Carnival of Mathematics 148

Monday, July 31, 2017 · 5 min read

Welcome to this month’s edition of the Aperiodical Carnival of Mathematics! The Carnival is a monthly roundup of exciting mathematical blog posts. Last month, it was hosted by Lucy at Cambridge Maths. This month, it is my honor to host it here at Comfortably Numbered. But first: a story…

A long time ago, there lived a great and powerful king. His kingdom was rich and his subjects content; ambassadors from afar would often come to his palace gates to offer presents from their distant homes. One day, such a traveler approached the king and challenged him to a game of chess. The king, always welcoming to his guests, accepted the challenge. He sent a minister to fetch a chessboard and pieces for the game. Soon, they were playing an intense game; the court watched in a hushed silence as the great king and the foreign traveler concentrated on the board. At last, the king won.

Now, this particular traveler was quite an arrogant young man. Outraged that anyone could defeat him, he rose and smashed the chessboard with his fist. The carved wood shattered into a number of pieces, splitting at the delicate joints between the black and white squares. The pieces clattered noisily on the marble floor.

Normally, such behavior would be punishable by death. But this king was a merciful king: he understood the insecurities of youth, and he believed in the healing powers of recreational math. So, he decided to give the traveler a second chance. He said,

“Young man, do you notice anything about the shattered pieces of the chessboard?”

The traveler, embarrassed, paused before answering,

“Your majesty, each piece seems to be a perfect square: there are three 4-by-4 squares, twelve 1-by-1 squares, and one 2-by-2 square.”

“Very good,” replies the king, “very good indeed. Now, if you can tell me how many ways there are to break a chessboard into (indistinguishable) square pieces, I shall spare your life.”

The next day, the traveler returned to the palace, escorted by guards. Standing before the throne, he handed the king a card upon which was written his answer: 148, which coincidentally happens to be this month’s Aperiodical Carnival of Mathematics!

Ayliean MacDonald entrances us by drawing a giant dragon curve! She says there was a bit of a learning curve involved, but fortunately for us, her video is a magnificent timelapse.

I love dragon curves, so I drew a gigantic one. I regretted starting it almost immediately. Let’s just say this was my first attempt and its a learning curve.

Rachel Traylor extends sets to “fuzzy” sets, which are just like normal sets, but, well, fuzzy!

What if we relax the requirement that you either be in or out? Here or there? Yes or no? What if I allow “shades of grey” to use a colloquialism? Then we extend classical sets to fuzzy sets.

Dr. Nira Chamberlain explains how important mathematical modeling is, and in particular, uses an extension of the gambler’s ruin problem to show how a model’s assumptions and limitations are related.

To me, mathematical modelling is about looking into the real world; translating it into mathematics, solving that mathematics and then applying that solution back into the real world.

Edmund Harriss exploits Desmos to draw some fantastic images.

One of my courses was to use Desmos to help develop thinking on functions and start to get to some of the ideas of calculus (without the need for the algebra). Here are the example calculators that I set up for the course.

David H. Bailey examines a surprisingly large collection of published papers that assert (incorrect!) values of pi.

Aren’t we glad we live in the 21st century, with iPhones, Teslas, CRISPR gene-editing technology, and supercomputers that can analyze the most complex physical, biological and environmental phenomena? and where our extensive international system of peer-reviewed journals produces an ever-growing body of reliable scientific knowledge? Surely incidents such as the Indiana pi episode are well behind us?

Jeremy Kun embeds Boolean logic in polynomials, thus revealing why finding the roots of multivariate polynomials is NP-hard.

This trick is used all over CS theory to embed boolean logic within polynomials, and it makes the name “boolean algebra” obvious, because it’s just a subset of normal algebra.

Ben Orlin evaluates whether or not you should ever buy two lottery tickets.

Gambling advice from mathematicians is usually pretty simple. In fact, it’s rarely longer than one word: Don’t! My advice is gentler…

Jimmy Soni and Rob Goodman emphasize that Claude Shannon’s wife Betty Shannon was a brilliant mathematician herself, and in fact instrumental in many of his successes.

Shannon valued the help. Though his ideas were very much his own, Betty turned them into publishable work. Shannon was prone to thinking in leaps—to solving problems in his mind before addressing all the intermediary steps on paper. Like many an intuitive mind before him, he loathed showing his work. So Betty filled in the gaps.

Peter Cameron elaborates on the conference held in honor of his 70th birthday, in Lisbon.

There is far too much, and far too diverse, mathematics going on here for me to describe all or even most of it. Nine plenary lectures on the first day! … I didn’t mention the film that the organisers have made about me (based mostly on old photographs). I am really not used to being in the spotlight to this extent!

Patrick Honner elucidates an erroneous exam question, and along the way tells us the story of a 16-year-old student who exposed the error and started a petition in response.

So, this high-stakes exam question has no correct answer. And despite the petition started by a 16-year-old student that made national news, the New York State Education Department refuses to issue a correction.

Danesh Forouhari extracts a fantastic math problem from properties of the factorization of 2016.

As we were approaching end of 2016, I was wondering if I could come up with a math puzzle which ties up 2016 & 2017, hence this puzzle.

Rachael Horsman extrapolates a lesson about measuring to explain the entire number line!

As we write the framework, activities such as pacing off form critical junctions between various areas in mathematics … Making these explicit to teachers and pupils helps cultivate their understanding of the connections that make up mathematics.

Vijay Kathotia enlightens us by revealing where the roman numeral for “10” might have come from.

What is the Roman numeral for ten? If you answered ‘the letter X’, it may not be quite right. It may well be what you write – but do you know why? There are at least two stories for explaining how X came to represent ten.

Lucy Rycroft-Smith engages UK-based artist MJ Forster in an interview about how math influences his art.

His latest series of paintings seem inherently mathematical; but just how explicit is the mathematics in his art, and how does he feel about the subject?

Brian Hayes estimates pi using rational numbers and an HP-41C calculator!

Today, I’m told, is Rational Approximation Day. It’s 22/7 (for those who write dates in little-endian format), which differs from pi by about 0.04 percent. (The big-endians among us are welcome to approximate 1/pi.)

Finally, the world expresses its sorrow: Fields medalist Maryam Mirzakhani passed away this July after a long battle with cancer. In keeping with the Carnival’s tradition, I offer you two blog posts from the mathematical community.

Ken Regan: on the significance and brilliance of her work

She made several breakthroughs in the geometric understanding of dynamical systems. Who knows what other great results she would have found if she had lived: we will never know. Besides her research she also was the first woman and the first Iranian to win the Fields Medal.

Terence Tao: a more personal perspective

Maryam was an amazing mathematician and also a wonderful and humble human being, who was at the peak of her powers. Today was a huge loss for Maryam’s family and friends, as well as for mathematics.

If you would like to learn more about her life and work, I encourage you to read some of the articles and watch the video on the AMS’ tribute to Maryam Mirzakhani.

This concludes the 148th edition of the Carnival of Mathematics. Please do join us next time, when Mel from Just Maths hosts the 149th!

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