (They have a blog, too. I haven’t browsed it much yet, but it deals with the geometry and neuroscience of creating VR videos. Cool!)

Hawksley posts about recreational math and the mathematical art she and others make. I am indebted to her for posting a template and instructions for making a 6-card ball with pyritohedral symmetry group out of playing cards. I made one while my students were taking their final exam last semester. Here’s my finished product:

A 6-card ball I made using Andrea Hawksley's template. Click for instructions. Image: Evelyn Lamb.

A 6-card ball I made using Andrea Hawksley’s template and instructions. Image: Evelyn Lamb.

The ball was her gift to attendees of the Gathering 4 Gardner conference, which she posted about here. If you’re like me, you’ll be swooning at the hair tie creations and the G4G gifts from other participants. You’ll also swoon over her other posts about mathematical art, particularly her “topological” origami.

Unukalhai, an origami sculpture in Andrea Hawksley's Star Polyhedra series. Image: Andrea Hawksley.

Unukalhai, an origami sculpture in Andrea Hawksley’s Star Polyhedra series. Image: Andrea Hawksley.

Two of my favorites are her posts on non-Euclidean chess. What happens if we design “chutes” between random squares on the board? Could we handicap better players to make more interesting games? How does a bishop move if we tile our hyperbolic chessboard with squares that meet six to a vertex? How does a rook move on a hyperbolic chessboard tiled with pentagons that meet four to a vertex? It’s interesting to think about the ways that the game would be different with these different choices. I’d love for someone to write a program that plays chess on these boards to figure out how the strategy changes as we change the board.

Finally, one of Hawksley’s most recent posts is on making Fibonacci lemonade, a layered drink that gets sweeter as you go down the glass, with lemon to sugar ratio gradually approximating the Golden ratio. As she writes, “This drink may be the world’s first tastable example of the relationship between the Fibonacci sequence and the golden ratio!” It’s not bad to look at, either.

Fibonacci lemonade. Image: Andrea Hawksley.

Fibonacci lemonade. Image: Andrea Hawksley.

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