Inscribed squares: Denne speaks

I recently learned that Elizabeth Denne of Smith has made some recent progress on the inscribed square problem in joint work with Jason Cantarella (Georgia) and John McCleary (Vassar). Since she knows this problem, its history, its variants, and her own contributions much better than I do, I asked her if she’d be willing to make a guest post: and here it is!

The problem:
A) Given a simple closed curve in R^2 (Jordan curve) are there four points on the curve which are the vertices of a square?
B) Given a closed curve in R^2 are there four points on the curve which are the vertices of a square?

Currently I am collaborating with Jason Cantarella (UGA) and John McCleary (Vassar College) on this problem.
First off, both problems are still open for a continuous map of S^1 into R^2. Secondly, I know of no counterexamples to the problem and believe the answer to problem A is yes. It appears (as John Cowan (7/25/07) commented) that problem B is not interesting, as parts of the curve could be “cut off”. So I’m not sure of the literature on B. I do think the problem is interesting for the following reasons. Don’t “cut off” parts of the loop and still try and solve the problem. In the Jordan curve case (A) when considering vertices ABCD in order about the square , they will match their order along the curve. In the case of problem B, this won’t necessarily be the case. (Just think about the Figure 8 example that John Cowan suggests.)

So from now on I’ll just consider problem A. All bloggers went straight to the heart of the issue. Namely, once you know that there is an inscribed square for some kind of Jordan curve (polygonal, smooth) you can’t necessarily show that every Jordan curve has an inscribed square. The reason that inscribed squares in the approximating curves may shrink to zero in the limit.

I’ll now describe what the known results are, repeating some of what others have written.

A good line from _On Beauty_

A good joke depends on its punchline, and a good comic sentence in a novel relies on its last word. One good example is the Michael Chabon sentence I quoted last week. Another is a great sentence from a New York Times magazine piece by the not-widely-enough-beloved Matthew Klam:

In Boston, the day before the convention started and after a long, glittering night following the Wonkette to fancy parties, I came back late and found Josh Marshall in my hotel room, lying sideways on a cot, blogging.

And here’s one from the book I’m reading right now, On Beauty by Zadie Smith. I have an annoying habit of avoiding looking at things that are very heavily promoted as being great. I never saw Pulp Fiction or read anything by Jonathan Safran Foer or Dave Eggers. But often things are heavily promoted as being great because they are great! So it is with Zadie Smith, it turns out. Anyway, here’s a winner of a sentence:

“But you’ve already privileged the term,” says the professor’s daughter, whom Katie, who is not given easily to hatred, hates.”

[some sentences about why I like the last words of these three sentences deleted because I didn't see how to say anything on this topic in a way that interested me more than pressing on reading On Beauty.]

What Wittgenstein would have said about this week in Orioles baseball

“What we cannot speak about without vomiting we must pass over in silence.”

When Paul Shaffer was in the Pixies

The Pixies play “Trompe Le Monde” on David Letterman in 1992, backed by Paul Shaffer and the house band. Is it appalling that I like the addition of the Paul Shaffer ELO-style keyboard solo to this song?

Also, did you know that Paul Shaffer co-wrote “It’s Raining Men”?

The day I ate four Big Macs

It was the fall of 1993, when I was living in Baltimore. McDonald’s was offering a special on Big Macs — two for two dollars. It was my first time living in my own apartment and I was still pretty interested in diminishing the amount of cooking I had to do, so this was an appealing offer. I figured, two Big Macs makes a nice meal, so I’ll bring home four, then have two for lunch and two for dinner. So I brought home my bag of Big Macs and ate the first two. And I thought to myself, you know, I could certainly handle one more Big Mac. So I ate another one. Then I was pretty full, but I thought: look, I now no longer have enough Big Macs for dinner, and it’ll be better hot anyway. So why not see if I can eat four? But then I thought: well, this may be the only time I eat four Big Macs at once, so I ought to make it a little special. So I spread some butter on the top and bottom bun of the final Big Mac and fried it up in a pan like a grilled cheese sandwich. Highly recommended (but more enjoyable without the three prior Big Macs.)

How many Big Macs can people eat at a sitting? Through the miracle of the Internet we can answer this question scientifically. Google hits for:

• “I ate two Big Macs”: 151
• “I ate three Big Macs”: 4
• “I ate four Big Macs”: 8
• “I ate five Big Macs”: 2
• “I ate six Big Macs”: 5
• No hits for seven or eight Big Macs. 2 hits for “nine Big Macs,” but both are from the screenplay to Super-Size Me, and the reference is to nine Big Macs over the course of a whole day.

Conclusion: six is the most Big Macs you can eat at once. Also, people prefer to eat an even number of Big Macs.

Update: search for “Big Mac eating contest” reveals an unverifiable message board comment asserting that Nick Suplizio, of Dubois, PA, ate seven Big Macs in 15 minutes. Nick Suplizio, if you someday find this post by Googling your name, I bow to you.

The last great problem

Hey, The Believer was kind enough to put a link to my old article, “The Last Great Problem,” on their front page. I wrote this article because Heidi Julavits asked me to write an article about books about the Riemann Hypothesis and books about people who died trying to climb Mount Everest, which I thought was a really strange idea, but which turned out to be a great idea, and now I think Heidi Julavits is a genius. And The Believer is really good and you should read it. Here’s Steve in The Believer on how and where to read contemporary poetry. And here’s Douglas in The Believer on the Fall’s Peel Sessions.

Putting things on top of other things

We have lots of food in the house from the farmers’ market, but I didn’t exactly get around to cooking a full meal this weekend, so I ended up eating semi-arbitrary combinations of the things I had lying around. Some experiments:

• Chunks of watermelon with chunks of feta on top: Yummy. But not really an experiment; apparently they eat this in Israel all the time.
• A small red pepper stuffed with feta and sungold tomatoes and roasted for 10 minutes:  not as sweet as I expected and very messy when the water poured out of the tomatoes.
• Chunks of smoked trout wrapped in sauteed swiss chard to make a kind of mock eggroll: not bad, but too salty.
• Slice of whole wheat bread with taramasalata and roasted strips of poblano pepper: I ate this right before bed and it hit the spot.
• Fried a bunch of sage in olive oil, then fried an egg on top of the sage, then put the egg in a bowl with a chopped up Roma tomato and crumbled feta. Delicious. Also not really an experiment: this was proposed by Mark Bittman in the newspaper the other day, but with oregano instead of sage and bread instead of a tomato.

The future belongs to the Bangles

I was wondering whether I’d be the youngest person at the Bangles show at the Orpheum Friday night, but it wasn’t so; there were plenty of other people my age. And you know who else goes to see the Bangles? Ten-year-old girls and their moms! Is it because the moms liked the Bangles when they were ten and want to introduce their daughters to the band? Or is there a Bangles song famous enough that current ten-year-old girls know about it? I think the former: my rule of thumb for figuring out what these kids today listen to is to look at playcounts on last.fm, and the Bangles have just 585,000, a bit ahead of Frankie Goes to Hollywood but well behind the Eurythmics, Duran Duran, and a-ha.

The Bangles lead off the set with three songs they didn’t write: “Hazy Shade of Winter” (Simon and Garfunkel), “Tear Off Your Own Head” (Elvis Costello), and “Manic Monday” (Prince). The Bangles are masterly interpreters of others’ songs; I think this is an underrated rock virtue. Their recording of “Hazy Shade of Winter,” for example, makes the original sound like a cruddy demo. Friday night’s version was a little too easygoing, but as the night went on the Bangles powered up and played their hits as rock numbers, guitar solos and all. They seemed to be having a good time, and so was I.

The band closed the set with “Walk Like an Egyptian,” another song they didn’t write, but which is impossible to imagine performed by anyone else. (The songwriter, Liam Sternberg, offered it to Toni Basil first! What a revolting alternate history that suggests.) Is this song the best #1 of the ’80s? I think only “Let’s Go Crazy” gives it serious competition.

Things that are great about “Walk Like an Egyptian”:

• Songs that sit on one chord for almost the whole verse are typically great — e.g. “Spirit in the Sky” (played on the soundsystem before the concert) and “Walk Like an Egyptian.”
• The way the Bangles trade off the lead vocal on the song; so that when you hear their famous harmony at last (“Way oh way oh, way oh way oh”) you’ve still got the individual pieces fresh in your mind.
• The part where they whistle (sadly, replaced in concert by a weird mini-cover of “Mrs. Robinson.”)

• And the lyrics of this song are great, great, great. They convey like nothing else the sense (kind of new at the time, at least to my teenaged self) that from now on we were going to see people all over the world acting in the same strange ways at exactly the same time. “The future belongs to crowds.” If you happen to be writing a book about 1980s culture, happiness, and consumerism, you should probably call it All the Kids in the Marketplace.

For the encore, the Bangles go garage and play a Seeds cover, “Pushin’ Too Hard” (which I learn from my diligent research was always a Bangles live staple) then close with the obligatory “Eternal Flame,” in which they relax and let the audience do most of the singing.

Here’s a 1986 profile of the Bangles to give some sense of how people saw them when they were big.

Here’s what the Bangles sounded and looked like 25 years ago, opening for the English Beat:

And just for fun, here’s a strange video of the Seeds themselves performing “Pushin’ Too Hard” on a sitcom sometime in the late ’60s. Great dancing.

Frank Lloyd Wright in bad decline

Not all of Frank Lloyd Wright’s buildings are as carefully preserved as the ones we saw last week at Taliesin. At Letter From Here, read about the FLW boathouse in James Madison park, unceremoniously demolished in 1926. And from Nadine Goff’s flickr stream, here’s a Frank Lloyd Wright house slowly decaying in the middle of downtown. (Follow the link for more information and location, not to mention more great Madison photographs.)

©  Nadine Goff, all rights reserved.

Best of the Dane County Farmers’ Market

Tomorrow is market day — if you’re new to Madison, the size of the place can be intimidating. So here are some of my favorite stops, starting from the corner of Pinckney and Mifflin and proceeding counterclockwise.

• Harmony Valley Farm: large, varied, yummy, non-cheap selection of organic produce. Their chiogga beets (and indeed all their beets) are delicious, as will be the big fat yellow and red carrots when they arrive. Mrs. Quomodocumque, who eats arugula the way you or I eat potato chips, says their bagged arugula is the best at the market.
• Butler Farms sells sheep’s milk cheese off a table — they’re the only people at the Market who make Camembert, and their Camembert is a nice treat.
• About halfway along Mifflin is a big friendly stand, whose name I don’t know, which is a good place for reasonably priced “non-fancy” vegetables — your broccoli, your zucchini, your cauliflower, your beans.
• Stella’s Bakery, just before the Carroll Street corner, is famous for hot spicy cheesebread — you’ll hear the barkers hollering about it halfway up the street — and the cheesebread is indeed great, though much better if you stop shopping and eat it while it’s hot. If you want a somewhat more manageable snack, the empanadas are also hot, greasy, spicy, and cheesy, but much smaller than a whole load of cheesebread.
• Carol Gitto is usually right at the corner, though she moves around a bit; she’s one of the few vendors who usually has ground cherries. I’d never heard of ground cherries before I moved to Wisconsin — now we try to keep a bowl of these weird delicious little things on the table at all times. They look like small tomatillos, but the taste is somewhere between very sweet citrus and pineapple.
• About halfway up Carroll there’s a stand selling fresh and smoked trout, raised in an artesian well. If you, say, have a little kid in the house and try to keep PCBs and mercury out of your food, and if you prefer sustainable fisheries, too, your fish options are pretty strongly constrained — but trout from a well passes every test. (If you know this to be false, please don’t tell me.)
• Tomato Mountain sells great heirloom tomatoes at a high but fair price, and lots of preserved products too — the sungold preserves, in particular, make an amazing sandwich spread or improvised pasta topping.
• Fantome Farm, almost to the corner of Main Street, sells fantastic chevres at a high price — get some when company’s coming. The cheese is different every week and depends on proprietor Anne Topham’s whim; the extra-aged hard cheeses are some of the richest and stinkiest available in town.
• At the corner of Carroll and Main there’s an organic vegetable stand that’s a good place to get weird varietals — exotic melons, non-standard species of lettuce and arugula, and so on.
• Now turning the corner onto Main, your first stop can be Don’s Produce, where I buy a lot of my tomatoes. They don’t have a wide variety of heirlooms, but their basic tomatoes are inexpensive and routinely good; they always have a lot of yellow tomatoes, which I like best for some reason.
• Capri Cheesery is one of my favorite spots in the whole market: Felix Thalhammer, a congenial Swiss guy who looks more like a professor than a dairy farmer, makes a wide variety of organic goat cheeses — not cheap, but worth buying weekly. I like the Mediterranean-style feta (marinated with thyme), the St. Felix (a harder, aged cheese) and the feta pesto spread.
• The Land of Oos jam stand makes just about everything into jam; they have a lot of exotic choices (apple basil, blueberry jalapeno, etc.) but to my mind the freezer strawberry jam is unbeatable — tastes more like fresh fruit than any other jam I’ve had.

I’m sure I’ve missed some here — will supplement after tomorrow morning’s visit. In particular, I skipped the whole Pinckney Street side of the market, but by the time I’m 3/4 of the way around it’s usually time to take CJ home for his nap, so I’m less familiar with the stands there.