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*.

This has been a weird week. People have been walking into traffic, trees and parked cars at an alarming rate while they compete to catch little animated beasties that are sort-of kind-of actually walking down the street next to us. I’m talking about Pokémon Go. The new augmented reality game that has you *catching ’em all*, all over the place.

The original Pokémon was a two player video game, which I am afraid to admit, I never actually played. Nevertheless, I was curious about the nature of the game: what sort of Pokémon strategies are there? And I found the most interesting thread on math overflow all about Pokémon research.

The basic strategy of the game is that each play picks 6 Pokémon from a stable of 718, and each Pokémon has 4 out of 609 possible moves. The opposing Pokémon face off against each other one by one and have various levels of strength and vulnerability to attack. So there truly are a finite number of strategies with two independent decision makers, meaning a Nash equilibrium does exist. Of course this is a relatively gigantic space of play, so it’s pretty difficult to actually model the possibilities, and it’s not a sure thing that the Nash equilibrium will be what plays out.

Another fun aspect that makes the Pokemon game hard to model and predict is that you are in a game with imperfect information. Your opponent may know that you’re holding Charmander, but your opponent has know way at face value to determine which moves you’ve chosen for him.

The new version, Pokémon Go, seems to be a bit less focused on an end point. Instead, it appears that you just like crazy catching Pokémon until you eventually level out and have a super powerful character that can do whatever he wants. In which case you would be the baller of the Poke world…but that is about it.

Business are also having some fun with the Pokémon Go craze. Apparently you can use Pokecoins to buy lures which entice Pokémon Go players into your brick-and-mortar establishment for the bargain price of one dollar and 17 cents per hour.

For some poor souls, registering an account has already been an exercise in game theory. Pokémon Go is so out of control popular that the number of people trying to resister every minute of every day has overwhelmed their servers beyond their capability. The Pokémon Go people have asked users to wait an hour and try again, but this of course, will never work and is a perfect example of a Tragedy of the Commons.

]]>I doubt I’m the only person who sees the front cover of a math book or a conference poster and wants to know more about the picture. That’s why I was excited that when the Society for Industrial and Applied Mathematics came out with their new journal on applied algebra and geometry (SIAGA), Berkeley graduate student Anna Seigal published a series of posts illuminating the mathematics behind seven images on the SIAGA poster.

Seigal is half of Picture This Maths, a blog she and University of Aberdeen graduate student Rachael Boyd use to talk math with each other and, in the words of their “about” page, “shed some light on what doing a PhD in maths actually involves.”

As the name implies, Picture This Maths tends to use pictures as a focal point of a post. Because I’m a good internet user, I’m especially partial to cat pictures, so Seigal’s recent post on using tensor approximation to compress images was welcome. She notes that tensor approximation is, as far as she knows, not currently used to store photographs, but it’s important to be prepared with this option in case the ongoing cat picture deluge requires us to get creative about how we store them.

Many of the posts at Picture This Maths look at what I would describe as applied abstract math—the math used is often more on the theoretical/”pure” side than what people typically think “applied math” is about. Seigal’s posts about the new applied algebra and geometry journal fall under this category, as do Boyd’s posts about persistent homology.

Homology is a tricky thing to explain, and I generally only think about it in an abstract, theoretical mathematical context. Homotopy is more intuitive for a lot of people, but especially in high dimensions, homology can be a more useful and more computable algebraic object to assign to a topological space. I like the way she explains why we use it. “So why do we do this? We might want to know something about a topological space, but maybe we can’t simply draw the space as it lives in a very high dimension. But the homology of a space is a sequence of groups which tells us about holes of all dimensions: and we know lots about groups!” Sometimes it seems like everything in math is figuring out how to ask a question we can answer. What is this manifold? I don’t know, but has this many holes in these dimensions.

The applied version of homology is persistent homology. I’ve encountered the idea before, but I never felt like I understood how it would be useful in practice. I still don’t think I could spot a good place to apply persistent homology in the wild, but Boyd has a nice post that describes how it can show up in viral gene transfer, Twitter connections, and non-transitive dice. If you’re a theoretical mathematician who can only handle low doses of applied mathematics, her post on Coxeter groups and the Davis complex (and its adorable diagrams) will make you feel at home again.

All in all, Picture This Maths reminds me of Math3ma, another favorite graduate student math blog. I think undergraduate math majors, math grad students, and other people who like looking under the hood in math will enjoy reading this blog.

]]>There is a classic urban legend about the algebraic geometer who was detained, arrested, and possibly stripsearched, depending on which account you read, for discussing “blowing up points on the plane,” while waiting in the security screening line on the way home from a conference. Of course we know the blowup is a common tool in algebraic geometry, but I’ll be darned if it doesn’t sound exactly like a plot to take down an airplane. I’m not sure how true the story is. It exists in the oral tradition of mathematical urban legends, and more recently on the math overflow thread, but I’d be curious: if you were the guy who was detained/arrested/stripsearched, let us know.

In a similar turn, but unfortunately entirely true, a man was detained earlier this year for doing math on a plane. He was an economist working out some PDE’s when his seat-mate pegged him as a suspected terrorist. Now there was a lot at play here, certainly this is a blatant case of racial profiling, confusion about what math is, and mostly just the presence of a crazy lady on a plane.

But on that theme, who among us hasn’t — likely prompted by some furious last minute TeXing, or on-board paper refereeing — had to come out to their neighbor as a mathematician. Now the above reaction is really rare, and I suppose all professions get their strange reactions (I’m always glad I’m not a gastroenterologist…or a priest) but something about being a mathematician makes people really spill their guts to you in a very unique way. And surely I’m not alone when I say, IT’S SO UNCOMFORTABLE. Complete strangers want to instantly put it all out on the seat-back tray table: explaining how bad they are at math (“No like really, I’m horrible! Watch, give me two numbers, I can’t even add them!”), apologizing for not trying harder, blaming the bad teachers, praising their good teachers, digging into their genetic legacy for some sort of explanation (“My grandfather on my mother’s side had such a head for math, but I don’t take after him, oh no, definitely not, I’m all grandma.”), and yearning for some kind of absolution that I simply can’t give.

And then there are the people who look at you like you just farted in their confined airspace and obviously hate your guts, kinda like this.

Either way, it’s a strange thing. And I should be clear, I have no aversion to coaching students through serious math anxiety, or helping anxious people learn math, in fact that’s one of my favorite things to do. I also think it’s fine to not be good at stuff (don’t go spreading this around or anything, but there’s plenty of stuff that I’m not good at). I’m just always surprised to get so much emotional baggage dumped on my head by a complete stranger, and my desire to contain it all is overwhelming.

So, with that, safe travels to everyone this summer and enjoy all that fun airplane stuff.

]]>And that’s how *The Intrepid Mathematician* got me hooked. Anthony Bonato, a math professor at Ryerson University in Toronto who specializes in network theory, writes this blog dedicated to the teaching, learning, living and loving of math, as well as his recent foray into science fiction writing.

But it’s his take on the learning and living math part that really got me. Which brings me back to the quote about the kid who got a day off from math. In the post “Let’s not do math today,” Bonato writes about this fabled kid who is treated to a math-free day — which of course, is insane, because who wants a day without math!? He uses this as a launching pad to discus how deeply our cultural attitudes around math affect the way our children acquire numeracy, citing relatively recent research from the OECD’s Programme for International Student Assessment.

Bonato has a good insight into some of the stumbling blocks of early mathematicianhood. He goes on to tackle math anxiety (in particular, some of the gendered aspects of it) in “Math Anxiety and Gender.” In the piece he points out that the content of mathematics is unique in that it is totally free of gender, race, country, or class and it should in some respects be the most accessible of all subjects. But of course we know that for a variety of reasons, this is not the case. Bonito goes on to point out, “Like any other subject, however, mathematics is taught and studied by people.” His post is a great primer on the idea of how the human hand seeps into the mathematics, one which should be followed up by a mega binge-read of mathbabe’s commentary on algorithms and accountability.

Bonato also delves into some of the really tricky parts of understanding and appreciating math. In “Is mathematics an art or a science” he considers the evolution of concepts in art and science, and whether or not they seem particularly mathy.

A challenge with thinking of mathematics as an art is that it hard to appreciate it unless you have the proper training. Most people enjoy music, a good novel, or a well-crafted painting. It is more challenging to convince a friend to read a brilliant paper or sit through a lecture by a leading mathematician.

This is so true. But what is it about math that makes the paywall so high? Is it just a matter of jargon? Or is it that the concepts themselves are actually so difficult? This is something I love to think about, and Bonato covers it in “This is your brain on mathematics” citing some current research that observed brain activity while thinking about math and problem solving, versus linguistic concepts.

As students of mathematics know, a great deal of linguistic recognition is needed to learn the subject. If I state Tutte’s conjecture as “Every bridgeless graph has a nowhere-zero 5-flow,” then the sentence is meaningless unless you understand the context of the phrases “bridgeless graph” and “nowhere-zero 5-flow.” However, understanding the real meaning behind Tutte’s conjecture requires mathematical, not just linguistic knowledge. In fact, no one really understands it, as it is an open problem!

And that is really the fun part of math, when you say “I don’t understand,” that can mean any number of things! Ah, the joy of confusion. For more from *The Intrepid Mathematician,* check the blog for updates every Wednesday or follow Bonato on Twitter @Anthony_Bonato.

Emily Grosvenor is the tiling enthusiast behind the push to make Tessellation Day a thing. She recently successfully crowdfunded Tessalation, a book about a girl named Tessa who sees patterns everywhere. The book will be available for non-backers on June 17, Tessellation Day. (Disclosure: I backed the project on Kickstarter.) The date was chosen because it is the birthday of M. C. Escher, one of the most famous tessellators.

Luckily, it’s easy to celebrate Tessellation Day. Just tessellate! Appropriately enough, tessellations.org has a tessellation tutorial to get you started. John Golden also has a page of tessellation resources on his blog math hombre, and John Baez has some cool posts about tilings on his AMS blog Visual Insight. Update: Emily Grosvenor also just published a list of 23 simple ways to celebrate World Tessellation Day.

The easiest shapes to base a tessellation on are equilateral triangles, squares, and regular hexagons—the regular shapes that fill the plane all by themselves—but there are lots of other shapes that can form the foundation of your tessellation. I’m particularly fond of pentagons, and Laura Taalman has instructions for 3D printing all the tessellating pentagons if you’d like to make yourself a desk organizer or other plastic object from irregular pentagons. To break free from the repetition of those tilings, the Penrose tiling is probably everybody’s favorite aperiodic tessellation. You can learn how to knit yourself a Penrose tiling from Woolly Thoughts.

One way to make your tessellations a little more exciting is to move them to the hyperbolic plane. Escher of course made a lot of good Euclidean tessellations, but I’m partial to his Circle Limit paintings, which tile the Poincaré disc model of the hyperbolic plane. There are good articles about Escher’s use of mathematics and the Circle Limit series in particular by several authors, including Doris Schattschneider, Bill Casselman, and Doug Dunham.

To elevate your tessellations, you might try venturing into the third dimension. It’s not hard to make a 3-dimensional tessellation, or honeycomb, by taking a 2-dimensional tessellation and making it into a prism, but there are other ways of filling 3-space with repeating polyhedra. One of my favorites is the combination of truncated cubes and cubes in this experimental water bottle by Portuguese design firm Pedrita.

If you want to really get wacky, you can combine hyperbolic geometry and the third dimension and learn about how to visualize hyperbolic honeycombs from Roice Nelson and Henry Segerman.

How will you celebrate World Tessellation Day? Share with the hashtag #WorldTessellationDay or #WorldTessDay.

]]>