Math

I last edited this draft back in 2020. I vaguely recall wanting to write more here and probably revisit the diagrams, which were made entirely using Asymptote. But it’s a good post and worth publishing. I made my first Möbius strip during my sophomore year of high school in math class. Since then, I have been fascinated by their construction and mere existence. I share them with every person I can, basing my own presentation on a mix of my math teacher’s introduction, provided so many years ago, and a wonderful talk given by Matt Parker at the Royal Society. ...

Terence Tao is a gift to the mathematical community. He is an excellent collaborator, a talented communicator, and one of the broadest and sharpest minds working today. Eighteen years ago, when he was just past thirty years old, he wrote the essay What is Good Mathematics?. I discovered this when Steven Strogatz invited Tao onto his podcast The Joy of Why to discuss how well this essay holds up. Both of these are great, but start with the opening section of Tao’s essay, where he lists twenty-one ways to measure mathematics as being “good.” It displays his impressive clarity of thought and writing ability and evinces how the professional mathematics scene isn’t what one may have expected.

In early high school, I remember all the hubbub about requiring metal baseball bats to align to the BBCOR standard. Given the pronunciation of this (“Bee-Bee-Core”), I always assumed it was a regulation about what specific materials must be used to make the bat. That’s only true insofar as the standard actually defines a material property. BBCOR stands for “Batted Ball Coefficient of Restitution”. In other words, the standard tells you how elastic the collision between ball and bat is allowed to be. This standard was designed to dull metal bats in an effort to protect pitchers, the most likely players to be grievously injured by a batted ball. It was adopted by the NCAA in 2011, and most youth leagues that I’m aware of followed their lead. I used BBCOR-certified bats throughout high school, and continue to do so in my adult league. ...

I was in a position to provide some talking points for my company’s upper-level math textbooks. It was written in the aftermath of customer-induced pique regarding how we sell ourselves. While it’s focused on my company, the core idea of a problem-first approach extends beyond what we do in particular. ...

Thanks to Mikhail for posing this problem. On average, how many rolls would it take to see every face on a particular die? ...

Hot on the heels of Ultimate Zone Rating is another tool with the same goal: measure a player’s defensive contributions in terms of runs saved. This other statistic, Defensive Runs Saved (DRS), doesn’t just have a better name, but in my experience is favored over UZR in most situations. ...

Leading into the home stretch, let’s discuss some defensive statistics. As a category, these represent recent innovations in baseball, as defense is the most dynamic aspect of the game. Players can change where they are positioned, which affects their ability to get to a ball. As we gather more data — ball speed off the bat and launch angle — we can better analyze defensive ability. But it’s still messy. ...

We’ve already addressed the concept behind WAR (Wins Above Replacement) in the abstract and for offensive players in particular. Today, we’ll consider how we quantify runs for pitchers and overview calculating their WAR. ...

Relief pitchers can enter the game in very different situations, and that should be accounted for when evaluating their performance. Coming in to clean up a 12-0 victory (or defeat) is different than coming in during a close, tense game with a lot on the line. We measure this disparity using Leverage Index (LI). ...

ERA+ is a funny, messy statistic. Unlike OPS+, it doesn’t tell us how much better a player is than the league: it instead tells us how much worse the league is than the player. This ignores our intuition and causes unnecessary confusion. Furthermore, it makes it harder to use as a tool for direct comparison: someone with a 200 ERA+ is not twice as good as someone with a 150 ERA+, while that would effectively be true with OPS+ (and similar offensive statistics.) ...