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!

How to compute arctangent if you live in the 18th century

Michael Lugo wrote a great post, following an idea of Andrew Gelman, about what would have happened if Pythagoras had known linear regresson.  Punchline:  he would have found a linear formula for the hypotenuse with an R^2 of 0.9995, and would surely not have seen any need to pursue the matter any further!

I thought this was mostly just a joke, until the mail brought me a copy of the very interesting A Wealth of Numbers from Princeton University Press, an anthology of popular writing about math stretching from the 16th century to the present.

From Hugh Worthington’s 1780 textbook, The Resolution of Triangles:

THE THIRD CASE is, the sides being given, to find the angles, and the rule is as follows.  “Half the longer of the two legs added to the hypotenuse, is always in proportion to 86, as the shorter leg is to its opposite angle.”

In modern language:  given a right triangle with legs a and b, and hypotenuse 1, how do you find the angle x adjacent to a?  Nowadays we would just say “x = arctan b/a.”  But this kind of computation was presumably not so easy in 1780.  Instead, Worthington offers the approximation

b/x = (a/2 + 1) / 86

which (after converting to radians, as good manners requires) gives

x = (86*pi/180) b / (a/2 + 1)

Of course, when the hypotenuse is set to 1, we have b = sin x and a = cos x.  So the approximation is

x = (86*pi/180) (sin x) / (cos x / 2 + 1).

This turns out to be a pretty awesome approximation!

How do you think they came up with this?