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Travels with Claude: A Natural Celebration of Pi

[3/23/2026: At least one person complained that the post was too long (perhaps because of spending too much time watching cat videos), so I had Claude generate a couple of summaries, which I put at the end of the post. I made a couple of small corrections.]

Another Pi Day, March 14, has come and gone, and as always I am somewhat uncomfortable. On the one hand, it’s great to see a public celebration of mathematics. At the same time, there is something about celebrating π on 3/14 that seems, well, inelegant. Pi is, after all, a universal constant. As a number, it doesn’t depend on who you are, where you are, what language you speak, how much money you have, or your political or religious affiliations. It is exactly the same on the earth as it is in any galaxy in our universe. For that matter it is exactly the same as it would be an any possible universe anywhere (or nowhere).

Once you start writing π down, however, you have to make some choices. We represent numbers in “base 10” because humans have 10 fingers. In base 10, π starts out as 3.14159 and winds it way on toward infinity from there. But in binary (base 2) it starts out 11.00100; in octal (base 8) it starts 3.11037; and in hexadecimal (base 16) it starts 3.243F6. So celebrating π on 3/14, rather than 3/11 or 3/24 or some other day already depends on making a choice.

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An Illuminating Proof of Schur’s Orthogonality Relations for Group Representations

Issai Schur’s orthogonality relations for characters of group representations are surely a mathematical wonder. Unfortunately, the standard proof, even from great expositors like JP Serre and Michael Artin, typically involves a bunch of index gymnastics that never meant much to me. Back when I was a mathematician, I saw a great proof (for compact Lie groups) in a paper by Raoul Bott. You need to know a little bit about tensor products to understand it, but otherwise it’s fairly simple.

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The Kuhn Piano Moves On

My grandfather, Sam Kuhn, married my grandmother, then Minette Stroock, around 1920. They were from well-to-do families, and received a Steinway baby grand piano as a wedding present. By the time I first remember seeing it in the 1960s, my grandmother and her decorator had replaced its original black finish with a bright red, meant to resemble a Japanese lacquered box.

When Sam died in the 1970s, the piano moved to my family’s living room in Princeton. Since I was the only person who (sort of) played the piano in my generation, it came to me when I had a place for it, in the mid-1990s. I moved it four times since. Our current house is not small, but the living room couldn’t accommodate both it and living. So it lived in a corner of the dining room starting in 2010. In addition to be impractically large, time had taken a toll on its mechanics since being rebuilt in the 1970s, and the lacquer red had faded unevenly to a kind of orange. It took a lot of patience, but I was finally ready to part with it last year.

To bring it up to a fully playable standard would have cost about twice value. We were reluctant to put in that much investment, and so were the potential donation recipients we considered. So we were very fortunate to be able to donate it to the North Bennet Street School (NBSS), where students in their Advanced Piano Technology training program learn how to bring it and pianos like it back to fully-functioning life. Once it’s served in that capacity, it will be sold to help keep the school going.

We were delighted that we were able to see it and some of the folks working on it when we visited NBSS’s open house. This is what the piano looked like on Dec 7, 2024:

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Do I Really Need to Use Unsalted Butter?

TL;DR 1 stick of salted butter (8 tbsp) has about 1/3 tsp of salt.

Maybe the same thing happens to you: you want to bake something. The recipe calls for unsalted butter. You have all the other ingredients. You have salted butter. You just don’t have unsalted butter. You look at the recipe. It calls for salt. Are you really going to go to the store and get unsalted butter? Should you just use salted butter? Maybe it will end up too salty! Why do they do that anyway!? Since we don’t eat that much butter at this point, it’s kind of a pain to keep two kinds on hand just in case.

This happened to me recently, and I finally decided it’s time for me to figure out how much salt there is in salted butter.

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Changes to Wordle that didn’t get much attention

tl;dr After the NY Times purged some solutions and available guesses back in February, they quietly reintroduced them as available guesses, and they also shuffled around some of the solutions, without eliminating any.

Wordle was born on June 19, 2021. Perhaps Josh Wardle wanted to celebrate the birth, because the first answer was “cigar.” Being birthed of a nerd, this was known as “Wordle 0.”

Joy often comes with pain, and Wordle is no exception. Wordle was pre-programmed with 2315 answers, so the last Wordle was to have been Wordle 2314, on October 20, 2027. (If you’re into spoilers, you can find all of them here.)

Around Feb 15, 2022, shortly after the New York Times purchased Wordle, they eliminated some words that they felt were obscure or offensive. People noticed because some folks had the original version in their browser cache, and they got a different answer (“agora”) from the folks who wound up with the revised NYT version (“aroma”). “Agora” was the first of six solutions that were eliminated. The other five were “pupal,” “lynch,” “fibre,” “slave,” and “wench.” With 6 fewer words, Wordle will draw its last breath on October 14, 2027.

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E&M Codes 2021: Good News, for a Change

Just about exactly eight years ago, I wrote a series of posts that began with this paragraph:

If you’re a psychiatrist (or psychiatric RNCS) in the US reading this, you are almost certainly aware that all of our billing codes changed on Jan 1, 2013.  If you are like most of the psychiatrists I know—at least in private practice—you are at least somewhat freaked out by this.  If so, keep reading.  If not—for example, if you’re not a psychiatrist in the US—stop reading this immediately and go do something more interesting, like… well, like just about anything other than memorizing a phone book.  (There used to be things called phone books…  never mind.)

Unlike 2013, the billing codes themselves have not changed, but as of January 1, 2021, the documentation requirements for these codes have gotten significantly less complex and onerous. Like then, this post should be of no interest to you if your are not a US psychiatrist or psychiatric nurse. But if you are—and especially if a significant portion of your practice is psychotherapy—the situation is very different from 2013 because these changes are:

  • good news
  • not widely publicized
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Doomsday Calendar, Part 3

In the process of looking up John Conway’s “Doomsday Algorithm” for figuring out what day of the week “any” date falls on, I came across Chamberlain Fong’s description of a simplification of Conway’s algorithm that he and Michael Walters came up with, which they describe in a paper on arxiv. (“Any” is in quotes for reasons explained in Part 2.)

What does the Fong/Walters algorithm do differently? Let’s note what’s the same. Conway’s scheme of “Doomsdays” is still in place, and you still need to know the Doomsday for the start of the century. The difference is in computing the offset. There is no dividing by 12, remembering the quotient, dividing the remainder by 4, etc. (Apparently that part of the method was due to Lewis Carroll, as explained in Part 2 or in Fong’s Scientific American blog post.)

How does the alternate method , which they call “odd+11” work? If you love flowcharts, look here. But it’s pretty simple:

  1. Take the last two digits of the year.
  2. Is it odd? If so, add 11.
  3. Divide your number (which now must be even) by 2.
  4. Is your new number odd? If so, add 11 again.
  5. Reduce your number modulo 7.
  6. Subtract your number from 7 (“take the 7s complement”).
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Doomsday Calendar, Part 2

This internet thing, though. It seems to be all rabbit holes, all the way down. In my previous post, I wrote about John Conway’s “Doomsday Algorithm” for figuring out what day of the way “any” day falls on. It seems that there are quite a few people who have spent a lot more time thinking about calendars in general, and mental calendars in particular, than I have. Most of what’s in this post is based on this excellent piece by Chamberlain Fong.

The first helpful thing Fong does is to give some history. Conway’s method has two parts: the identification of particular days of the year as “Doomsdays,” which all fall on the same day of the week. Once you know what day of the week any give year’s Doomsdays fall on, it is easy to figure out the day of the week of an arbitrary date. The other part is to figure out what day of the week the given year’s Doomsdays fall on.

The first part of the algorithm seems to be entirely due to Conway. The second part, interestingly, was apparently worked out by Lewis Carroll—one of the few people in the orbit of mathematicians who may have been more playful that Conway. Martin Gardner, the legendary author of Scientific American’s Mathematical Games column, was a serious student of Carroll, and when he came across Carroll’s work on perpetual calendars, challenged Conway to come up with something simpler.

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John Conway’s “Doomsday Calendar”

[This wound up being Part 1 of 3. You can find Part 2 here, and Part 3 here. You don’t need to read all three parts, but Part 3 does actually describe a significantly simpler algorithm than the one Conway used.]

John Horton Conway, a great mathematician of unmatched playfulness, died recently at age 82 of Covid-19. His biographer, Siobhan Roberts, wrote an excellent New York Times obituary. Conway was something of a showman, in a good way, so I should give a spoiler alert up front, especially these days, that this has nothing to do with Doomsday as in the end of the world.

Photo by Denise Applewhite, Princeton University Office of Communications

I never met Conway, but in the mid-70s my math camp buddy Rob Indik had a summer in which he commuted to the UI Chicago campus “with Vera [Pless] driving, Conway in front, and me riding in the back and trying to ride along with Conway’s free flow of ideas.” I remember Rob regretting that he couldn’t remember a mental “perpetual calendar” that Conway had taught him. In the computer age we have no call for perpetual calendars, but back when dinosaurs roamed the earth they were novelty items: a small object with a few dials, which would let you figure out what day of the week any date falls on. A mental one sounded like fun, and I occasionally thought—not very hard—about how you might go about it. I never got very far. [I see that this is not the definition of “perpetual calendar” favored by Wikipedia, but that the American Heritage Dictionary has me covered.]

With Conway’s death, I decided to look it up, and found that it’s around the web in several places, but without a very clear explanation of why it works. So I thought I would explain both how it works and why it works.

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