Hello, and welcome to the 13th Carnival of Mathematics. Since the previous Carnival was posted on Friday the 13th, and since this one is the 13th Carnival, I'll start with some links involving the number 13. The pdf file you can download by clicking (EDIT: you have to cut and paste this link in two parts (it's too long for one line)...for some reason, I can't get it to work: public.carnet.hr/ccacaa/CCA-PDF/cca2004/v77-n3/
CCA_77_2004_447-456_pogliani) is a paper involving various facts about the number 13, although it digresses far and often. This site contains a lot of facts about 13, though some stretch pretty far to be labeled seriously interesting; a few that really are kind of interesting: the squares of 13 and its digit reversal (31) are digit reversals (169 and 961); the sum of the primes up to (and including) 13 is 41 (the 13th prime); "twelve plus one" and "eleven plus two" are anagrams (that's my favorite).
Browsing the On-Line Encyclopedia of Integer Sequences for something interesting, I came up with:
- 13 is a Fibonacci number (the sequence starting with 0,1, and where each subsequent term is the sum of the previous two), a tribonacci number (the sequence starting with 0,0,1, and where each subsequent term is the sum of the previous three), and in the "skipped Fibonacci sequence" (the sequence starting with 0,0,1, and where each subsequent term is the sum of the previous one and the one two before that). In sequence terms, it's in the sequences defined by a(n)=a(n-1)+a(n-2), and a(n)=a(n-1)+a(n-2)+a(n-3), and a(n)=a(n-1)+a(n-3).
- 13 is also a tetranacci number (each term being the sum of the previous four), but only if the first four terms are 1,1,1,1 (not 0,0,0,1, which generates another sequence that is also called the tetranacci sequence elsewhere in the database—there aren't really standards for these things, I guess).
- 13 is the smallest sum of the squares of distinct primes.
- 13 is the smallest emirp (a term that just learned from the database: a prime whose reversal is also a prime).
I had always thought that the fear of the number 13 (triskaidekaphobia) derived from the 13 people present at the Last supper (Jesus and 12 disciples). But Wikipedia reports that Hammurabi's Code (dated 1760 B.C., which would be fully 1790-ish years before the Last Supper) omits 13 in its numbered list much like airplanes omit rows labeled 13 and buildings omit floors labeled 13 (news flash to the occupants of apartment 1403: you really are on the 13th floor, you know). A more interesting possibility raised in that article: a year has almost exactly 13 lunar (28-day) cycles (13 times 28 is 364), and the moon is associated in many cultures with femininity (for the obvious 28-related and therefore 13-related reasons). Thus a masculine culture might have thought they had something to fear from a mystical feminine connection to nature.
But on to the Carnival proper.
A math Ph.D. student named Michi works on homological algebra. I didn't know what that was either. And I admit that my understanding of formal algebra isn't deep enough to really understand his post explaining it. But that's my failing entirely—it's well-written and I'm sure it will be enlightening to many.
One link that I just found myself is a very nice proof of the Pythagorean theorem via java applet that roughly follows Pythagoras's own reasoning. But with the animation, it's crystal clear—impressive from a teaching point of view, and it fits in exactly with my own interests (shamelessly plugging the AHSMP in the sidebar).
Another page that describes a more advanced idea from a basic point of view is this one that discusses the poorly-understood-in-lower-math constant e. Very clear and intuitive.
An undergraduate gives a brief overview of ANOVA, a technique that I think a lot of people use without really understanding very well.
This blog discusses the Babylonian method for estimating the square root of 2 in the context of attractors and fixed points. I hope I have time read the whole series—I don't understand those ideas very well, and the explanations seem good.
A blog of math and logic puzzles reviews an important counting technique for finding sizes of intersecting sets.
A blog specializing in K-12 math has a post with a simple puzzle, but does a nice job of explaining how extending the problem can create some thought-provoking math.
And finally, a retired teacher with a blog seems from this post to have a philosophy similar to mine: general concepts and complete mastery of basic skills are important; the varying philosophies aren't truly in opposition. In this post, he presents a nice geometry problem—it has several solution methods, and seeing how they're all linked is instructive.
(Due to some difficulty finding time tomorrow, when this carnival is technically supposed to be posted, I am posting it on Thursday night at about 11pm EDT. If your submission isn't in this carnival, I apologize; it wasn't left out intentionally. Please submit your link again to the next carnival.)