Not Those Midterms

What our midterms need is a sensible counting tool. Like this Sharp Elsimate EL 428.

I was reminded recently of a time a few years ago when I sent my students an email on November 4, 2014 with the following addendum “P.S. Don’t forget to vote in the midterms today.” The next day I was greeted with some genuine and totally unanticipated confusion. Students asking “What do you mean? Are we voting on when our midterm happens? I thought it was going to be take home!” And I realized the majority of them had no idea what a midterm election was. And now, in retrospect, it seems quaint that there was a time so recently that midterm elections weren’t a full blown three ring circus.

Anyhow, I don’t need to remind you that we had midterm elections last week. This time I’m quite certain my students could tell there were midterms — beyond the one they took in Calculus — going on. I’m sure that even the alien probe Oumuamua could tell from outer space that we were having midterms last week.

The midterms were a big deal this year, so big, in fact, that the voting eligible percentage of the population that turned up was the highest in 50 years. And these really high turnouts meant some really long lines, many rightfully irate voters, and lots of people who were rendered unable to vote. Laura Albert, who writes the blog Punk Rock Operations Research, says, “legally, you can only vote once. But if you vote early, you can enable more than one vote to be cast.” Albert frames the process of voting in terms of queuing theory and explains the different strains on the system that can cause a blockage in the flow of voters and what can be done about them. One of the strains that can be alleviated pretty easily is the number of people entering the queue. This can be done, she says, by voting early. Lots of people took this advice, and in what was a kind of crazy — but also kind of unsurprising — turn, more people voted early in Texas this year than voted at all in 2014!

Gerrymandering has been a hot topic over the last several years. I blogged about Hacking, Cracking, and Packing last year, Fivethirtyeight did a series of podcasts for The Gerrymandering Project, and the Metric Geometry and Gerrymandering Group led by Moon Duchin and Justin Solomon has been busy looking for mathematical and computational tools to solve the redistricting problem. If you’re looking for a quick easy demo of gerrymandering, I liked this tweet from @_WTProject about creatively chopping up a 25 district city. It would be a fun starter project for students thinking about gerrymandering for the first time, and it can be generalized in all kinds of fun ways.

Once the districts have been decided and the votes have been cast the problem of counting the votes still remains. That should be easy right? lol.

I was reminded of this great article, “How to Count” from Brian Hayes that appeared in American Scientist back in 2001, a gentler time, but a time when vote counting was decidedly on our minds. He gives a nice overview of classical counting in Platonic mathematics where “chad are never hanging, and every count is full, fair and accurate. But that’s not the world we live and count in.” Hayes talks about cascading counting errors and particularly how voting machines might be vulnerable to such errors, propagating and amplifying errors in arithmetic as counts are consolidated and tabulated.

As I write this post today Sunday November 11th votes are still being counted, and hopefully the counts are error free. And while the Florida election officials tabulate the midterm results, I’ll be counting up some midterm results of my own. No, not those midterms. These midterms.

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Join In The Fun For #Noethember

Portrait of Emmy Noether

Portrait of Emmy Noether (U.S. Public Domain image)

The Inktober design challenge was created in 2009 by Jake Parker, an illustrator, writer and teacher based in Provo, Utah. Worldwide, thousands of artists participate in this endeavor, which challenges them to create ink drawings (pencil sketches under the ink are allowed) for the month of October and post them online with #Inktober. “Note: you can do it daily, or go the half-marathon route and post every other day, or just do the 5K and post once a week,” according to the challenge rules. There are also official prompts, but using those isn’t a requirement of the challenge.

This year, Constanza Rojas-Molina used Inktober as inspiration for a design challenge dubbed #Noethember after Emmy Noether. Rojas-Molina is a mathematician at the University of Düsseldorf, illustrator (under the pseudonym E. A. Casanova) and blogger at “The RAGE of the Blackboard,” which she named after the RAGE theorem. She shared her idea for #Noethember in an October 1, 2018 tweet:

“I won’t do Inktober this year, but I’m planning on doing Noethember in, of course, November, and sketch about Emmy Noether. Could I get 30 facts about Emmy Noether? Who would like to join? ,” she wrote.

Soon, Katie Steckles shared an offer to host the challenge on the Aperiodical. She wrote an October 19 post explaining the challenge. “It’s the same idea [as Inktober] – 30 days, 30 drawings, but this time each day the theme is a fact or story about the life and work of mathematician Emmy Noether.” In an October 30 post, she shared this year’s topics. For instance, the prompt for today (November 7) is “At Erlangen, Noether was one of only two women in a university of 986 students, and was only allowed to audit classes rather than participate fully. She required the permission of individual professors whose lectures she wished to attend.”

Thus far, it has been fascinating to see the varied approaches participants have used to tackle each day’s prompt. Nicholas Jackson, a mathematician at the University of Warwick in the U.K. has tweeted photos of his pencil drawings. Lele Saa, an art director, illustrator and artist based in the U.K., has chosen to depict the characters in the prompt — including Noether — as foxes. Check out Saa’s #Noethember Twitter thread here. Rojas-Molina has tweeted photos of her ink creations. Here’s an example.

Folks of all experience levels have been encouraged to participate by using the hashtag when sharing their work on Twitter or Instagram. The bloggers at the Aperiodical are retweeting their favorites and will create a roundup of some of the best creations at the end of the month.

Posted in History of Mathematics, Math Communication, Mathematics and the Arts, people in math, Recreational Mathematics | Tagged , , , , , , | Leave a comment

The New Issue Of Chalkdust Magazine

The latest issue of Chalkdust Magazine dropped last week, and it’s filled with as much mathematical goodness as a fresh unopened box of Hagoromo “Fulltouch” chalk. It’s a proper glossy magazine — also available as a PDF — featuring profiles of notable mathematicians, articles about fun mathematical concepts, puzzles, and more. There is something in there for everyone.

The cover of Chalkdust Magazine, Issue 08.

Chalkdust, the magazine, fills an interesting niche in mathematical publishing. The content is a mixture of mathematical articles and other types of mathematical curio such as games and advice columns, and it’s written (mostly) by students in the Math(s) Department at University College London. Consequently, it’s a great resource for student, or anyone who likes to read a little but of casual math on their lunch break.

The latest Issue features a fantastic profile of Eugenia Cheng written by Chris Bishop. I knew that Cheng was a mathematician, an author times two, and a YouTube star, but she is involved in so much more. It’s inspirational seeing someone like her doing such positive outreach for the field.

There are also games! This issue features Hilbert’s hotel: the boardgame, and there’s even a chance to win prizes by completing the “famously fiendish” Crossnumber Puzzle, which is exactly what you would expect, a crossword puzzle with numbers instead of words. The prize is an assortment of goodies from Maths Gear, a website you should definitely know about before the holiday season descends.

Aside from the magazine, Chalkdust is also a weekly blog. There they cover all kinds of stuff from current events in math, book reviews, and even mathematical tattoos. I do love to read about mathematical tattoos.

Content in both the magazine and the blog come from a variety of contributors. Including the students at UCL, there are also contributions from Tae-Danae Bradley, who writes the blog Math3ma, and from Colin Beveridge who writes the blog Flying Colours Math, as well as many other teachers, lecturers, professors, graduate students and mathematical people from all over.

If you or your students would like to submit an article to Chalkdust – which I think would be a very cool idea – you can do that! You can order paper issues of Chalkdust Magazine for a very fair price, or access the magazine in PDF form. You can follow the magazine on Twitter @chalkdustmag.

By the way, since I mentioned Hagoromo “Fulltouch” chalk at the outset. For the unfamiliar, Hagoromo chalk is the best chalk there ever was, gloriously smooth, like writing with a stick of butter. The original Hagoromo factory closed in 2015, but the rumor I hear – which is corroborated on this Math Overflow thread – is that the technology has been transferred to another plant, and you can still ge your chalk fix. But I’ve never actually tested the new stuff, and I can’t vouch for it. Let me know if you have on Twitter @extremefriday.

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A Tour of Robert Kaplinsky’s Online Resources

Image: US Department of Education.

Robert Kaplinsky is a math educator and presenter. He also co-founded Open Middle, a website that encourages problems which require “a higher Depth of Knowledge than most problems that assess procedural and conceptual understanding,” according to the Open Middle website. These “open middle problems” support the Common Core standards and give students opportunities to discuss their thinking.

Kaplinsky is also the creator of #ObserveMe, according to his Twitter profile. That movement encourages teachers to observe one another and provide suggestions for improvements. In a post on his blog, he provided a template that teachers can use to list a few points they want feedback on from colleagues who observe their classrooms. Many teachers have used the hashtag to share photos of their signs on Twitter. (In a different blog post, he also shares a collection of suggestions for “Troubleshooting #ObserveMe” based on some common problems teachers have encountered after joining the movement.)

His blog has a wide range of posts that appeal to different audiences. Some are targeted towards a K-12 audience of teachers, parents and students (such as this series about the Common Core standards). Posts such as “What Do Van Halen, King Solomon And Formative Assessment Have In Common?” are framed in the context of education, but could also appeal to folks who enjoy reading engaging posts that are sprinkled with novel connections. (I won’t spoil the surprise link between King Solomon and Van Halen, but it’s a fascinating one.) Here are just a few of his other thought-provoking blog posts:

Kaplinsky’s website also offers more than 70 “real world problem-based math lessons” for grades K-8 and the subjects of algebra 1, geometry and algebra 2. The lessons, which are free to download and use with students, are centered on topics ranging from “How Can We Make Stronger Passwords?” to “How Many Soda Combos Are There On A Coke Freestyle?” At the bottom of each lesson is information about which content standard(s) the lesson relates to.

Recently, Kaplinsky offered a series of webinars on “Why We Should Reconsider Using Word Problems (And What We Should Be Doing Instead).” He created one for elementary school teachers, one for middle school teachers and one for high school teachers. You can replay them here.

Have you joined the #ObserveMe movement for your own teaching? Are there other resources for math teachers that you would like to see covered in future blog posts? Please always feel free to reach out to me in the comments, on Twitter (@writesRCrowell) or via email (!

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Math Games That Make You Think

Screenshot from Nicky Case’s interactive game about networks.

In the echo chamber, social media kinda world that we’re living in, network theory is playing an increasingly important role. So I was delighted, this morning, to spend several minutes playing an interactive game by the talented Nicky Case called The Wisdom an/or Madness of Crowds. The game takes you through the steps of building a graph, connecting edges, and watching how ideas, contagions, and influences spread through networks. It’s particularly surprising how easy it is to build a network that creates the majority illusion. That’s the one where you’re connected to just the right number of people with just the right kind of beliefs that you think everyone around you thinks the same thing.

This is just another fine example of a Case curated experience that teaches, entertains, and inspires. Case keeps a blog with some behind the scenes narratives, explanations of the explorables and also some short stories. I really fell in love with the post A Way-Too-Philosophical Behind-The-Scenes Post, which gave voice to every feeling of futility I’ve ever had. In it Case riffs on several graphs of “stuff I’ve made,” plotting number of people reached vs value, wondering where the sweet spot lies. Case strips down this analysis with one nice “valuable-ness” curve.

As an academic mathematician this is a balance that makes me think, and often bums me out a lot. I spend so much time working on research papers refining arguments dealing with the emotional upheaval of the referee process and then the subsequent concern that no single human eyes will ever pass over the finished product. Compared to this blog post, which I wrote in <2 hours and at least one person (you!) is reading right now. Granted it’s not likely having any huge impact on your life. But if I have n people reading this post and learning something marginally cool from it, and m people reading my Corollary 2, what ratio of m/n am I really comfortable with? Certainly less than 1 is fine, less that 100 is ok to, but less than 1000? It’s a weird calculus to try and sort out. And then wondering…what’s it all for?

Truth, that’s what it’s all for. Well, truth and tenure. But mostly truth.

A bunch of happy polygons looking for a place to call home.

I first became acquainted with Nicky Case’s work when I encountered Parable of the Polygons. This interactive post, co-created by Vi Hart, is based on a paper by the Nobel Prize winning game theorist Thomas Schelling. By dragging around little polygons you learn how a small amount of racism can easily result in hugely segregated neighborhoods. It’s a great interactive for young brains because it gives such a concrete demonstration of a pretty fundamental concept in game theory and social science. Also it’s just fun to play.

If you want to hear Case talk about this project, check out the episode of The Other Half on Racism and Segregation (forgive the blatant logrolling, but I just think Case has a lot of interesting things to say!)

You can support Nicky Case and this incredibly fun work through their Patreon page and following Case on Twitter at @ncasenmare. And if I find anything you might like on the internet this week, I’ll tweet at at you @extremefriday.

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On Michael Atiyah and the Riemann Hypothesis


Riemann has been back in the news lately, thanks to an announcement that his nearly 160 year old hypothesis might be solved.
Public domain image courtesy of Wikimedia CC.

At the 2018 Heidelberg Laureate Forum (HLF), Sir Michael Atiyah gave a lecture in which he claimed to have found a proof for the Riemann hypothesis. If Atiyah’s proof holds up, then the nearly 160 year problem concerning the distribution of primes will finally have a solution.  It’s on the Clay Mathematics Institute’s list of seven Millennium Prize Problems and just one of those — the Poincaré Conjecture — is listed as solved on the institute’s website. However, the $1 million prize for a proof of the Riemann hypothesis is yet to be awarded, and some folks doubt that the long-open problem is finally solved.

It’s unsurprising that the world is watching Atiyah — and waiting to hear the verdict about the proof he proposed. Atiyah received a Fields Medal in 1966. In 2004, he and Isadore Singer were jointly awarded the Abel Prize for their discovery and proof of the Atiyah-Singer index theorem. Yet many other supposed proofs for the Riemann hypothesis have been proposed, only to fall apart under further scrutiny.

The Clay Mathematics Institute’s “Official Problem Description” for the Riemann hypothesis is 11 pages long (with nearly two full pages of references), while Atiyah’s current write-up of the proof is five pages long. “Atiyah attributes much of the theoretical work that underpins the proof to a paper of his own that has been submitted to the Proceedings of the Royal Society A,” Frankie Schembri wrote for Science. “That paper has yet to be published,” she added. In fact, Atiyah’s list of references includes just three works: the aforementioned unpublished paper, his 2018 Abel lecture at the ICM in Rio de Janeiro and Friedrich Hirzebruch’s 1966 Topological Methods in Geometry.

News of Atiyah’s claim has reached far and wide since his announcement. Articles about the development appeared in Science News, Gizmodo, Popular, the Irish Times and more. Bloggers also covered the announcement, including Katie Steckles and Christian Lawson-Perfect for the Aperiodical, John D. Cook for his consulting blog, as well as Dick Lipton and Ken Regan for Gödel’s Lost Letter and P=NP.

John Baez also wrote a lengthy thread about Atiyah’s claimed proof on Twitter, starting on September 23 (the day before Atiyah’s lecture at the HLF). He wrote “I bet that Atiyah’s claimed proof…will not convince experts. In 2017 he claimed to have a 12-page proof of the Feit-Thompson theorem, which usually takes 255 pages: He showed it to experts, and… silence.” What’s more, “In 2016 Atiyah put a paper on the arXiv claiming to have solved a famous problem in differential geometry. The argument was full of big holes: So, I’m not holding my breath this time. But of course I’d be happy to be wrong,” Baez added.

Steven Strogatz tweeted “Uh oh. I have a bad feeling about this. Famed mathematician Michael Atiyah claims proof of Riemann hypothesis,” along with a link to the September 21 New Scientist article about Atiyah’s claimed proof. When asked by Twitter user Jonathan Horrocks “Why bad? I take anything Atiyah says very seriously,” Strogatz responded “I take him seriously too. It’s the same bad feeling I have when a diver attempts an extremely difficult dive: afraid, yet hoping for success.”

The skepticism surrounding the claimed proof doesn’t appear to phase Atiyah. “Nobody believes any proof of the Riemann hypothesis, let alone a proof by someone who’s 90,” he said to Gilead Amit​, author of the New Scientist article. “People say ‘we know mathematicians do all their best work before they’re 40,'” Atiyah said, adding “I’m trying to show them that they’re wrong. That I can do something when I’m 90.”

Regardless of whether Atiyah’s proof holds, he has already done something phenomenal: reignited a worldwide conversation about the Riemann hypothesis. That alone is something to celebrate.

What do you think? Share your thoughts in the comments below or reach out to me on Twitter @writesRCrowell! As always, I’m also happy to hear ideas for future blog posts.

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The Thing Last Week With That Sexist Paper

Once again the mathematical world is rocked with scandal. Let me get you quickly up to speed. It started when a controversial paper on the variability hypothesis was accepted to the Mathematical Intelligencer. Shortly thereafter, University of Chicago mathematics professor Amie Wilkinson sent an email to the editor in which she “criticized the scientific merits of the paper and the decision to accept it for publication.” The paper never appeared in the Mathematical Intelligencer, but did eventually turn up (one author less) in the New York Journal of Mathematics.

Without going to deeply into the details, the paper proposes mathematical models to support the variability hypothesis, which essentially says that if one gender is more selective in mating, then the opposite gender will have greater genetic variability. For us humans, it’s thought that females are more choosy with their partner since they spend more time (for reasons biological, social, historical, etc) nurturing the offspring, and consequently the males end up with more variability. This is why — allegedly — there are more men with very high and very low intelligence, and women just tend to clump around the middle. You probably remember when Larry Summers suggested something along these lines a few years ago. That landed like a ton of bricks.

And so too has this recent paper. Beyond the weird publishing bait and switch, the disappearing coauthor, and the sort of dubious job of editing and refereeing, the paper seems to be just fraught with baggage of every possible sort. First, there’s the sexist baggage that comes promoting a controversial theory in what seems like a pretty unsubstantiated way. Terry Tao wrote a short post for What’s New about the whole affair, and if you’re not convinced of the existence full-blown gloves off misogyny, go spend five minutes in his comments section. Timothy Gowers also takes a deep dive into the whole mess on his blog. Again, check out the comments over there but take a deep cleansing breath before you start.

In the meantime, Hill has published a post on Quillette giving his own blow-by-blow of the events. Not only that, but Hill has set up a public dropbox full of documentation and correspondence relevant to the episode.

Oh, the intrigue!

And even later in the week an inspired rejoinder came from Lior Pachter, a computational biologist who writes the blog Bits of DNA. He starts by really taking the math community to task saying, “imagine the hubris of mathematicians spewing incoherent theories about sexual selection when they literally don’t know anything about human genetics or evolutionary biology, and haven’t read any of the relevant scientific literature about the subject they are rambling about.” Ouch!

But Pachter has a good point. Mathematicians and biologists don’t communicate enough, despite the fact that there are so many fundamental and interesting conversations we could be having. He also goes on to make many very thoughtful points about the dangers inherent in “hijacking” the publishing process to push a personal agenda. Specifically a sexist one. The post is great, and the comments section will have you either grimacing, fist pumping or high-pitched squealing, depending on your temperament.

So the conclusion I draw based on the posts I’ve seen is that it’s a confluence bad math, bad biology, an epic fail on the part of the publishing process, and some people just need to delete their account. You disagree? You can let me know @extremefriday.

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On The ‘Math Section’ Blog

Elias Wirth

Photo courtesy of Elias Wirth

Swiss mathematician Elias Wirth created the “Math Section” blog earlier this month. Even though the blog is new, he’s already written several interesting posts, like this one about using the mean value theorem to catch speeding motorists. In an interview conducted over email, Wirth shared more about his blog and recommended resources for other math bloggers. (Note: The following interview has been edited for length and clarity.)

Rachel Crowell: What would you like to share about your own math background, including your interests and research?
Elias Wirth: I just finished my bachelor’s degree in mathematics at the University of Berne and am now looking forward to beginning my master’s degree in applied mathematics at the ETH Zurich. During my years in Berne, I developed a fascination with analysis, complex analysis especially, and applied mathematics. As a preparation for my master’s degree at the ETH, the applied mathematics professor and I then decided that I was going to write a thesis titled Linear Multistep Methods to improve my knowledge of numerical analysis.

At the moment, I am mostly interested in finding wonderful applications of mathematics in everyday life. Partial differential equations are also something that I want to spend more time on because the topic seems to be extremely diverse.

RC: I read that you launched your blog in 2018. How long has it been since you first launched?
EW: The blog was released on September 2. Everything is new to me, but the worst part is over and I can now focus on writing articles.

RC: What inspired you to start a blog?
EW: I already mentioned my bachelor’s thesis. I wrote the first draft in a week and a half, not because I had to,  but because I wanted to. During the writing process I was able to enter a state of concentration that was really satisfying. I woke up in the morning and started writing and would not stop until I went to bed. (During my last semester, I didn’t attend many lectures. Thus, I was able to basically take two weeks off and focus on writing.) I had never before learnt so efficiently and with so much joy. The idea of starting my own blog probably originated during that time. All that I needed to really go for it was one last push. This final push came from my co-worker telling me about his fashion blog and how much fun it was to write for others. It was at that moment that I knew I just had to go for it.

RC: What are some of the things you have learned about communicating about math since you first started your blog?
EW: There are mainly three things that I learned. First, I learned that it is really important to write a good title. People get flooded with information on a daily basis and the title needs to capture their attention. Second, I am surprised by the number of people that are really interested in reading about applications of mathematics. A lot of people from all over the world are eager to broaden their knowledge, which is just wonderful. Third, there are a lot of people that have already established their own blog, website or Facebook page that are very friendly. They are willing to support a new blog like mine if they feel like one is serious about informing people about mathematics.

Continue reading

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Seeing The Future From The Past

Hop in my Delorian and we’ll travel back in time and appropriately tweak our predictive models.  Imagine courtesy of William Warby via FlickCC.

I just finished reading The Signal and the Noise, a book about predictions by the American statistician and blogger turned big time data journalist Nate Silver.  I highly recommend it.  The book came out in 2012 and there was some sort of meta-instructive quality to reading a book whose main theme is using the past to predict the future, written in a past that was (mostly) unaware its own future.  Still with me? 

For example, reading anything written about correctly interpreting political polling in the pre-2016 world induces the reader to struggle under the weight of her own misconceptions.  This in turn causes her to stare off into the distance for one dim moment and think about all of the futures that could have been but at the same time were never destined to be.   Then she shakes it off and keeps reading.

A theme that Silver returns to repeatedly is the impact of our own biases on our ability to interpret and deliver statistical predictions.  Our biases almost inevitably seep into our mathematical models, like which variables we choose to include, how heavily we choose to weight them, and how willing we are to adjust models as we move through time.  Bias has been a hot topic for Google lately, who has been in some hot water for its biased human-programmed (and not actually a magical oracle at all) search results that autocomplete and prioritize some really racist search results.  This could be compounded as Google and others become more reliant on AI that’s trained on fallible human data and the output gets simultaneously farther from an actual human brain but somehow more deeply steeped in human thought. 

The blog Overcoming Bias, written (mostly) by Robin Hanson explores a lot of the places where bias interferes with our understanding of predictive modeling.   In his most recent post Hanson explores the idea of introducing what he calls news accuracy bonds to combat the spread of fake news.  The basic idea is that each article comes paired with a token, or bond, and a reader can get the bond by provably demonstrating that the article is false.  The article goes into more of the details on how this might work, but the basic idea is that an article with a high value bond is more likely to be true.  A high bond values indicates a high degree of certainty on the part of the publisher.  

A related idea has been put forward by Facebook to introduce reputation scores, where effectively users’ scores get decreased every time they post content that the company considers suspicious.  It sounds a little bit spooky — and maybe a bit too reminiscent of a certain episode of Black Mirror — but the idea is similar to Hanson’s in that number attached to an article or user profile will act like a confidence interval by the platform or publisher.  

What do you think?  Would you trust Facebook to responsibly sort dubious information?  Do you ever think about how your biases from the past make it into your assessment of the present or predictions of the future?  If you need me I’ll be in the garage tinkering with my time machine, or as usual, you can find me on Twitter @extremefriday

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On Tricurves

rainbow triangles

If the straight edges of these colorful triangles were made curved, they would become tricurves, a shape which Tim Lexin described in a recent blog post. Image courtesy of Solkoll via Wikimedia Commons.

Tim Lexen, a mechanical engineer in Cumberland, Wisconsin, wrote a post about tricurves for the Aperiodical. As their name implies, tricurves are sort-of triangle cousins which have three sides, but instead of having three straight edges, each of their sides are curved.

“Of the two ancient tools, I preferred the compass over the straightedge. I was fascinated with the classical geometric constructions, the intersecting circles and arcs. As a simple personality test, preferring a compass over a straightedge might mean something: maybe roundabout-holistic-intuitive more than straightforward-linear-realistic. At any rate, the pursuit of curves eventually led me to this topic,” Lexen wrote.

He invites us to imagine a plane which is tiled with repeating identical triangles. The picture starts to get interesting when he begins adding curves.

“But something unexpected happens when we add curves. For tiling we need equal amounts of concave and convex arc. The only way to do that is with two shorter concave sides joining a longer convex side,” he wrote. His post shows a variety of tricurve constructions, from the 30-60-90 tricurve to the 45-135-180 one.  He shows beautiful examples of periodic, non-periodic and radial tiling of tricurves.

Last year, Paul Bourke wrote a post inspired by Tim Lexen on his website. “There are a number of ways one can define a tricurve, the one used here is to start with an arc of some angle, replicate two identical curves ard [sic] rotate each about some angle about the ends of the arc. The Tricurve is the enclosed area,” Bourke wrote. Besides showing more examples of tricurves, he also links to a 2017 paper by Lexen about the shape, it’s tiling and variations, a method for finding its area, and more. (“This is intended as an informal paper. I am freely presenting the idea, for what it is worth; and I am soliciting feedback from any interested readers,” Lexen noted, adding that he can be reached via email at

In a different post for the Math ∞ Blog, Lexen wrote about these and other curved shapes from the perspective of designing a flat puzzle that’s more interesting than one “with dozens or hundreds of identical pieces [which] may sound a little dull and predictable.” In fact, Cherry Arbor Designs now offers a wooden tricurve puzzle.

(While a few of the above mentioned posts provided a link to the National Curve Bank entry on tricurves, I was disappointed to find the link broken. This left me wondering if another curve bank has taken its place or if we are now living in a curve-bank-less world…Anyone know of similar current resources? If so, please reach out in the comments below, on Twitter @writesRCrowell or via email

John Golden used GeoGebra to create a related applet for tiling lenses. Check it out here.

Circling back to the Lexin’s original statement about preferring the compass over the straight edge, my question to you is “Are you Team Compass or Team Straight Edge?” Let me know in the comments below or on Twitter.

As for myself, I’m Team Straight Edge. This position dates back to my preschool years when I told my sister “I may not be a good draw-er, but at least I can draw a straight line.” (My drawing skills have since improved, at least marginally.)



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