## Carnival of Mathematics 86

Welcome to the 86th Carnival of Mathematics! $86 = 2 \cdot 43 = 222_6 = 20+21+22+23 = 3^2 + 4^2 + 5^2 + 6^2$ is semiprime, nontotient, and noncototient. It is also happy since $8^2 + 6^2 = 64 + 36 = 100$ and $1^2 + 0^2 + 0^2 = 1$. In fact, it is the smallest happy, nontotient semiprime (the only smaller happy nontotient is 68—which is, of course, 86 in reverse—but 68 is not semiprime).

However, the most interesting mathematical fact about 86 (in my opinion) is that it is the largest known integer $n$ for which the decimal expansion of $2^n$ contains no zeros! In particular, $2^{86} = 77371252455336267181195264$. Although no one has proved it is the largest such $n$, every $2^n$ up to $n = 4.6 \times 10^7$ (which is quite a lot, although still slightly less than the total number of integers) has been checked to contain at least one zero. The probability that any larger power of 2 contains no zeros is vanishingly small, given some reasonable assumptions about the distribution of digits in base-ten expansions of powers of two.

“Eighty-six” is also apparently some sort of slang term in American English, but it really has nothing to do with math, so who cares? Onward to the carnival! I had a lot of fun reading all the submissions, and have decided to organize them somewhat thematically—though they don’t always fit perfectly, so don’t assume you won’t be interested in a post just because of my categorization!

### Art

Christian Perfect has started a series of posts on the theme of “Arty Maths”, with links to artistic images and videos with a mathematical bent. Above is a cool example, some sort of stellated polyhedron made out of money by Kristi Malakoff (you can find more here).

Katie Steckles submitted a link to Robby Ingebretsen’s blog post First Digital 3D Rendered Film (from 1972) and My Visit to Pixar. Katie says,

This is possibly the earliest example of a computer animation, and one of its two creators, Edwin Catmull, who went on to found Pixar, is credited with “having work[ed] out [the] math to handle things like texture mapping, 3D anti-aliasing and z-buffering”. Fascinating to think he had to invent all of that in order to do this!

Robby’s blog post (and the extensive comments on it) give a lot more context and fascinating details. And, of course, you can watch the video itself!

Mike Croucher of Walking Randomly writes about some cool mathematically-themed stained glass windows, and wonders whether anyone knows of any others.

### Statistics/data analysis

Arthur Charpentier of Freakonometrics writes about Nonconvexity, and playing indoor paintball: if a bunch of people in a nonconvex playing area are all holding water pistols and shoot at the closest person, who doesn’t get wet?

Katie Steckles submitted a link to Data: it’s how stores know you’re pregnant, an article by Matthew Lane of Math Goes Pop! Ever wonder how companies can predict various things about you (such as whether you are pregnant!) based on your browsing habits and other publicly available data? This article explains some of the basic math underlying this sort of “data mining”.

John Cook of The Endeavour answers the question: What is randomness? in Random is as random does. It turns out that the best answer might just be to avoid answering at all!

### Geometry

Augustus Van Dusen of thinkingmachineblog submitted a post titled Superellipse, saying

I read an article about Sergels torg, a plaza in Stockholm, being an example of a superellipse. When I looked up superellipse on Wolfram math world, I noticed that the area formula involved gamma functions. I then decided to derive the result myself to see if it could be simplified and how it would reduce to the familiar formula for the area of an ellipse.

Frederick Koh of White Group Mathematics shares a geometric solution to an optimization problem that doesn’t initially seem like it has anything to do with geometry.

Zachary Abel of Three-Cornered Things has written a series of three “excursions into the miraculous and interconnected workings of the humble triangle”: Many Morley Triangles, Several Sneaky Circles, and Three-Cornered Deltoids. These are some of my personal favorites from this month’s Carnival: chock-full of surprising mathematics and beautiful illustrations and animations!

### Teaching

Colin Wright writes The Trapezium Conundrum: how should a trapezium (aka trapezoid if you’re from the US) be defined—with exactly one pair of parallel sides or at least one pair of parallel sides? More generally, how are definitions arrived at and agreed upon? The answer may depend on the audience.

Karen G. of school•a•rama muses upon the relationship between language and learning place value in her post Looking to Asia.

On her blog Math Mama Writes…, Sue VanHattum writes about Linear Algebra: Leading into the Eigen Stuff. Sue says, “I’m teaching linear algebra for the first time in over a decade. That has meant relearning it—a delightful experience.”

Paul Salomon of Lost In Recursion writes Exponents and the Scale of the Universe – a 21st Century math lesson, a fun story about how an initially dry lesson on exponents turned into a remarkable learning experience.

### Fun

Alistair Bird submitted a link to Enormous Integers, saying,

It’s still a common enough misconception that pure mathematics research must be about larger and larger numbers, but it’s still nice to sometimes play up to this stereotype, as John Baez’s blogpost on Azimuth about ‘Enormous Integers’ does. Comments are worth a look too.

Pat Ballew writes on Pat’sBlog about Pandigital Primes: “exploring pandigital primes and finding out how handy Computer programming skills might be”.

Quick, what comes next in the series $\pi/2, \pi/2, \pi/2, \pi/2, \pi/2, \pi/2, \pi/2, \dots$? The answer, as explained by Steven Landsburg on his blog, The Big Questions, may surprise you! (Thanks to Katie Steckles for the submission, via Alexandre Borovik.)

Paul Salomon of Lost In Recursion displays The “Lost in Recursion” Recursion. Can you figure out what’s going on without getting lost in the “The ‘Lost in Recursion’ Recursion” recursion?

### Stuff That Did Not Fit In Any Other Category But Is Still Awesome

Colin Beveridge of Flying Colours Maths submitted Secrets of the Mathematical Ninja: The surprising integration rule you don’t get taught in school, and writes,

When I stumbled across this rule, my reaction was ‘whoa.’ It’s quick, it’s extremely dirty, and it’s surprisingly accurate. The kind of thing the mathematical ninja dreams of.

Andrew Taylor writes a guest post, Electoral Reforms and Non-Transitive Dice, on The Aperiodical, explaining Why Choosing a Voting System is Hard in terms of a set of nontransitive dice.

Peter Rowlett, of Travels in a Mathematical World, opines in his post, What a nice job you have, that a popular ranking listing “mathematician” as one of the top ten best jobs shouldn’t just be accepted and repeated uncritically.

In her article How culture shaped a mathematician, Carol Clark gives a glimpse into the life and background of mathematician Skip Garibaldi. She writes:

Mathematicians see the world differently than me. I interviewed a mathematician to get a glimpse of that view, and learned how everything from fine art to popular films and books played a role in shaping that view.

The previous Carnival of Mathematics was hosted at Travels in a Mathematical World; next month, the 87th Carnival will be hosted by Mr. Chase at Random Walks, so start getting your submissions ready now! For lists of past and future carnivals, instructions on submitting, and answers to frequently asked questions, see the main Carnival of Mathematics site. The next Math Teachers at Play carnival is also coming up soon, with a submission deadline of this Friday.