Image: David Simmonds, via flickr.

Welcome to the 103rd Carnival of Mathematics! The number 103 is prime, and it's the "older" twin of a pair of twin primes as well—or is it the "younger" twin because it comes later? Regardless, 101 and 103 are twin primes. According to Tanya Khovanova's site Number Gossip, 103 is not only prime and happy but also odious and deficient. (I guess it's a bit self-contradictory, too—I'm certainly not happy when I'm feeling deficient.) I wrote about happy numbers last year. 103 is odious because it has an odd number of ones in its binary expansion: 1100111. It's deficient because its positive divisors (other than itself) sum up to a number that is less than 103. (All prime numbers are deficient: their proper divisors only sum up to 1.) Now that we've gotten the niceties about 103 out of the way, we can get on to the fun part: math blog posts. Boy, were there a lot to choose from this month!

I am of course far too modest to point you to my own posts about forgotten trig functions and the linguistic and mathematical complexities of open and closed sets or to the two interviews I've posted in the past month, one with Wesleyan mathematician Constance Leidy and one with University of Michigan mathematical biologists Victoria Booth and Trachette Jackson.

Around the same time I was covering compactness in my class, Jeffrey Shallit was writing about how off-putting it can be for a beginner mathematician to see a phrase like "by the usual compactness argument" in a paper. I understand his frustration, but I also see the point that we can't explain every basic notion that we use in every paper we write, and when you're writing a math research paper, "basic notions" can be quite advanced. Where is the right balance?

The first-ever Heidelberg Laureate Forum happened in late September, and there were some great posts about it by Julie Rehmeyer, John D. Cook, Dana Mackenzie, and others. There were so many good ones that I can't decide which ones to highlight. Just head over and read all of them. Bonus points (and posts) if you can read both German and English.

John D. Cook had the opportunity to talk with Sir Michael Atiyah at the Heidelberg Laureate Forum and posted the full interview on his blog The Endeavour.

Christian Perfect of the Aperiodical wrote about his quest to make an enneahedron (9-faced polyhedron) with the same symmetries as something called the Herschel graph. I was worried that he wouldn't manage to make it work, but it turned out OK in the end! He made a Vine of his completed polyhedron. (If the Vine doesn't load here, you can watch it at the post linked above.)

Complex Projective 4-Space had a few nice posts this month. My favorite was Desert Island Theorems, about the eight theorems you'd bring with you if you were marooned on a desert island. I'd bring the Uniformization Theorem for sure! A post about the Category of Scones was also popular.

Which words have the most anagrams? A programming challenge from Mike Croucher at Walking Randomly. In the comments, someone shares a list of 15 Portuguese words that are anagrams!

This month The Rennaisance Mathematicus had posts about the corresponding accountant, the man who invented pi and the Earls of Macclesfield and how Mary Somerville was not "the first scientist" and Ada Lovelace did not inspire "the first modern computer." In the comments to the latter post, the author Thony Christie says, "I try to write about women in the history of science on their own merits in the context of their times and not within a framework of some sort of great men narrative."

Justin Lanier meditates on "partial cubes" and other variations of standard three-dimensional forms at Math Munch.

How to "level up" in mathematics, by John Baez. What lessons in being excellent can we glean from pro swimmers? Speaking of Baez, he recently started a new blog called Visual Insight for the American Mathematical Society. I love the {6,3,3} honeycomb in hyperbolic 3-space. (Full disclosure: I also write for the AMS, specifically the Blog on Math Blogs.)

Billions and billions of cosines! Finding fixed points of trig functions with your graphing calculator.

How tall would your Twitter timeline be if it actually extended off your computer screen? And what does that have to do with the Doomsday Argument? The always entertaining Randall Munroe of xkcd explains.

Martin Gardner's birthday was in October, so many recreational math enthusiasts host "Celebrations of Mind" this month in his honor. "Shecky Riemann" writes about the recently released autobiography of Martin Gardner on Math Tango, and Sol Liederman interviewed Colm Mulcahy, who actually knew Martin Gardner, for his podcast series Inspired By Math.

You might have heard about the problem of finding the smallest uninteresting number. What about the smallest uninteresting sequence? What does that even mean? Click over to find out.

Oluwasanya Awe figures out whether his team could have been first if a competition had been scored differently. Linear, quadratic, and exponential scales gave very different rankings in this tournament.

Patrick Honner wrote a post about four big ideas in algebra in response to a challenge by Grant Wiggins, who thinks algebra is a dumb course. (For the algebraists in the audience, both posts are about algebra as it is currently taught in most high schools, not the advanced mathematical research area.)

Who doesn't love an unexpected application of linear algebra? The Physics ArXiv blog has a nice post about a recent paper on Google's translation algorithm. "To translate one language into another, find the linear transformation that maps one to the other."

And for even more advanced algebra, check out this post about Life on the Modular Curve, as visualized by a new video game. Yes, please!

Thank you to everyone who wrote these posts and submitted them! Stay tuned next month for the 104th Carnival of Mathematics, hosted by Shecky R. at Math-Frolic.