Un-Junking your Charts

Junk Charts is a blog by Kaiser Fung, who describes himself as “the Web’s first data visualization critic.” People have been criticizing and prescribing solutions for misleading data visualization for a long time. (How to Lie With Statistics was first published in 1954, when a gallon of gas was 22 cents, a movie ticket was 70 cents, and the average new house was $10,250.00.) I don’t know whether Fung was literally the first to do it on the Web, but his blog has been around for over a decade and has an extensive archive of interesting posts for your perusal.

A found graph displays the density of cats in the vicinity of each part of the park bench. Credit: Evelyn Lamb

When I first saw the title Junk Charts, I assumed it would be a blog that pointed out and made fun of bizarre and misleading graphs and charts. That’s all good fun, but this blog generally takes a less adversarial approach. Fung often examines data visualizations that are pretty good and shows how he would make them even more effective. For example, a recent post shows his suggested tweaks for a Washington Post graphic about voter polarization. The original graphic isn’t ugly or misleading, but the new one makes certain statistics jump out more readily.

Kaiser Fung’s redesigned display of information about how the two parties have diverged in the past few decades. Credit: Kaiser Fung. CC BY-NC-SA 3.0

Some posts start with less successful original material, such as this post discussing a flawed chart about politician approval ratings. On Pi Day 2014, Fung started the #onelesspie initiative to replace pie charts with better charts. (Except when they are self-descriptive, pie charts are mostly bad. Embrace non-pies!) The #onelesspie posts in later years have been entertaining.

The only good pie chart is a self-descriptive one. Credit: Randall Munroe, xkcd. CC BY-NC 2.5.

I don’t have much experience creating data visualizations or working with statistics, so I’ve enjoyed the perspective Fung brings in Junk Charts. Synergistically, while I was writing this post, the Information Is Beautiful website unveiled their Information is Beautiful Awards longlist for this year, which has lots and lots of interesting and visually arresting data displays. I can use my gradually developing chart sense when public voting opens later this month.

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Hacking Cracking & Packing

The original gerrymander courtesy of Wikimedia Commons.

Sometimes the boundaries of voting districts can look really suspicious. If you’ve ever seen Illinois’ 4th Congressional District, you know what I mean. Sometimes there are good reasons for this; communities with common interests may want to vote together. But sometimes the reasons are bad; partisan politicians might be cracking and packing certain demographics. That is, cracking up certain demographic groups and scattering them through the districts and then packing all of the remainder into one often strangely shaped (perhaps dragon shaped?) district to minimize their votes. These are the classic tools of gerrymandering.

In the state of Wisconsin this practice has got particularly bad, and last week the supreme court heard oral arguments in the case of Gill v. Whitford. This case seeks to determine exactly how bad the partisan map rigging in Wisconsin is, and hopefully the outcome will be some sort of consensus on how to recognize and rectify the systematic disenfranchisement that comes with hardcore gerrymandering.

Jordan Ellenberg, a resident of Wisconsin, wrote about the case for The New York Times and explained why this might be of interest to us as mathematicians.

Over the summer, the Metric Geometry and Gerrymandering Group at Tufts University led by Moon Duchin, ran a summer camp where participants developed tools for detecting and understanding gerrymandering. As Duchin often points out, sometimes a weird looking district looks weird for a reason, so it’s important to find out why things look the way they do.

A few reasonable measures have been proposed. One being used in the Wisconsin case is called the Efficiency Gap, and it measures the net wasted votes as a share of the total votes in the state. Wasted votes are all votes cast for the losing candidate and all extra votes for the winning candidate, beyond what was needed to win. But as Olivia Watch illustrates in this graphic explainer for The Nib, efficiency gap can’t tell the whole story.

Another way to measure gerrymandering is to consider, in some systematic way, all possible redistricting schemes in a given state and compare them to what is being used. If the one being used is a significant outlier compared to the others, then it’s probably gerrymandered. A paper that recently appeared on the Arxiv uses this method to expose the badness of Wisconsin’s districting.

Recognizing and measuring gerrymandering is of course a totally different task from actually redrawing the lines in a fair and unbiased way. There are many schemes for this, but my favorite one to explain to strangers (and then watch them get all in a lather) is the shortest split line algorithm. It sort of disregards human interests, but hey, it makes really good looking maps!

This will not be the first time that math has been used to fairly distribute representation. Apparently Thomas Jefferson devised a sensible algorithm for assigning seats in congress by state. I imagine Jefferson might have something to contribute to our current discussion.

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The arXiv, Curated

The arXiv: a mathematician’s favorite preprint server and semiproductive procrastination enabler. Don’t get a morning newspaper? You can enjoy your breakfast over the arXiv submissions for your favorite area of math. Stuck on that lemma? Might as well surf on over and see if you missed any important breakthroughs in your field. The arXiv contains multitudes, and that’s exactly what the website arxivist.com aims to help you with.

Anton Lukyanenko, a mathematician at George Mason University, started arxivist while he was a postdoc at the University of Michigan to help people sort through the flood of arXiv submissions and find papers that might be of interest to them. There can be dozens of arXiv submissions every day in any given sub-area, and a mathematician (/physicist/computer scientist/quantitative biologist/etc.) who browses through a few of them can end up overwhelmed. He says he found himself checking for certain keywords and author names and wondered if the process could be automated. After setting it up for himself, he decided to work on making a public version of the website. “It was fun to do a hands-on project alongside with my much more theoretical research,” he wrote in an email.

To use arxivist, you sign in using your Google account and start rating papers by giving the site’s suggestions thumbs-up or thumbs-down. The more you rate, the better the site’s recommendations will become. If you’d like, you can sign up for a daily email with some suggestions for the day. The site is currently in beta, but it has worked smoothly for me so far.

I’ve had fun rating papers on arxivist and seeing the suggestions respond to my ratings, but my favorite part is probably the “arXiv catch of the day” from the arxivist Facebook page. As the name suggests, it’s a fun paper to peruse every day. If you enjoy commutative mathematical phrases as much as I do, “Quasihyperbolic geodesics are hyperbolic quasi-geodesics” should be right up your alley.

Lukyanenko says there are some improvements to the user interface and algorithms in the works, but right now he’s interested in building the number of users and hearing what features are most appealing to them.

“It’s been really cool seeing the arxivist spread,” says Lukyanenko. “It’s also been exciting (and a bit terrifying) to hear that some of my more senior colleagues are using the system.”

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Exploding Dots For Global Math Week

A man learns math by exploding dots. Image courtesy of Francisco Barberis via FlickrCC

If you hang around the #MTBoS long enough you can’t help but notice something called exploding dots. Today in a quite moment I took some time to dig in, and I am not disappointed.

Exploding dots is the focus project of Global Math Week, happening Oct. 10-17, 2017. Kind of like Hour of Code, the aim of the Global Math Project is to get hundreds of thousands of people all over the world doing math together at the same time. The architect behind exploding dots is MAA mathematician-at-large James Tanton who hosts G’day Math a blog full of problems, lessons ideas, and mathematical essays.

Today I watched this video of Tanton presenting exploding dots to a general audience. First off, he is a mesmerizing lecturer and the video is worth watching if just as pedagogy inspo. But on top of that, this thing he presents is just so much fun. The idea behind exploding dots is a simple visual representation of base 2, base 10, and eventually base x. The basic idea is that you fill a row of boxes with dots, and the dots represent 1, x, x2 and so on depending which box they lie in. If one box gets full, the dots explode, and move over to the next box. Using this representation, Tanton builds arithmetic from the ground up, starting with addition and multiplication, and then adding in subtraction and division — even polynomial long division!

I won’t say any more about the precise details, since it’s better to explain visually. Just go watch the video.

One thing I appreciate immensely about his presentation is that he’s very clear that the point of this isn’t to get answers (we can easily do multiplication on our iPhones), but rather it’s figuring out how to develop a system intuitively and rationally to make it do what you want.

The only thing that is just niggling my brain a little bit is the problem with convergence of power series. In the video, Tanton shows how easy it is to use his exploding dots method to write


giving an infinite power series representation to the rational function 1/(1-x). But this really gives the impression that you can put any number in for x and the equality holds, like for example, x=2. But this would give


in other words


which of course can’t be true, and I imagine it wouldn’t take long for a clever student to pick up on that. I understand that Tanton is presenting this as an elementary alternative to the usual presentation, which is great. But I’m just curious if there’s some obvious way in his construction to see when a function is actually going to be equal to its power series representation. I’m sure the internet will let me know.

You can sign up to be part of Global Math Project and check out all of the great lessons and resources for you, your students, your children, or any unsuspecting friends who are foolish enough to go to happy hour with you on October 10th.

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Michael Pershan’s Problem Problems

I have enjoyed math teacher Michael Pershan’s work for a long time. I follow him on Twitter, and I wrote about his website Math Mistakes a few years ago because, darn it, mistakes are interesting! A couple years ago, he started another blog, Teaching with Problems, at a URL I love, problemproblems.wordpress.com.

I never taught the elementary and middle grades Pershan teaches, and I’m out of the classroom altogether now, but I am always excited to see one of his posts in my blog feed. I find his writing extraordinarily thoughtful, and he is humble, passionate, and thorough in his posts. In a recent, slightly meta, post, he wrote about one reason that might be. Some math bloggers write quick posts that deal with one smaller idea at a time, and that’s great. But he prefers to “slowly, painstakingly, dutifully carve out posts.” He loves taking the writing seriously, and it means great, but not always frequent, posts from him. It also means you want to read them slowly and thoughtfully rather than skimming.

I had somehow missed the launch of Teaching with Problems, but I started reading it after finding — and being blown away by — this post about a student he calls Rachel. She is a smart kid who has a strong command of math concepts and a lot of trouble with basic arithmetic. I don’t want to try to summarize the post. Just go read it. Another 0f my favorites is this post about whether third graders think fractions are numbers. Point: “NO a fraction is not a number a fraction is only part of a number.” Counterpoint: “Fractions are a certain category of numbers because without numbers fractions would just be lines.” 

Last month, I spent a lot of time thinking about ancient Mesopotamian mathematics because researchers published a new paper about Plimpton 322, a tablet I was familiar with from my math history teaching days. (I wrote about why I don’t agree with their interpretation here.) So when I was looking through the problemproblems archive, I was happy to see that Pershan had coincidentally written about Plimpton 322 and Mesopotamian mathematics as a teaching tool in July. In his post, he writes about how some of the earlier mathematicians who studied Plimpton 322 and other tablets imposed their more algebraic view of mathematics onto the tablets in anachronistic ways and homes in on a dilemma of looking at ancient mathematics from the point of view of a modern math teacher: “The historical question is whether this mathematics would have been meaningful to the ancients. The pedagogical question is whether it could be meaningful to our students.” He ends the post pessimistic about whether the geometric Mesopotamian methods can help students with the algebraic concepts and notes that perhaps, “It’s only when you understand both that you can look back and see the connections between them.”

A week or so ago, the New Yorker shared one of their old articles, the one about how political science professor Andrew Hacker thinks math is about nothing, in a tweet. It caused a bit of a dustup in the math Twitter world, as it tends to. As high school math teacher Patrick Honner pointed out after attending a debate last year between Hacker and mathematician James Tanton, it’s frustrating that we’re listening to Hacker and not math teachers here. I was thinking about that as I read Pershan’s blog and thought about writing this post. When he writes about whether ancient Mesopotamian tablets can help teachers communicate the difference of squares method to students, he has a much more realistic understanding of what students can make of that than I do. If you’re looking for math teachers to listen to, he’s a great one to add to your list.

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

That Neural Net That Predicts Sexual Orientation

Image via Flickr CC courtesy of Stefano Mortellaro.

What does a computer see when it looks at a face? Image via Flickr CC courtesy of Stefano Mortellaro.

A neural network is one way to achieve machine learning. Modeled after the human brain, a neural net teaches a computer how to do some task by processing a huge set of training data. The data passes through the network training thousands of nodes how to react to future data of that type. Some machine learning can lead to interesting if hilarious results, some of which Evelyn blogged about earlier this year.

This week a more questionable use of neural nets hit the newsstands with the announcement of a Journal of Personality and Social Psychology publication, Deep neural networks are more accurate than humans at detecting sexual orientation from facial images, and I hope you don’t mind me saying that the mere existence of this technology is frightening in the utmost.

As their training data the authors used several thousand photos of men and women who self-identified as homosexual or heterosexual from an internet dating site. After the training, they found that their computer could correctly detect sexual orientation 81% of the time for men and 71% of the time for women. This was compared to a 61% accuracy for men and 54% accuracy for women when detected by humans employed by Amazon Mechanical Turk.

There are many reasons why these numbers could have emerged, several are summarized on the blog ScatterPlot in a guest post by the sociologist Greggor Mattson. My first thought was that it may well have something to do with the provenance of the training set. Jesse Bering blogged about a similar but low-tech version of this type of study for Scientific American several years ago. However, the authors seem to dispatch with this idea in Study 5 when they feed Facebook photos into the trained computer.

According to the authors, the success of the neural net may have something to do with the (their words) gender-atypical features of gay men and women. And this, they claim, has something to do with prenatal hormone exposure.

A good analysis of this (erroneous) claim and publication overall was given in a blog maintained by the professors of a (really incredible looking) University of Washington course Calling Bullshit: Data Reasoning for the Digital Age. In it, the authors claim that even if we assume the neural net was set up in a totally reasonable way, and that all algorithms are mathematically sound, it’s still easy to see that the conclusions the authors draw are (in their words) not parsimonious. I appreciate their willingness to treat the technical business as a black box and nevertheless analyze the good-sense of the findings. “black boxes should not be deterrents,” they argue, “one doesn’t need extensive technical training in order to think critically about even highly technical analyses.”

But the mathematical pith shouldn’t always be ignored, since it often it takes a bit of pulling apart of the apparatus to see where things go wrong. At the same time I am also sensitive to the fact that the mere presence of math can sometimes bully people into believing.

I just finished reading Weapons of Math Destruction, the book about dangerous algorithms by the blogger Cathy O’Neil of mathbabe.org. First of all, I can’t endorse this book strongly enough. But also, this book really hammers home the idea that we mustn’t just accept things on face value because they are rooted in math. The assumptions that go into programming an algorithm are just as biased and fallible as humans, and the way we interpret the outputs of algorithms (or neural nets in this case) also require some critical thought.

With all of this, it’s just sobering to recall that whether the conclusions are specious or not, the tool now exists. And in this year 2017 we should know enough to believe that even the most critically flawed tools of math can be used against us.

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Public Domain Math

Many pieces of mathematics — for example, simple geometric shapes and some mathematical formulas — are uncopyrightable or unpatentable. You can’t copyright a square or patent the area formula for a circle. Anyone can use them. But this post is not about the intricacies of patent or copyright law as they apply to mathematics, as fascinating as that can be. This is about different public domain math.

An image from Max Brückner’s 1900 book Vielecke und Vielflache: Theorie und Geschichte (Polygons and Polyhedra: Theory and History). Credit: Public domain, via Internet Archive

The Public Domain Review is a website and nonprofit project that highlights weird and wonderful work that is in the public domain. (The definition of public domain varies by country; Public Domain Review labels their posts with further information if necessary.) I love seeing their posts in my blog feed because they are so varied and interesting. Some are beautiful, some are strange, some are funny. And some are math.

C. H. Hinton used multicolored cubes to illustrate the tesseract. Credit: Public domain, via Internet Archive

For instance, this article by Jon Crabb looks into some interesting late nineteenth and early twentieth century writings on dimension. Edwin A. Abbott’s Flatland: A Romance of Many Dimensions is fairly familiar to mathematicians. It’s some of the most effective math communication I’ve ever read, and as a bonus, you get vicious satire of Victorian social structures. I was unaware of C. H. Hinton’s 1904 treatise The Fourth Dimension, which runs away with the idea of a fourth spatial dimension. But it turns out I had already interacted with him, or at least a piece of his legacy: he coined the term tesseract for the four-dimensional analogue of the cube (also called a four-dimensional hypercube). Hinton believed the fourth dimension had psychic as well as physical implications, and his ideas about the fourth dimension influenced artists and writers including Marcel Duchamp and Gertrude Stein. Hinton’s book uses colored cubes to visualize the many cubes in a tesseract. Hinton was married to Mary Ellen Boole, one of the remarkable daughters of George and Mary Everest Boole.

Then there’s one of my favorite examples of design meeting mathematics: the 1847 Oliver Byrne edition of the first six books of Euclid’s Elements. And Ernst Chladni’s figures illustrating the nodes of vibrating plates. And Étienne Léopold Trouvelot’s astronomical illustrations, including a beautiful depiction of a total solar eclipse.

Credit: Public domain, via New York Public Library

Browsing through the mathematics tag, it’s fun to see work from people with enduring legacies as well as some things that are a little out thereMathematics shows up in some unexpected places. You can try to find frieze and wallpaper groups in an 1863 book of French textile samples or identify the curves in a 1919 book of Japanese wave and ripple designs. You can learn the correct proportions for Buddha and Bodhisattva depictions from an eighteenth-century book from Nepal. You can take a peek at early twentieth century data visualization in the infographics W. E. B. DuBois and his students created depicting various facets of African American life.

The Public Domain Review publishes a lot more than just math and science. It’s a worthy addition to the blogroll for all the interesting artifacts it brings to light, from math and science to art and religion.

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Some Stories of Journals Behaving Badly

Turns out money can buy a lot of things, even an expression for π as an algebraic number. Image via flicker CC: reynermedia.

Hoax papers have a long and time-honored history. Ten years ago a group of students from MIT wrote a program that randomly generated totally nonsensical computer science papers. One of their bogus papers was accepted by a conference and it caused enough of an uproar to prompt IEEE to pull its sponsorship of the conference. But this wasn’t before Springer had already accepted 120 papers generated by the program.

Needless to say, this caused academics and the bloated body of academic publishing to take a long look at themselves and wonder what the heck was going on with their standards. It’s no secret that predatory publishers exist. These are publishing outfits that usually charge a high fee for publication, may (or may not) promise some sort of referee process, and often have names like “Journal for Advances of Algebra, Number Theory, Biology, Chemistry and Electrical Technology.” Sometimes they’re easy to spot, sometimes they aren’t. Sometimes they ask you to submit a pdf to a gmail address, but sometimes they don’t.

But as we saw above with the 120 papers retracted by the venerable publishing house, Springer, detrimental and predacious practices aren’t just restricted to the obviously pay-to-play scam journals.

The blog Math Scholar recently wrote an interesting essay about a slew of predatory journals that published papers (eight of them!) claiming that π=(14-√2))/4. The author rightly argues that this is a terrible thing. The author speaks of the “collapse of peer-review,” and I just need to express my own mildly divergent view at this point. The peer-review process has not collapsed; peer-review is still a very good and necessary thing. The problem is just that journals are promising peer-review and not doing it. It’s a bit like using nacho cheese Doritos as evidence against the declining quality of cheese. We can’t besmirch all cheese based on the fact that those promethean deities of food science over at Frito-Lay use the word “cheese” on their devil corn chips.

Fields Medalist turned blogger Tim Gowers is at the forefront of a movement to call out the predatory practices of the so-called legitimate journals. In a recent post Gowers makes the case for uprooting existing journals and replanting their entirety — editorial boards, content, reputation –in an open-source context leaving behind empty hulls called “zombie journals.” This move was recently carried out by the journal formerly known as the Journal of Algebraic Combinatorics.

Unsurprisingly, math is not the only field feeling leery of its publishers. The field of gender studies also saw an interesting dust-up this summer with the publication of a hoax article (ostensibly) about toxic masculinity followed up by a take-down of the field of gender studies and its publishers by the authors of the hoax. The whole saga is interesting in our context because it points to the blurry line between legitimate and illegitimate publishing. The blogger Ketan Joshi does a nice job teasing out some of the universal-yet-nuanced complications of this particular hoax. In particular, Joshi gets at the question: how damning should a hoax paper be to the publishes versus the field itself? It’s an interesting read.

And just for fun, in case you have any secret messages that you need to send, the same group from MIT that wrote the hoax paper generator also have a program that can encode a secret messages as a bogus spam conference announcement. And that, my dear most esteemed sir/madam, is certainly one way to guarantee that nobody will be interested in reading your email.

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Back-to-School Blogs, 2017 Edition

Today, I’m taking my chances with traffic and driving up to Idaho to try to get in the path of eclipse totality. (Fun fact: according to my back-of-the-envelope calculations, if everyone in the country went to the path of totality, its population density would be a bit higher than that of Salt Lake City. Get more eclipse math from Bedtime Math and NASA. Don’t have eclipse glasses? Ask the past how to behold it safely.) While I’m on my sojourn, my spouse and millions of other teachers and students will be getting back to the classroom. Last year around this time, I posted a roundup of some of my favorite math education blogs. It’s a good list, and you should check it out.

Welcome back to school! Image: US Department of Education.

This year, I want to add a few more recommendations for good math resources for parents, teachers, and students.

Not awful and boring ideas for teaching statistics I’m a sucker for this blog name. Are you teaching statistics? Do you want some ideas that aren’t awful and boring? Here you go! The author, Jessica Hartnett, is a professor in the psychology department at Gannon University. She finds interesting data in the news and other places and gives her advice about how to use it in a statistics class. She’s recently posted about a Harry Potter sorting quiz, unpopular wedding songs, and the statistics that indicate that ride-sharing apps might curb drunk driving. (I’ve idly wondered about this myself while waiting for a ride after pub trivia night.)

Math Hombre John Golden is a math teacher and math education professor at Grand Valley State University. Follow his blog for posts about teaching teachers and making cool math art, games, and designs.

On This Day in Math Pat Ballew rounds up math-adjacent births, deaths, and events every day. I also enjoy the quotes he includes at the top of each post.

MatthewMaddux Education University of Saskatchewan math education professor Egan Chernoff compiles this feed of articles and blog posts related to math and math education. It helps me keep up with the latest news, especially in mainstream media outlets I don’t normally check for math stories.

inclusion/exclusion My co-blogger Anna Haensch wrote about this new AMS blog here a few months ago. It’s not specifically about teaching, but as math professors make up a lot of the intended audience, it discusses aspects of teaching that intersect with its mission of addressing diversity and inclusion in mathematics. In light of the heartbreaking, infuriating events in Charlottesville earlier this month, Brian Katz wrote about how to discuss justice on the first day of class.

Have other suggestions? Feel free to add them in the comments. If you’re a teacher, parent, or student, have a great beginning of the school year!

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With Profound Sadness

It was an incredible day in 2014 when Maryam Mirzakhani became the first woman to win the Fields Medal. I remember feeling absolutely overwhelmed with emotion and thinking to myself, alright, beginning today winning the Fields Medal is officially something that women do. It went from something impossible to something possible, just like that. It felt like the breaking of such a monumental glass ceiling, and like the opening up of this entire alternate universe of possibility. It was huge. At the time, Erica Klarreich wrote about Mirzakhani’s early life and work for Quanta Magazine.

It was with almost unbearable sadness that news of her passing broke last month. Mirzakhani died at only 40 years old, and the world became dimmer one shining star.

The weeks since have seen an outpouring of writing celebrating the life and work of Mirzakhani and mourning her death. Fellow Fields Medalist Terrence Tao shared a post the day after Mirzakhani’s passing, highlighting her contributions to the field and his own experiences in meeting her. Blogger and mathematician John Baez also wrote a thoughtful piece about Mirzakhani’s life and mathematics. The blog Mathsbyagirl featured a tribute post to Mirzakhani with links to expository articles on the key areas of Mirzakhani’s research.

Our own inclusion/exclusion blog posted about how Mirzakhani shone in all of her various roles, as mathematician, mother, trailblazer, and role model, with reflections on her life by various notable women in math. From Tai-Danae Bradley, “While reading through the many beautifully written tributes to Maryam, I am especially touched by one theme that pervades them all: her character. Words like persistent, determined, and resolute appear time and time again. And her humility and modesty seem to have garnered as much attention as her mathematical accomplishments.”

Mirzakhani working through ideas on a large sheet of butcher paper. This image is a screen shot from the Simon’s Foundation short film on Mirzakhani.

RAGE of the Blackboard, a blog exploring the bridge between scientists and artists, featured an illustration of Mirzakhani and spoke about the importance of drawing in her work. Mirzakhani often said in interviews that she enjoyed doing math by writing, drawing, and doodling on large pieces of butcher paper.

The AMS has collected a full list of tributes and obituaries for Mirzakhani, including words from AMS president Ken Ribet and a short video from the Simon’s Foundation.

This month has also seen the passing of several other notable women in math. Cathleen Morawetz, who did pioneering work related to airflow at supersonic speeds, died last week at the ago of 94. Earlier in July, Marina Ratner, who found acclaim later in life, died at the age of 78. Mathematician Amie Wilkinson wrote for the NYTimes about the shared aspects of Ratner’s and Mirzakhani’s works, despite very divergent lives. This month also saw the passing of Marjorie Rice at the age of 94. Rice was an amateur mathematician who made a big discoveries in the study of pentagonal tiling.

It is with tremendous sadness that we bid farewell to these mathematicians who inspired us and gave us so much. The impact of their work will surely continue to resound for many decades to come.

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