I was a twitter skeptic for a very long time. And now my love affair with twitter is so deep, it’s hard to even remember what my reservations were in the first place. I first joined in 2013, compelled by my hard-tweeting cohort of the AAAS Mass Media Fellowship. And what I discovered was the most pleasant online community of mathematicians and educators chatting all day about…well, math and education!

I get some of my best lesson plan ideas from Twitter, I learn about new math research papers, I get links to every possible math blog in the universe, and perhaps most relevant to you, I get all of my ideas for this blog. So I wanted to take this week to tell you about some of my favorite math tweeters and hashtags.

So first things first, a few hashtags to get you on your way.

This week the MAA hosted the #loveyourmath challenge. Tweeters posted on a different topic each day of the week, including, favorite math puzzles, proudest math moments, and mathematical arts and crafts. I didn’t find the challenge itself particularly inspiring, but if you’re new to the medium, it’s a good introduction to the mathematical twittersphere.

When I’m looking for fun problems and puzzles, I love to peruse the tweets on #mathchat. It tends to skew more towards K-12 teachers, but there are fantastic videos and links to technology and lesson ideas that I would never otherwise find.

Now on to people. At the very least, you can follow the big mathematical societies. I frequently check in with the AMS, the MAA and the AWM to see what they’ve found on the internet in a given week.

If you’ll forgive the blatant log rolling, my co-blogger Evelyn Lamb maintains a highly entertaining twitter account. She also wrote about her favorite mathy ladies on twitter for her blog at Scientific American several years about.

Mathematicians and tweeters John Allen Paulos and Marcus du Sautoy were listed among Science Magazine’s top 50 science stars of twitter. Other top mathematicians who tweet are Steven Strogatz, Laura Albert McLay, Matt Parker and Hannah Fry, to name just a few.

Some of my favorite math tweets over the years have also come from the collective hive mind behind The Aperiodical: Christian Perfect, Peter Rowlett, and Katie Steckles.

Finally, there are accounts that are dedicated to spitting out intermitted math facts just when you need them most. Like Algebra Fact, Prime Numbers and the not exactly mathematical but still fun in a mathy sort of way, Wolfram|Alpha Can’t.

So here are just a few ideas to get your started. Happy tweeting!

]]>When it comes to 3D printing, Henry Segerman and Laura Taalman are two of my inspirations. I wrote about Taalman’s MakerHome blog a couple years ago, and now she posts at Hacktastic, a blog about “design, math, and failure,” in her words. Segerman’s book Visualizing Mathematics with 3D Printing was published recently. I had the privilege of reading it early and blurbing it, and of course I recommend it. As a mathematician, of course I want to print mathematical objects, but something I appreciate about both of them is getting an idea of what 3D printing can do that other visualization media often can’t, like knots and links printed in place without seams and hinged negatively curved surfaces. People like Segerman, Taalman, and Mike Lawler, who has been basing some of his math lessons with his kids on their work, help me understand how I can put that 3D printer to good use if I ever decide to get one.

Another emerging technology is virtual reality. Recently I’ve been admiring the way Emily Eifler, Vi Hart, and Andrea Hawksley of eleVR are helping me think about how to play with virtual and augmented reality. I’ve never been particularly gung-ho about VR. It’s always felt like something for a certain type of tech geek or gamer whose interests and mine are not terribly aligned. But a series of blog posts and videos they’ve made recently have broadened the way I think about what you can do with VR.

They’ve posted about room makeovers, VR makeup, multi-person activities and games you can play with a VR headset and brush for drawing, and experiments in combining VR with real physics. They also have a guest post from artist Evelyn Eastmond about using VR to enhance the experience of meeting people online, and I enjoyed Eifler’s meditation on context in VR and the way you can use this technology to change and enhance people’s experiences in the world.

More immediately mathematically, they have teamed up with Segerman and Mike Stay to create Hyperbolic VR, which immerses the user in hyperbolic 3-space. I also recently saw a video and blog post about a VR app called Hypercube that allows the user to manipulate 3D shadows of 4D objects using VR equipment.

The eleVR team’s social VR experiments remind me a little of one of my favorite podcasts, Flash Forward by Rose Eveleth. In each episode she imagines a possible future scenario—meat is outlawed, the Earth acquires a second moon—and talks about what might actually happen to cause that scenario. On the last episode of the recently concluded second season, she had an algorithm write the script for the fictional future scenario part of the show and then tried to figure out what the scenario would be and how to interpret it. I’ve seen some funny stuff come from Markov chains and neural networks, but this was a new level of creativity and interactivity for the listener, and it got me thinking about how else the techniques could be used.

Thank you to the brave explorers who help the rest of us tap into our creativity!

]]>I recently became aware of the mathematical artist Lun-Yi London Tsai. Tsai has a master’s degree in math, and it is clear that he has studied a great deal of math in his life. His mathematical paintings and drawings are like a snapshot of chalkboard right at the end of a brilliantly delivered lecture. There’s a chaotic energy, a blend of math symbols as art, surfaces and shapes common to mathematics as well as the artists own interpretation of the concepts and processes.

I really like Tsai’s work because, as a mathematician, I always have these strange ideas in my head of what certain math concepts look like. Not the numbers and notation, but the actual carrying out of infinite processes. There’s such a beauty and order to it all. And I’m no artist, but I often find myself doodling tangent planes, manifolds and chain complexes when I’m sitting in meetings.

And there’s nothing quite like the gobsmacked feeling you get when you walk in on a chalkboard covered with really intense looking math. There are several tumblr dedicated to precisely this art, and in fact the photographer Alejandro Guijarro recently traveled the world photographing the chalkboards of quantum physicists for his art exhibit *Momentum*. And in a very cool way, I feel like Tsai’s art — particularly the charcoal drawings — recreate that sensation.

Here is Tsai in an excerpt from from an interview for Germany’s Jahr der Mathematik.

I don’t actually have a favorite number. When people find out I do art and math, they guess that my favorite number must be Phi, the golden section. But I actually don’t like that number much, because I think it limits people’s concept of what math and art is about—there’s so much more to math and art than just the golden ratio! I’ve only made one painting of this number and I called it “Goodbye, Golden Ratio.”

It’s always nice to recall that bridge between the theoretical and the visual, and I’m always excited to find new mathematical artists. Are there mathematical objects that you love to draw, professionally or just for fun? Are there mathematical processes for which you have a distinctly visual interpretation? Tweet it at me @extremefriday.

]]>Physicist Paul Ginsparg started arXiv while he was a researcher at Los Alamos National Laboratory, and now it lives at the Cornell University library, where Oya Rieger is the program director. In recent years arXiv has been concerned with funding and has launched a membership model where university libraries and the Simons Foundation (disclosure: I have written for the Simons Foundation) contribute to help maintain the site. (Perhaps surprisingly, arXiv does not receive funding from the NSF, even though it is a vital component of research for many mathematicians, physicists, and computer scientists.)

This year, in honor of its 25th anniversary and as part of an upcoming site overhaul, arXiv conducted a survey to see how they might better serve their users. The most divisive issue seemed to be comments sections. In an arXiv paper about the survey, Rieger writes that people were hesitant about the idea, and even people in favor noted that implementation of a good comment system would be tricky. Like Izabella Laba, I think internet comments sections are often not worth the pixels they’re printed with, and I share her concerns about women being subject to unnecessary scrutiny about their credentials or even abuse.

But on to more important things, by which I mean trivial questions about the scientific/mathematical vernacular. Earlier this summer I set up a highly non-scientific online survey to address a burning question: what do we call the preprint repository with the web address arXiv.org? The official arXiv preference is to refer to it without the definite article, but I usually hear people saying “the arXiv” when they talk about posting things there. I this post I’m following my policy of referring to people (or in this case entities) the way they wish to be referred to, but it sounds weird to me without the definite article, and I’m curious about what the rest of you say.

So far 65% of survey respondents say they tend to use the definite article when referring to the website with the address arXiv.org. The answers so far do not appear to correlate with respondents’ scientific fields, native languages, or status as arXiv volunteers. Some respondents wonder if “the” might be a marker of unhipness: “Older folks talk about ‘the Google’; maybe folks younger and cooler than me drop the ‘the’ from ‘the arXiv’?” So far, age does not seem to correlate with direct article use, but perhaps in 10 or 20 years it will only be backwards Millennials and older who will be saying “the arXiv.” I did not think to ask about people’s geographical location, but another respondent notes that Northern and Southern Californians have different customs when it comes to highways: “people from northern Cali call a freeway by its number: “take 80 to 580″ while SoCal people say, take the 405 to the 5.” If you would like to chime in with your responses, theories, or silly comments (“people should not be taking the Christ out of arChristiv.org”), the survey is still open and will be until September 30. I’ll update this post with the final results then.

How should you celebrate arXiv’s birthday? By reading open-access math papers, obviously! (Or physics, computer science, or whatever your favorite arXiv-using discipline is.) You might consider Grigori Perelman’s papers solving the Poincaré conjecture, which are only available on arXiv, Paul Ginsparg’s reflections from five years ago on the occasion of arXiv’s 20th birthday, or if you’re feeling whimsical, an economic analysis of the ramifications of destroying Death Stars. If you want to go all-out, you could submit a paper to arXiv or one of the overlay journals such as Discrete Analysis that have sprung up recently. You could even surf over to bioRxiv, a biology preprint server, sociology’s SocarXiv, or another discipline-specific preprint server, or take a walk on the wild side with viXra.

For a hit of nostalgia, try partying like it’s 1991. The *Billboard* number-one hit song in August 1991 was (Everything I Do) I Do It For You by Bryan Adams, big hair was in, and *Star Trek: The Next Generation *had just finished its fourth season. That sure sounds like a party to me.

This is not the point of Barany’s post at all, in fact it’s barely even mentioned beyond the title and a passing ~~dig at~~ mention of Jordan Ellenberg. Instead, what it seems Barany is endeavoring to establish is that *mathematicians* are not everywhere. And this, he says, should be considered when making policy decisions concerning public support of advanced mathematics.

Barany discusses the role of mathematics in society and culture from the Babylonian times through post-World War II.The ancients, he says used math as “a trick of the trade rather than a public good,” adding, “for millennia, advanced math remained the concern of the well-off, as either a philosophical pastime or a means to assert special authority.” Historically, Barany explains, math was the domain of the elite, it was a source of power that was only available to the highest born. Mathematicians inhabited a rarefied domain, they were acting as advisors to heads of state and maintained an untouchable air of the occult.

This otherness and elite status meant that math wasn’t available to all people, and Barany argues that this has bled through to mathematics today. And I completely agree. Anecdotally — since I can’t find an actual demographics survey to back this up — advanced math is largely occupied by people who come from the economically advantaged side of things. But here’s where Barany’s argument starts to lose some strength: isn’t that the case for *any* course of advanced study? Attending graduate school *at all* is a tremendous luxury, whether studying math, science, language, or art, it’s a pretty brazen thing to want to sit around for 4-6 years to get paid very little to think about things.

And some historians, most notably the blogger Thony Christie, even take exception with the picture Barany paints of math and society. Christie suggests that Barany may her oversold the elitism and “otherness” of mathematicians, pointing out that math played a huge role in the scientific revolution of the seventeenth century. Confirming what I suspect, which is that math isn’t all that different from the other sciences in terms of its growth and expansion over the last several centuries.

In a conversation on twitter, Barany defends his position to Steven Strogatz — legendary bringer of math to the people — saying there are two core questions that people try to answer with “math is everywhere” namely, (1) why should the public support advanced math, and (2) why should the public learn basic math? Barney contends that “math is everywhere” is not an appropriate answer to either. I disagree.

I think math being everywhere is exactly the antidote to the ancient rhetoric of “when will I ever use this?” Math is everywhere just as much as anything is everywhere. Science is everywhere, art is everywhere, language is everywhere, and for some reason in those cases, being everywhere is sufficient to convince folks that they will use the subjects. It is everywhere, and therefore knowing it will help you understand everything. So I think a good dose of “math is everywhere” is a great way to motivate why the public should learn basic math. And I think we can mostly agree that learning basic math is a good and important thing.

And since the best way to help your kids develop good habits is to set a positive example, I obviously think policymakers should opt in favor of supporting advanced math, just as they would support any advanced science research. Because even though it often seems pointless, basic research is the provenance of great discovery.

]]>Dave Richeson wrote two posts about this illusion by Kokichi Sugihara, one of the top illusions from this year. In it, (topological) cylinders appear to have either square or circular cross-sections depending on what angle you view them from. He used Geogebra to show how to derive a curve that has the requisite properties and made a template so you can put your own deceptive cylinder together.

Naturally, Richeson’s posts led me down an illusive, illusory, illustrative, or perhaps just illusional, rabbit-hole. The illusions on the contest website highlight various perceptual habits most people’s brains share—our preference for right angles, the way we infer motion from changes in light, and the importance of context in identifying shade—but not so many of them have obvious mathematical connections. Then I got to a color illusion that really grabbed me.

In June, Vi Hart, mathemusician and virtual reality researcher, posted a long, interesting rabbit-hole of a post about color perception on the eleVR blog. It starts with that late-night dorm room question “is my red your red?” and considers how our color perception might influenced or be influenced by virtual reality. What color effects will we be able to learn about and play with as VR gets better and more widespread?

The mathematics of color is fascinating and perplexing to me. I was strongly indoctrinated into the red-blue-yellow primary paint color paradigm as a child, and it’s been hard to unlearn that enough to understand how we actually see color in light. Nick Higham, applied mathematician at the University of Manchester, has a post about mathematics and color that explains some of the nuances.

To me, the most astonishing thing about these illusions is that even when you know the mathematical and perceptual reasons you see what you do, you can’t help but see it. Right now I’m stuck on these drifting Gabors.

]]>And this brings us to the number 137, which is the 16^{th} lazy caterer number. It’s the maximum number of pieces that you can get from cutting a circular pizza straight through 16 times — so really they should call it the 16^{th} smart pizzeria number. In general the n^{th} lazy caterer number is given by the equation ^{n^2+n+2}⁄_{2}, and together they form the lazy caterer sequence

1, 2, 4, 7, 11, 16, 22, 29, 37, 46, 56, 67, 79, 92, 106, 121, 137, 154,…

of which 137 is the 16^{th} term (assuming we call 1 the 0^{th} term). So those pizza cutters at my pizzeria should go on doing exactly what they’re doing, but always be sure to aim for the n^{th} lazy caterer number when they start slicing, y’know, just to make things fair.

Now, on to the main dish of this carnival: the posts of the month!

- I really liked this piece from Brian Hayes about the derivation of wire gauging. Seriously, before today I had spent approximately 0 minutes of my life talking about gauged wires, but this post is so much fun I just made my poor mother listen to me explain to her all about 36 gauge wire and the 39
^{th}root of 92. Trust me, just go read it. - Kevin Knudson sent us a great piece about visualizing music mathematically. He describes a software that interprets the different tonal and percussive qualities of music to plot out a multidimensional character profile. I can’t get the real time video to load, but the still photos are already really cool. Plus, Michael Jackson.
- On the more technical end of things, a post from Mark Dominus explores how to decompose a function into its odd and even parts. It would be a fun discussion to have in an algebra or calculus class someday, I also like that Mark explains one piece of his discussion by saying “…as you can verify algebraically or just by thinking about it.” Ah, the old proof by just thinking about it trick.
- Gonzalo Ciruelos explains an algorithm for determining the roundest country. Harder than it sounds, and also, geez, the island nation of Nauru is really round. Check out the post for a ranking of the roundest countries.
- In case you’re wondering what’s going on with the ABC conjecture, this post from David Castelvecchi gives us a nice plain English update on what the key players in the fight to verify Mochizuki’s proof are up to these days.
- And for the crafty maker types, Nancy Yi Liang submitted this how-to guide for an incredible laser cut dress. The dress is a graphical visualization of some arcsin functions, and it’s custom made to fit!

Thank you for so many wonderful submissions! You can check information on past and future carnivals at The Aperiodical.

]]>There are a boatload of blogs by math educators at every level of schooling, so I’m not going to attempt anything approaching a comprehensive list. These are a few that stand out to me personally, but there are tons of other great ones out there as well, and I’d love your suggestions in the comments. I also have to recommend Math Teachers at Play, a monthly math education blog carnival, or the Twitter hashtag #MTBoS (math Twitter blog-o-sphere) as places to find like-minded people to talk with and blogs to follow. And of course you can check out the Math Education category right here on our blog.

**Kiddos**

Bedtime Math Why should stories get all the fun? Bedtime math gives parents and kids daily prompts to help engage with math in an open-ended way outside the classroom.

Let’s Play Math! Denise Gaskins, who organizes the Math Teachers at Play blog carnival, also has a blog with ideas for math play with young kids.

Mike’s Math Page Mike Lawler shares videos of him talking and playing with math with his two kids as well as other posts about doing math with his kids.

Natural Math Maria Droujkova is probably best known for her advocacy of the idea that 5-year-olds can do calculus. She and her crew have courses, books, a blog, and an online forum for people who want to have fun doing math with little kids

**Tweens and Teens**

Continuous Everywhere but Differentiable Nowhere Sam J. Shah’s blog is one of my favorite math teacher blogs because of the number of times he has genuinely nerdsniped me.

dy/dan Dan Meyer is a former high school math teacher who now works for Desmos. He blogs about math education, especially good lesson planning and how to get students thinking mathematically without frustrating them too much.

Finding Ways It’s hard to describe Fawn Ngyuen’s blog. She posts about her life story, teaching math, and social justice, but it’s all mixed up together. If you don’t read anything else I link to in this post, read her poem irrelevant.

José Vilson I hadn’t heard of José Vilson until I started seeing tweets about his keynote address at TwitterMathCamp in July. It was clear from those tweets that his presentation got people thinking and talking about race and justice in their classrooms. He is the founder of EduColor, an advocacy group for equity in education, and his perspective is valuable for anyone who wants to work for justice in their schools.

Math=Love Sarah Carter, a high school math teacher, shares classroom materials and fun activities. I enjoyed using the 1-4-5 square puzzle with my high school math program students last summer.

Math Munch Every week, this blog has a post with three fun finds from the mathematical internet. They have a mix of types of math they highlight, but there is usually something interactive in every post, and they share interviews with a lot of cool mathematicians too.

Math With Bad Drawings Ben Orlin’s accurately-named blog explores issues in math and math education with thought-provoking metaphors and entertaining drawings.

Mr. Honner Patrick Honner is a high school math teacher who has recently been focusing on the many problems with the New York state Regent’s exam math questions (hello, overly pointy sine wave graph). He also posts more uplifting fare, including gorgeous math photos.

Teaching/Math/Culture Ilana Horn researches math teaching and learning in secondary schools. She focuses on inclusion and cultural awareness.

**College and beyond**

Launchings David Bressoud posts monthly in this MAA blog about college math teaching, especially research related to the calculus sequence.

Math3ma I’ve written about math grad student Tai-Danae Bradley’s blog here before, but her excellent post for people getting ready for their qualification exams is so good I just have to share it again.

On Teaching and Learning Mathematics This AMS blog has posts about math teaching from K-12 but primarily focuses on college classrooms. I have especially enjoyed their posts about active learning and creating positive environments that are conducive to learning.

PhD plus epsilon This is another AMS blog. It is not exclusively devoted to education, but the early-career mathematicians who write it often post about their teaching struggles and successes.

Finally, if all of this other stuff gets too heavy, there’s always Math Professor Quotes to give you a laugh.

]]>The Carnival of Mathematics is organized by the excellent British math(s) blog The Aperiodical. The carnival is itinerant, traveling around the math blogsphere from host to host. I hosted carnival #103 at my Scientific American blog Roots of Unity, and next month’s carnival will be number 137.

I try to visit the Carnival of Mathematics every month to find gems I’ve missed and blogs I’ve never read before. I’ve lost count of how many cool math blogs I’ve added to my feed from past carnivals.

The other math blog carnival I know about is Math Teachers at Play, coordinated by Denise Gaskins and hosted by math education bloggers from around the internet. As the name suggests, it focuses more on math education from preschool to high school, and it looks like a great resource for teachers and parents.

Blog carnivals need two things: attentive hosts and enthusiastic submitters. That’s where you come in. Please submit to the carnival! We are looking for posts from roughly the past month that have amused, challenged, or delighted you. Anything mathematical is fair game. Don’t be shy—self-promotion is encouraged!

Read the most recent carnival at Math Misery, check out past carnivals at the Aperiodical, and click here to submit a post to next month’s Carnival of Mathematics!

]]>The counterintuitive study lands amidst several other data-driven attempts to understand police use of force in a racial context, and as usual, FiveThirtyEight has done their due diligence in weighing the pros and cons of various studies and methodologies.

Fryer’s study, which culled data from a small sample of police narratives from arrests in the city of Houston, attempts to tabulate various uses of police force according to race. Statisticians across the internet have had a lot to say about the study, namely, that its findings are implausible, the statistical methods lacking, and shame on the New York Times for hyping up a purportedly *paradigm shifting* study that hasn’t even been peer reviewed yet — here I paraphrase the sentiment of The Statisticians.

What is seems Freyer is attempting to do is disentangle racial bias and statistical discrimination. But what’s getting lost in the data, according to Josh Miller, is the huge and variable amount of bias already present in police stops. If an officer is more inclined to perceive a black individual as threatening when they actually aren’t, Miller points out, then the actual average threat level of the blacks being arrested versus the whites might skew low. And in this case, he says, it’s no longer the least bit counterintuitive that blacks are 20% less likely to be shot. It actually makes total sense.

In one portion the study also normalizes across several parameters: arrest demographics, year of arrest, threat level of the encounter, to name a few. This is standard practice when massaging data for statistical analysis, but Uri Simonsohn points out that this normalizing actually erases all potentially salient points of the study. Particularly the perceived threat level, which the researchers refer to as “encounter characteristics.” When normalized for threat level, whites and blacks are equally likely to get shot by police. Simonsohn is quick to add that this is not robust enough to be a result on its own, but is certainly enough to cast serious doubts on the results of the study.

Somewhere along the way in life I learned that when you read a grabby headline that claims to flip the common narrative on its head, you should read the article that goes with it. And you should read it carefully, and you should be circumspect. Because as much as Malcolm Gladwell would have us believe that everything we thought was true is actually the opposite, it’s just not usually the case. True, crazy counterintuitive things happen *sometimes*, but flawed experimental design and sampling error happen *all the time*.