Category Archives: offhand

Large-scale Pareto-optimal topologies, or: how to describe a hexahedron

I got to meet Karen Caswelch, the CEO of Madison startup SciArtSoft last week. The company is based on tech developed by my colleague Krishnan Suresh. When I looked at one of his papers about this stuff I was happy to find there was a lovely piece of classical solid geometry hidden in it!

Here’s the deal. You want to build some component out of metal, which metal is to be contained in a solid block. So you can think of the problem as: you start with a region V in R^3, and your component is going to be some subregion W in R^3. For each choice of W there’s some measure of “compliance” which you want to minimize; maybe it’s fragility, maybe it’s flexibility, I dunno, depends on the problem. (Sidenote: I think lay English speakers would want “compliance” to refer to something you’d like to maximize, but I’m told this usage is standard in engineering.) (Subsidenote: I looked into this and now I get it — compliance literally refers to flexibility; it is the inverse of stiffness, just like in the lay sense. If you’re a doctor you want your patient to comply to their medication schedule, thus bending to outside pressure, but bending to outside pressure is precisely what you do not want your metal widget to do.)

So you want to minimize compliance, but you also want to minimize the weight of your component, which means you want vol(W) to be as small as possible. These goals are in conflict. Little lacy structures are highly compliant.

It turns out you can estimate compliance by breaking W up into a bunch of little hexahedral regions, computing compliance on each one, and summing. For reasons beyond my knowledge you definitely don’t want to restrict to chopping uniformly into cubes. So a priori you have millions and millions of differently shaped hexahedra. And part of the source of Suresh’s speedup is to gather these into approximate congruence classes so you can do a compliance computation for a whole bunch of nearly congruent hexahedra at once. And here’s where the solid geometry comes in; an old theorem of Cauchy tells you that if you know what a convex polyhedron’s 1-skeleton looks like as a graph, and you know the congruence classes of all the faces, you know the polyhedron up to rigid motion. In partiuclar, you can just triangulate each face of the hexahedron with a diagonal, and record the congruence class by 18 numbers, which you can then record in a hash table. You sort the hashes and then you can instantly see your equivalence classes of hexahedra.

(Related: the edge lengths of a tetrahedron determine its volume but the areas of the faces don’t.)

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Mariano Rivera was elected to the Hall of Fame, the first player ever to appear on every single ballot. Why has this never happened? Because there are a lot of ballots and thus a lot of opportunities for glitchy idiosyncrasy. In 2007, eight voters left Cal Ripken, Jr. off. What possible justification could there be? Paul Ladewski of Chicago’s Daily Southtown was one of the eight. He turned in a blank ballot that year. He said he wouldn’t vote for anyone tainted by playing during the “Steroids Era.” In 2010, he voted for Roberto Alomar.



Walking to the gym today I was thinking, 10 degrees Fahrenheit — sure, it’s cold out, but when I lived back east I would have considered this brutally, unfairly cold. Now it’s an unexceptional level of cold, easily managed by putting on gloves and a hat.

Then I got to the gym and looked at my phone and it was actually -3.

Small baseball triangles

This all started when CJ asked which three baseball stadiums formed the smallest triangle.  And we agreed it had to be the Brewers, the White Sox, and the Cubs, because Milwaukee and Chicago are really close together.

But it seems like cheating to use two teams in the same city.  The most elegant way to forbid that is to ask the question one league at a time.  Which three American League parks form the smallest triangle?  And what about the National League?

First of all, what does “smallest” mean?  There are lots of choices, but (perhaps inspired by the summer we played a lot of Ingress) we asked for the triangle with the smallest area.  Which means you don’t just want the parks to be close together, you want them to be almost collinear!

I asked on Twitter and got lots of proposed answers.  But it wasn’t obvious to me which, if any, were right, so I worked it out myself!  Seamheads has the longitude and latitude of every major league ballpark past and present in a nice .csv file.  How do you compute the area of a spherical triangle given longitudes and latitudes?  You probably already know that the area is given by the excess over pi of the sum of the angles.  But then you gotta look up a formula for the angles.  Or another way:  Distance on the sphere is standard, and then it turns out that there’s a spherical Heron formula for the area of a spherical triangle given its edgelengths!  I guess it’s clear there’s some formula like that, but it’s cool how Heron-like it looks.  Fifteen lines of Python and you’re ready to go!

So what are the answers?

We were right that Brewers-White Sox-Cubs form the smallest major league triangle.  And the smallest American League triangle is not so surprising:  Red Sox, Yankees, Orioles, forming a shortish line up the Eastern Seaboard.  But for the National League, the smallest triangle isn’t what you might expect!  A good guess, following what happened in the AL, is Mets-Phillies-Nationals.  And that’s actually the second-smallest.  But the smallest National League triangle is formed by the Phillies, the Nationals, and the Atlanta Braves!  Here’s a piece the geodesic path from SunTrust Park in Atlanta to Citizen’s Bank Park in Philly, courtesy of GPSVisualizer:

Not only does it go right through DC, it passes about a mile and a half from Nationals Park!

Another fun surprise is the second-smallest major league triangle:  you’d think it would be another triangle with two teams in the same city, but no!  It’s Baltimore-Cincinnati-St. Louis.  Here’s the geodesic path from Oriole Park at Camden Yards to Busch Stadium:

And here’s a closeup:

The geodesic path runs through the Ohio River, about 300m from the uppermost bleachers at Great American Ball Park.  Wow!

Now here’s a question:  should we find it surprising that the smallest triangles involve teams that are pretty far from each other?  If points are placed at random in a circle (which baseball teams are definitely not) do we expect the smallest-area triangles to have small diameter, or do we expect them to be long and skinny?  It’s the latter!  See this paper: “On Smallest Triangles,” by Grimmet and Janson.  Put down n points at random in the unit circle; the smallest-area triangle will typically have area on order 1/n^3, but will have diameter on order 1.  Should that have been what I expected?

PS:  the largest-area major league triangle is Boston-Miami-SF.  Until MLB expands to Mexico City, that is!

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The great qualities with which dullness takes lead in the world

He firmly believed that everything he did was right, that he ought on all occasions to have his own way — and like the sting of a wasp or serpent his hatred rushed out armed and poisonous against anything like opposition. He was proud of his hatred as of everything else. Always to be right, always to trample forward, and never to doubt, are not these the great qualities with which dullness takes lead in the world?

(William Makepeace Thackeray, from Vanity Fair)



Why is a Markoff number a third of a trace?

I fell down a rabbit hole this week and found myself thinking about Markoff numbers again.  I blogged about this before when Sarnak lectured here about them.  But I understood one minor point this week that I hadn’t understood then.  Or maybe I understood it then but I forgot.  Which is why I’m blogging this now, so I don’t forget again, or for the first time, as the case may be.

Remember from the last post:  a Markoff number is (1/3)Tr(A), where A is an element of SL_2(Z) obtained by a certain construction.  But why is this an integer?  Isn’t it a weird condition on a matrix to ask that its trace be a multiple of 3?  Where is this congruence coming from?

OK, here’s the idea.  The Markoff story has to do with triples of matrices (A,B,C) in SL_2(Z) with ABC = identity and which generate H, the commutator subgroup of SL_2(Z).  I claim that A, B, and C all have to have trace a multiple of 3!  Why?  Well, this is of course just a statement about triples (A,B,C) of matrices in SL_2(F_3).  But they actually can’t be arbitrary in SL_2(F_3); they lie in the commutator.  SL_2(F_3) is a double cover of A_4 so it has a map to Z/3Z, which is in fact the full abelianization; so the commutator subgroup has order 8 and in fact you can check it’s a quaternion group.  What’s more, if A is central, then A,B, and C = A^{-1}B^{-1} generate a group which is cyclic mod its center, so they can’t generate all of H.  We conclude that A,B, and C are all non-central elements of the quaternion group.  Thus they have exact order 4, and so their eigenvalues are +-i, so their trace is 0.

In other words:  any minimal generating set for the commutator subgroup of SL_2(Z) consists of two matrices whose traces are both multiples of 3.

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Ringo Starr rebukes the Stoics

I’ve been reading Marcus Aurelius and he keeps returning to the theme that one must live “according to one’s nature” in order to live a good life.  He really believes in nature.  In fact, he reasons as follows:  nature wouldn’t cause bad things to happen to the virtuous as well as the wicked, and we see that both the virtuous and the wicked often die young, so early death must not be a bad thing.

Apparently this focus on doing what is according to one’s nature is a standard feature of Stoic philosophy.  It makes me think of this song, one of the few times the Beatles let Ringo sing.  It’s not even a Beatles original; it’s a cover of a Buck Owens hit from a couple of years previously.  Released as a B-side to “Yesterday” and then on the Help! LP.

Ringo has a different view on the virtues of acting according to one’s nature:

They’re gonna put me in the movies
They’re gonna make a big star out of me
We’ll make a film about a man that’s sad and lonely
And all I gotta do is act naturally
Well, I’ll bet you I’m a-gonna be a big star
Might win an Oscar you can’t never tell
The movie’s gonna make me a big star,
‘Cause I can play the part so well
Well, I hope you come and see me in the movie
Then I’ll know that you will plainly see
The biggest fool that’s ever hit the big time
And all I gotta do is act naturally

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On not staying in your lane

This week I’ve been thinking about some problems outside my usual zone of expertise — namely, questions about the mapping class group and the Johnson kernel.  This has been my week:

  • Three days of trying to prove a cohomology class is nonzero;
  • Then over Thanksgiving I worked out an argument that it was zero and was confused about that for a couple of days because I feel quite deeply that it shouldn’t be zero;
  • This morning I was able to get myself kind of philosophical peace with the class being zero and was working out which refined version of the class might not be zero;
  • This afternoon I was able to find the mistake in my argument that the class was zero so now I hope it’s not zero again.
  • But I still don’t know.

There’s a certain frustration, knowing that I’ve spend a week trying to compute something which some decently large number of mathematicians could probably sit down and just do, because they know their way around this landscape.  But on the other hand, I would never want to give up the part of math research that involves learning new things as if I were a grad student.  It is not the most efficient way, in the short term, to figure out whether this class is zero or not, but I think it probably helps me do math better in a global sense that I spend some of my weeks stumbling around unfamiliar rooms in the dark.  Of course I might just be rationalizing something I enjoy doing.  Even if it’s frustrating.  Man, I hope that class isn’t zero.

dBs, Dentists

Two songs that seem very much of a common kind, though I find it hard to articulate exactly why. Maybe the high, incredibly clear vocals. Maybe that they’re songs about love that aren’t about pleasure.

The dBs, “Black and White”

The Dentists, “Charms and the Girl”

The Dentists song was one of those lost songs for me for years, something I’d heard on a mixtape some WHRB friend had made — all I remembered was that opening, the note repeated again and again, then the four-note circle, then the notes repeated, then the four-note circle, then the tenor vocal coming in: “I have heard / a hundred reasons why…” When I finally found it again, thanks to tech magnate and indie-pop culture hero Kardyhm Kelly, it was just as great as I’d remembered.  It’s not on Spotify.  Is that what we define as “obscure” these days?

If I have caught a foul ball it is only by standing on the shoulders of giants

“Combined cap and baseball mitt,” A patent by Richard Villalobos, 1983.


This is meant for fans, not players. The idea is that if a foul ball comes towards you, you may not have time to grab a glove you’ve stashed at your feet. Rather, you quickly slip your hand into the cap-mounted glove and snag the foul with your hat still attached to the back of your hand.

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