The Intrepid Mathematician

%22A classroom of kids cheer with joy. Later that day a boy runs in the front door of his house and says_ 'Mom, teacher said we didn’t have to do math today!'%22

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.

Anthony Bonato, a math professor at Ryerson University, is The Intrepid Mathematician.

Anthony Bonato, a math professor at Ryerson University, is The Intrepid Mathematician.

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.

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2, 4, 6, 8, It’s Almost Time to Tessellate

This Friday, June 17, is the inaugural World Tessellation Day. I am normally skeptical of attempts to create new holidays, but I am so fond of filling up the plane with shapes that I just can’t help myself.

No word on the birds, but bees love to celebrate Tessellation Day. Image copyright Alex Wild. Used with permission.

No word on the birds, but bees love to celebrate Tessellation Day. Image copyright Alex Wild. Used with permission.

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.

A Penrose tiling. Image: Inductiveload, via WIkimedia Commons. (Public domain.)

A Penrose tiling. Image: Inductiveload, via WIkimedia Commons. (Public domain.)

The easiest shapes to base a tessellation on are equilateral triangles, squares, and regular hexagonsthe regular shapes that fill the plane all by themselvesbut 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.

Image: Pedrita.

Image: Pedrita.

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.

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Enchanted

There are only 12 posts on Jim Propp’s blog Mathematical Enchantments so far, and they are all superb. Propp is a professor at the University of Massachusetts Lowell, and his blog is different from a lot of blogs I read. He only posts once a month, on the 17th, and his posts are quite long and in depth. Some math bloggers who take the long post approach can veer towards academic-ese, sometimes ending up with posts that read a bit like journal articles. (There’s nothing wrong with that, and those posts can be very valuable to researchers interested in those specific topics.) Propp’s posts don’t feel long, they feel leisurely. He takes a topic and slowly unfolds it, trying to make it accessible to people regardless of their mathematical background.

I decided to write about Mathematical Enchantments now because of his most recent post, The Curious Incident of the Boasting Frenchman. I try to hide it, but I have a strong curmudgeonly streak deep down inside, and one of my inner curmudgeon’s pet peeves is when people uncritically report that Fermat had a proof of Fermat’s Last Theorem that would not fit in the margin of his copy of Diophantus’ Arithmetica. In this regard, Propp seems to be a kindred spirit, though he is not very curmudgeonly about it.

Fermat’s Last Theorem is the statement that there are no positive integers x, y, and z so that xn+yn=zn for n an integer greater than 2. Fermat did indeed claim that he had a proof which could not fit in the margin, but the proof we know was only completed in the 1990s by English mathematician Andrew Wiles using techniques far removed from any Fermat would have known. Was Fermat bluffing, was he mistaken, or did he indeed have a proof that we have just failed to recreate for more than 300 years? Propp explores those options and includes information about the mathematical culture Fermat was a part of at the time. Along the way he walks us through the idea of infinite descent, one of my favorite proof techniques.

(Serendipitously, Pat Ballew just published a post about “almost integers,” numbers that are surprisingly close to being integers, like eπ√163, which is close enough to 262,537,412,640,768,744 to fool a lot of calculators. Ballew doesn’t mention them in his post, but some of my favorite almost integers are ones that come from Fermat “near misses,” numbers thatare close to satisfying Fermat’s equation for some n other than 1 or 2. Harvard mathematician Noam Elkies has a page of Fermat near misses for your perusal.)

One of the things I like about Mathematical Enchantments is that Propp often writes about subjects I’ve seen a hundred times, but I still learn from his posts. He often brings more depth or historical context to the discussion than the topics usually get. For example, too many proofs that 0.999…=1 rely on intimidation. Instead, Propp digs into how the expression 0.999… can mean anything at all, going back to Archimedes and forward to the hyperreals for different interpretations of infinitesimals. The math he covers is always interesting but never feels like a party trick, even though one of his posts is literally about a party trick—the trick of harnessing projective geometry and symmetry to rotate children through paintball teams optimally at a birthday party.

Each post on Mathematical Enchantments takes some time to read and digest, but they are worth the effort. Besides your own edification, they are great examples of mathematical exposition done both accessibly and deeply, and I know I am always looking for inspiration in that direction.

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The Ramanujan Movie

I saw the Ramanujan Movie and I loved it. “The Man Who Knew Infinity,” came out a few weeks ago, starring Dev Patel as Srinivasa Ramanujan and Jeremy Irons as G.H. Hardy, it was a beautifully told story of what Hardy would later call, “the one romantic incident in my life.” images The story is a truly captivating one, and the mathematics in play — partition functions, mock modular forms, and Ramanujan’s famous identities — fall secondary to the relationship between the two main characters.

As a mathematician, I tend to watch mathematical movies with one eye shut for fear of the gross misrepresentations that will befall my field and my fellow practitioners. I felt like this movie did a splendid job. It was light on the details, but gave just enough information on the partition function so that a lay audience could grasp the problem. But more importantly, thanks in large part to the consulting work of Ken Ono, the scenes were they were actually doing math felt so real to me. The way they worked together, the way they talked to each other and interacted, and even the montages of Ramanujan working alone in his room felt so familiar. And I’m so glad that for once a math movie decided to forgo that strange mathematician-writing-equations-on-a-piece-of-glass trop. Ramanujan used pen and chalk. Just like in real life.

Ramanujan, who never had any formal training in mathematics, arrived at Trinity with a notebook full of equations and identities without a single proof. To him, the proofs were trivial, since everything just seemed obvious to him. To Hardy, the results might as well not exist if they couldn’t be substantiated with proof. In one scene, Hardy lays into Ramanujan, telling him that they cannot proceed unless he begins to formalize rigorous proofs of his ideas. Ramanujan balks. I loved this scene, because this exact struggle happens on a smaller scale in my proof writing classes all the time. A student makes (what to him is) an obvious claim and I say, “prove it.” Then he said, “but it’s obvious.” And I say, “tell me why.” And he gets so mad. Eventually the student understands what I mean, but usually not without a mini revolt. I love to imagine that for at least a few moments Hardy and Ramanujan felt the same strains of the teaching-student dynamic that we all do.

As a side note, it was absolutely thrilling to sit in a movie theater full of people who all came out to see this movie about one of my favorite people. I know, it’s not about me. But wow, if I didn’t just want to tap the people in front of me on the shoulder and ask, “Isn’t Ramanujan just the greatest. Which identity do you love the most??”

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Beyond Euro-American Mathematics

A New York Times op-ed by  earlier this month calls out university philosophy departments for their lack of diversity. “We therefore suggest that any department that regularly offers courses only on Western philosophy should rename itself “Department of European and American Philosophy.”

The Monte Albán observatory in Oaxaca, Mexico. Image: Rob Young, via Wikimedia Commons.

The Monte Albán observatory in Oaxaca, Mexico. Image: Rob Young, via Wikimedia Commons.

Garfield and Van Norden take the position that philosophy deserves to be singled out in a way mathematics and science do not. They write,

Others might argue against renaming on the grounds that it is unfair to single out philosophy: We do not have departments of Euro-American Mathematics or Physics. This is nothing but shabby sophistry. Non-European philosophical traditions offer distinctive solutions to problems discussed within European and American philosophy, raise or frame problems not addressed in the American and European tradition, or emphasize and discuss more deeply philosophical problems that are marginalized in Anglo-European philosophy. There are no comparable differences in how mathematics or physics are practiced in other contemporary cultures.

Putting aside the fact that it is strange to ignore literature, art, history, or religion, departments that are frequently Eurocentric in ways that mathematics and physics are not, it may not be as clear as Garfield and Van Norden think that mathematics departments should not be criticized for being Eurocentric. The apparent universality of mathematics is one of the things that draws people to mathematics, but nothing takes place in a vacuum. Michael Harris, author of the book Mathematics without Apologies and a blog of the same name, writes,

What is or is not ‘comparable’ is in the eyes of the comparer, of course, and it’s no doubt true that cultural differences are no barrier to communication between contemporary mathematical practitioners in Asia and the rest of the world.  Historically, however, mathematics developed around the world in conjunction with a variety of metaphysical traditions, and this has inevitably affected the approaches to foundational matters.

In another post, he suggests that “the most interesting problem currently facing philosophy of mathematics is to clarify how or whether Chinese and European mathematics differ and how or whether these differences reflect differences in the respective metaphysical traditions.”

I taught math history for two semesters, so I’m hardly an expert on how the subject is taught in general, but I did struggle with how Eurocentric my own math history background and the vast majority of math history resources I came across were. Sometimes it seems like the dominant math history narrative is “Greeks (nevermind that many of the ‘Greeks’ were from North Africa and the Middle East, we call them Greeks so you’ll think of them as European) invented mathematics, it died out around 500 CE, and then Italians started doing it again in the 15th century.” If we’re lucky, the narrative might mention Al-Khwarizmi, whose name gave us the word algorithm and whose book Hisab al-jabr w’al-muqabala gave us the world algebra.

Unfortunately, my math history class fell into the Eurocentric model more than I wish it had. I felt I did not have the knowledge base necessary to teach a class specifically on non-European math well, but I did require my students to do projects on mathematics from “non-western” sources. (It’s difficult to figure out the right label here. I wanted my students to research mathematics from someone whose culture is not well represented in math history books. Non-European is not quite right, because many so-called Greeks were from Africa and Asia. Non-western is not quite right because mathematics from the Americas before European conquest very much counts. In the end, I went with “non-western” in scare quotes and a long explanation of what I meant.)

One difficulty we encountered in researching non-western math sources was that my students and I are all products of the same metaphysical tradition, as Harris would call it, in mathematics, and it was difficult for us to understand mathematics from other traditions on their own terms rather than viewing them through our own cultural lens. Another, as I’ll come back to later, was the dearth of documents available for them, especially if they were interested in math from pre-Columbian America, Africa, or Oceania.

Eurocentricism in mathematics is on my mind right now not only because of Garfield’s and Van Norden’s New York Times article and Harris’s response to it but because I’m on vacation in Oaxaca, Mexico, home to several impressive ruins from pre-Hispanic civilizations, including Zapotec and Mixtec. These civilizations are not as well known as the Aztec or Maya, but they, like those more famous Mesoamericans, were accomplished astronomers. (In the ancient world, astronomy and mathematics went hand in hand in a way they don’t today.) On a tour of Monte Albán, the remains of a Zapotec city, we saw buildings oriented exactly to the cardinal directions and an observatory that occasionally aligns with the sun perfectly. (Perhaps we should call it Albánhenge.)

Heartbreakingly, the destruction of indigenous populations and documents from indigenous cultures means we have very few resources for learning about the astronomy and mathematics of ancient Mesoamerican people. I learned this when I saw how limited the choices were for my math history students wanted to find Mesoamerican math sources for their projects, despite the sophisticated astronomical calculations they did. (Go ahead, try to understand the Maya calendar system!)

I would love to share some good online resources on non-Euroamerican mathematics, but sadly, I don’t have many. Offline, The Crest of the Peacock seems to be one of the best books about non-European mathematics out there, and the North American Study Group on Ethnomathematics publishes a Journal of Mathematics and Culture.

Online, the award-winning MacTutor math history archive has some articles about the mathematics traditions of different cultures. (If you’re wondering why they have an article specifically about the mathematics of Scotland, note that the site is hosted by the University of St. Andrews.) The Story of Mathematics, an online math history site, also has some articles about Maya, Chinese, and Indian mathematics. On blogs, the pickings are a bit slim. I do want to toot my own horn a bit and point you to my students’ math history blog, 3010tangents. There, my students wrote about a lot of topics, including the amazing navigation devices of the Marshallese, the number zero in Babylonian, Indian, and Maya mathematics, and The Nine Chapters on the Mathematical Art, a Chinese math text.

Do you have any more suggestions on where to learn about mathematics from cultures who are often left out of the history mathematics? Please share them below.

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Functions Too Cool For Facebook. But Don’t Worry, We’ve Got You Covered

This map captures the web of contributions from over 80 participants in the LMFDB project over several years.

This map captures the web of contributions from over 80 participants in the LMFDB project over several years.

Today is the official launch of the L-functions and modular forms database. The LMFDB is a database containing all the relevant information about millions of mathematical objects. Set up like a Facebook for mathematical objects — by objects I mean curves, functions, special equations and structures — the LMFDB lets us see which objects are related to each other, which ones share a common ancestor, and which ones can at least play nice.

But maybe you, like nearly all people who aren’t seeped in a daily brew of number theory, wouldn’t recognize an L-function if it walked into the room right now. Even so, I promise this database has some exciting implications for you. Yeah, you. Understanding how the social network of all these millions of objects looks can give a huge kick in the pants to the famous Riemann Hypothesis. But even for those of us who don’t run around muttering about zeroes on the critical strip, we still profit, perhaps unwittingly, from this and other really hard number theory problems every day when we use the internet. Knowing more about he universe of the LMFDB can help find vulnerabilities in encryption, keeping our private data and transactions safe.

The connection between elliptic curves and modular forms is just a small part of the L anglands Program, a vast web of conjectures proposed by Robert Langlands, at the Institute for Advanced Study, in the late 1960s. Image courtesy of David Dumas, Timothy Boothby, and Andrew Sutherland.

The connection between elliptic curves and modular forms is just a small part of the L anglands Program, a vast web of conjectures proposed by Robert Langlands, at the Institute for Advanced Study, in the late 1960s. Image courtesy of David Dumas, Timothy Boothby, and Andrew Sutherland.

But much more broadly — and perhaps more importantly — one of the motivating goals of so much mathematics of the last century has been to find a so-called grand unifying theory of mathematics which we call the Langlands Program. In the mathematical universe we deal with all kinds of seemingly unrelated objects, like those curves and functions and other things I mentioned earlier. “The connections between these classes of objects lie at the heart of the Langlands program,” explained the Fields Medalist Terry Tao in a blog post about the LMFDB today. The LMFDB teases out a lot of surprising relationships between theoretical objects, ones that aren’t so easy to see when you look at these things one at a time.

And even if you aren’t chasing the grand unified theory, if you work in certain areas of math, these objects come up all the time, and having an atlas to this mathematical universe can be incredibly helpful. As Emmanuel Kowalski wrote on his blog today, the LMFDB can help us understand their “random and possibly spooky” behavior.

Another huge boon of the LMFDB is that it stores billions of time intensive calculations for immediate retrieval — literally thousands of years worth of computations — saving our future selves huge time and effort. Tim Gowers, Fields Medalist and proponent of effort-saving tools, wrote about the LMFDB on his blog today, saying “I rejoice that a major new database was launched today.” This frees us up to do other things, like prove deep results.

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Straws Thingys and Other Mathematical Sculptures

I love an abstract math pondering session as much as the next mathematician (or at least within epsilon), but there’s something immensely satisfying about coming back down to earth and using your hands to make something. At some point last November, I stumbled onto Zachary Abel’s blog series about Straws Thingys, and I knew a Straws Thingy had to be mine.

A Straws Thingy. Image: Zachary Abel.

A Straws Thingy. Image: Zachary Abel.

A Straws Thingy is indeed a thingy made of straws. In this case, it happens to be a compound of five intersecting tetrahedra, popularized by mathematical origami guru Thomas Hull. After several months of forgetting to buy the recommended brand of straws from Target, I finally managed to make my very own Straws Thingy last week.

My Straws Thingy, before and after savagely breaking free from its scaffold. Image: Evelyn Lamb.

My Straws Thingy, before and after savagely breaking free from its scaffold. Sadly, the straws I bought only came in four colors, so my fifth tetrahedron is multi-colored. Image: Evelyn Lamb.

Abel is a graduate student in mathematics at MIT as well as a mathematical sculptor, and I just loved his Straws Thingy posts. HHe had challenged himself to write a blog post a day for the month of November (#NaBloPoMo, a more modest undertaking than #NaNoWriMo), so the instructions are conveniently served in bite-sized pieces.

I tend to be something of a mathematical social butterfly. I like to learn a little about something and then flit away to the next thing. Abel, on the other hand, is much more thorough. His month-long Straws Thingy series explores the subtle asymmetries of various Straws Thingys, eventually building to a five-dimensional hypercube of thingys (conveniently immersed in three-dimensions). 

If you have a mirror, you only need 16 Straws Thingys to make a 5-dimensional hypercube of them. Image: Zachary Abel.

If you have a mirror, you only need 16 Straws Thingys to make a 5-dimensional hypercube of them. Image: Zachary Abel.

He also gives us a peek behind the scenes to show us the origin story of the scaffold he designed to make the Straws Thingy easier to assemble.

A Straws Thingy scaffold, ready for action. Image: Evelyn Lamb.

A Straws Thingy scaffold, ready for action. Image: Evelyn Lamb.

If you want to make a Straws Thingy (or 32) of your own, his scaffold is available as a pay what you want download, and the instructions are easy to follow on his blog. I poked around the blog a little bit after building my Straws Thingy. He has a lot more fun posts about geometry and mathematical sculptures, including the Penny Pincher (made with $20.00 in pennies!) and instructions for a “potentially lethal” Impenetraball. Maybe I’ll get there eventually, but right now I still have a lot of leftover straws.

A Straws Thingy in Waiting. Image: Evelyn Lamb.

A Straws Thingy in waiting. Image: Evelyn Lamb.

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Math and Verbal Gymnastics

We are coming to the end of Math Awareness Month, whose theme this year was The Future of Predictions. A clever theme name, indeed. I do love when mathematics and verbal gymnastics come together. And on that theme of math and words, you should know that April is not only math awareness month, but it’s also National Poetry Month! In honor of this double-whammy, I thought we could take a moment to explore the intersection of math and poetry.

As it turns out, there are actually quite a few blogs dedicated to poetry and mathematics. If you, like me, are a mathematician who is new to mathematical poetry, a good place to start is with “Five Types of Mathematical Poetry” on the blog Mathematical Poetry.

If the strictly lexical type of mathematical poetry is what you prefer — that is, mathematical poems constructed from the written word and influenced by ideas in math — then I suggest JoAnne Growney’s blog, Intersections — Poetry with Mathematics. In the theme of math awareness month, Growney posted a beautiful poem by Joyce Nower, inspired by prediction, fate, and the tragic story of the mathematician Hypatia.

Another type of mathematical poetry melds the written word with mathematical symbols, not necessarily following any mathematical rules. The late Bob Grumman, a pioneer in visual mathematical poetry, described it as “poetry that does mathematics, rather than merely discusses mathematics.” In a post on the Scientific American Guest Blog, Grumman discusses the state of the art form and the work of fellow visual poet Karl Kempton.

Magic Square By BabelStone - Own work, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=16206206

Magic Square By BabelStone – Own work, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=16206206

Pure mathematics can also be seen as poetry. The patterns and repetition in numerical and symbolic mathematics do echo those in traditional lexical poetry. To the right you can see a magic square, an ancient example of pure mathematical poetry.

In the spirit of poetry and math awareness, let me close with a terrible haiku that I just wrote in honor of the close of the semester.

reaching the limit
harmonic series diverge
and so too do we

Share your #mathematicalhaikus with me on Twitter @extremefriday.

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Fold Your Way to Glory

Yesterday, I led a meeting of a Teachers’ Math Circle about the fold and cut theorem. This theorem says any region with a polygonal boundary can be folded and cut from a sheet of paper using only one cut. I learned about the theorem last year when Numberphile posted this excellent video featuring a virtuoso performance from Katie Steckles, who folds and cuts every letter of the alphabet from memory.

I was a little nervous about leading the program because I had prepared almost nothing to say. Everything I thought about saying was boring, so I decided the best way to approach the activity was to just get people started on it. Luckily for me, the group was ready to jump right in. I dumped a bunch of paper into the middle of the table, and people started folding.

I encouraged people to try the most symmetric shapes first, but other than that, I didn’t have to give them many suggestions. I was prepared for some frustration when they started trying the scalene triangle because it’s a big step up in difficulty, but several of them got the scalene pretty quickly, and no one seemed to give up. In general, the strange shapes people got when you mess up were amusing rather than frustrating.

My favorite fold-and-cut mistake. I was trying to make a rectangle with a smaller rectangle inside it. Image: Evelyn Lamb.

My favorite fold-and-cut mistake. I was trying to make a rectangle with a smaller rectangle inside it. Image: Evelyn Lamb.

Participants almost immediately started asking mathematical questions and trying to extend the activity: do we have an existence theorem? Must we always fold along every angle bisector? Is there a general theory of folding? I liked Anna Weltman’s suggestion of trying to make things without drawing on the paper, and I spent some time trying to fold stars without drawing them, but the teachers didn’t really bite on that. Instead, some of them started thinking about minimal folding numbers for different shapes, and some of them worked on developing a folding algorithm.

Erik Demaine is one of the pioneers of fold-and-cut theory and the mathematics of paper folding in general. His page about folding and cutting has links to all the gory mathematical details as well as some templates. I ended up bringing copies of his swan to the teachers’ circle. They are beautiful, but I had mixed feelings about bringing them because they have the fold lines marked on them already. I didn’t hand them out until one group had started talking about how to use angle bisectors and perpendiculars in their folding algorithm, and I thought the swan template might give them some ideas. Because I gave them only the template, not any explanation of how it was made, I think it didn’t take away too much of their fun.

My fold-and-cut swan now enjoys pride of place on my new hexagonal shelf. Image: Evelyn Lamb.

My fold-and-cut swan now enjoys pride of place on my new hexagonal shelf. Image: Evelyn Lamb. Swan template: Erik Demaine.

In addition to Demaine’s swan, I brought templates for lots of different shapes from Patrick Honner and Joel David Hamkins, who uses hole punching symmetry activities as a warm-up for cutting. I also got ideas from Mike Lawler, who has done fold and cut activities with kids, and Kate Owens, who ran a fold-and-cut workshop for teachers.

I’ve done a little bit of origami, but I’ve never gotten good enough to feel like I had geometric intuition for doing it. I’m still at the level where I follow directions and get what the book says I should. Making these fold-and-cut shapes, though, is an easy way to start thinking about paper folding mathematically and creatively. Thanks to the resources I mentioned above, you too can easily introduce people to the joys of mathematical paper folding.

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Packing Heat: The World Of Sphere Packings Is On Fire

Suppose somebody hands you a bunch of oranges and asks you to stack them on a shelf, I’ll bet I can guess how you would do it. You’d build a pyramid by laying down a base layer and then fill in the upper levels by placing oranges in each of the divots provided by the layer below. If you’ve done this before, you might have noticed that the oranges in the base layer create a repeating hexagon pattern. In case you don’t have a crate of oranges next to you right now to try this out, check out the photo below.

A hexagonal packing of oranges creates the base layer on the left, and then the pyramid continues to rise using the hexagonal close packing.  Image via Jeremy Jenum Flickr Creative Commons.

A hexagonal packing of oranges creates the base layer on the left, and then the pyramid continues to rise using the hexagonal close packing. Image via Jeremy Jenum Flickr Creative Commons.

This is called a hexagonal circle packing, and it’s the densest way to pack a bunch of circles together. By densest, I mean that any other way you pack together circles is going to have much more empty space left over. When you place the subsequent layers on top by filling in the divots, what you’re doing is creating a well-studied arrangement called the hexagonal close packing of spheres. Just like the hexagonal packing in 2-dimensions, the hexagonal close packing is the densest way you can pack 3-dimensional spheres together. This was a result proved by Thomas Hales in 1998.

These both belong to the broader family of n-dimensional sphere packings, and it’s been a long standing open problem to find the densest sphere packings in each dimension. While we have the nice orange stacking analogy to help us visualize dimensions 2 and 3, in higher dimensions we can’t visualize things in the same way. But here is the essence of the problem. In any dimensions, a sphere is just a set of points that are equidistant from some center point, and a dense sphere packing is just an arrangement of non-overlapping spheres that fills up as much ambient space as possible.

Maryna Viazovska unlocked the solution in 8 dimensions, paving the way for the 24-dimensional solution. Photo Courtesy of Oberwolfach photo archives.

Maryna Viazovska unlocked the solution in 8 dimensions, paving the way for the 24-dimensional solution. Photo Courtesy of Oberwolfach photo archives.

A few weeks ago Maryna Viazovska, currently a post-doc at the Berlin Mathematical School and the Humboldt University of Berlin, solved the sphere packing problem in 8-dimensions. Erica Klarreich, a math journalist for Quanta Magazine gives details on how Viazovska arrived at her solution, and some stories about the people she met along the way.

And then not a week went by before she and her coauthors, Henry Cohn, Abhinav Kumar, Stephen D. Miller, and Danylo Radchenko generalized her methods to solve the sphere packing problem in a 24-dimensions. On his blog, mathematician Gil Kalai gives some historical background for the 8- and 24-dimensional sphere packing problems.

In a video posted by the Institute for Advanced Study, Stephen Miller gets into the details of the proof, he says “there’s something very special about 8 and 24, we can’t expect every week to keep proceeding like this.” Although the dimensions 8 and 24 might seem totally random, the reason these solutions came so close on each other’s heels is that these sphere packings — unlike those in other dimensions, as discussed on the n-category cafe — are related to two special lattices, E8 and the Leech lattice. Having this connection to lattices, which, full disclosure, I’m obsessed with, means that there is a world of machinery in the realm of modular forms for dealing with the packings. In a very broad sense, solving the packing problem came down to finding some suitable modular function that satisfied an appropriate list of properties that are derived from methods in harmonic analysis.

Sphere packing problems, of course, have many interesting applications, but the one that has always fascinated me is the link between dense sphere packings and error correcting codes. Trying to pack n-dimensional spheres as close to each other as possible is like trying to find points (namely, the center point of the sphere) that are as close to each other as possible, while maintaining some prescribed amount of distance between them (namely, the buffer created by the sphere around each center point). This acts just like an error correcting code, in the sense that we want to find code words that are similar enough that we can build a language out of them, but far enough apart that they can be transmitted over noisy channels and not be totally degraded by interference.

Like all good problems, sphere packings touch on many branches of mathematics: number theory, geometry, analysis. The fact that this problem has so many approaches and that its solutions are simultaneously so diverse in flavor, John Baez points out so perfectly in his blog post, “hints at the unity of mathematics.”

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