I’ve gotten plenty of submissions that span the entire gamut of math-blogging: education, pure math, applied math, debunking bad math – it’s all there. Only the gender distribution could be made slightly more equal (and that’s an understatement). I’m linking to the posters in roughly increasing order of mathematical difficulty, but don’t let my opinions deter you from reading the posts closer to the bottom.
First, Denise of Let’s Play Math has a long collection of quotes exhorting people to study mathematics. George Washington, Martin Gardner, Henri Poincaré, Abraham Lincoln, Galileo Galilei, Bertrand Russell – they’re all there.
Barry Leiba of Staring at Empty Pages uses modular arithmetic to solve a digit puzzle. The puzzle is based on writing any positive integer – say 8293 – and jumbling its digits and subtracting: 8293 – 3982 = 4311. Then delete a nonzero digit. Based on the resulting number, it’s possible to tell which digit you deleted.
P. Sternberg of Discreet Math links to various companies producing non-orientable material, like Möbius shoes and Klein bottle wool hats.
Heath Raftery writes about a probability paradox. Two dice are rolled, of which one is a 4. Given that, what is the probability that the sum of the two dice is a 7? Without further qualification – for example, “At least one die is a 4” – it’s obviously 1/6. And yet Heath’s professor maintained that the answer was 2/11.
Charles Daney of Science and Reason gives a brief history of algebra from the Middle Ages till the Renaissance. Whereas most accounts begin with a relatively modern mathematician, like Fermat or Euler, Charles begins far earlier, with Al-Khwarizmi, who invented the word “algebra” and after whom the word “algorithm” is named.
In the bad math department, Tyler DiPietro has a long rant fisking the arguments of Sal Cordova, an intelligent design proponent who makes dishonest arguments from information theory and computability theory in order to try refuting evolution.
JD2718 has a puzzle in plane geometry. A parallelogram is defined as a quadrilateral where two opposite pairs of sides are parallel. But there are other equivalent definitions, and other conditions that look like they define parallelograms but in fact don’t.
Science Pundit Javier Pazos tells an anecdote about John von Neumann and the bee problem. The bee problem involves two trains initially spaced 40 km apart and approaching each other at 20 km/h, and a bee that flies at 40 km/h, starting from one train, flying toward the other, turning around when it meets it, flying toward the first train, etc. How far does the bee fly until the two trains meet?
Suresh Venkatasubramanian of GeomBlog has a long primer for the game theory of author ordering. When publishing a scholarly paper, each contributor wants to be as close as possible to first author in order to get more credit; Suresh explains which ordering strategies are stable and which appear the most utilitarian.
He also notes that the theory of algorithms is undergoing fundamental changes as computing power reaches saturation. In the past, it was sufficient for researchers to say that the run time of a process is proportional to some simple function, like x^2; but now, the constant that accompanies that function is becoming increasingly important.
John Kemeny introduces spigot algorithms, which can be used to systematically generate digits of some irrational numbers, including e and pi.
Jeffrey Shalit of Recursivity writes about the prime game. The initial observation is that the decimal representation of every prime p can be reduced to one of 26 primes by scrubbing off digits. That leads to a more general investigation of the concept of subsequences.
Eric Kidd programs infinity into Haskell, in order to solve such expressions as dividing infinity by 2 or adding 1 to infinity. The trick is to teach Haskell about the finite natural numbers first, and then write into it something strong enough to code for infinite numbers.
David Eppstein of 0xDE writes about subgreedy algorithms for Egyptian fractions. An Egyptian fraction is a representation of a fraction p/q as 1/n1 + 1/n2 + … + 1/n(k). A greedy algorithm is one that tries getting the best-looking result in one step without regard for efficiency in later steps; a subgreedy algorithm is one that is almost greedy, but tends to get far better results in the long run.
He also writes about a local version of the central limit theorem, which differs from the regular theorem in that it says repeated distributions look approximately normal in a small neighborhood of 0 rather than globally; and about coloring tilings in such a way that on the one hand the colors display a regular pattern, just like the tiling, but on the other no two adjacent tiles have the same color.
And finally, Mikael Johansson has multiple great posts about algebraic topology and related subjects; choosing three was a fairly hard decision. Of the three that made it into this edition, the easiest is the Borsuk-Ulam theorem, which roughly states that any given time, there’s a pair of antipodal points on Earth with the same temperature.
Also, A∞ for the Layman explains algebraic topology, beginning with simple definitions and ending with an overview of homological algebra. For the braver souls, there’s his post about carry bits and cohomology; group cohomology is an indispensable tool in algebra, and Mikael applies it to addition and multiplication modulo 10, or in other words carry digits.
The next edition will be posted in two weeks, on 2/23, on Good Math, Bad Math. Send submissions to Mark CC, or to me so that I’ll forward them to him.
On a final note, I should remind everyone I’m still looking for someone talented enough to make a decent logo or banner for the carnival.