“What is Math?” — As Told By Math Jokes

If you’ve spent any time learning a specialized subject along with a group of friends & peers, you’ve probably experienced Discipline-Specific Humor—that uncomfortable but oh-so-funny form of joke telling that only makes sense to you and your fellow students. Usually, they rely on a carefully-constructed pun (like this one), but occasionally they tell you something about the inner-workings of the discipline.

Since I’m a mathematician by training, math jokes are the ones I appreciate the most. I’m guessing you’ve all heard a few in your school days, and maybe you’ve even heard a few outside of the world of the math class.

What I’d like to do here is lay out some of my favorite math jokes, and spend a moment examining what exactly they tell us about mathematics and mathematicians. You may learn something new about the intellectual habits of the field, and you’ll definitely gain an appreciation for the people whose quirks give this subject some of its appeal. Here we go!

An engineer, a physicist, and a mathematician were on a train ride through Scotland. They passed by a field with a black sheep in it. The engineer said, “hmm, I guess sheep in Scotland are black.” The physicist said, “well, at least one sheep in Scotland is black.” The mathematician said, “there exists at least one sheep in Scotland, at least half of which is black.”

Mathematicians are known for being annoyingly precise. One famous case appears in Imre Lakatos’ book, Proofs and Refutations. In it, a professor and a group of students try to prove Euler’s formula: “The number of vertices of a polyhedron, minus the number of edges, plus the number of faces, equals 2.” The thing is, they go through a dozen proofs (and refutations of those proofs), and just as many reformulations of the theorem. After a while, you’re not sure that any mathematical statements are true at all! (Don’t stop there, though, because Lakatos brings it home eventually.) Good times. Next:

A physicist and a mathematician live in adjacent apartments. In the middle of the night, an electrical short starts fires in both of their kitchens. The physicist awakes first, sees the fire, grabs the fire extinguisher, and puts it out. The mathematician awakes later, sees the fire, then sees the fire extinguisher, and says “Aha! A solution exists!” and returns to bed.

Okay, I love this one. In mathematics, there’s a distinction between “constructive” proof and “existential” proof. Constructive proofs rely on the fact that you can find an example of something that has a desired property: for instance, you can prove the statement “there is a positive number x for which x²–x–2 equals zero” by pointing out that the number 2 satisfies this property. However, you can’t take this approach with the statement “there is a power of 2 that begins with the digits 50283086.” (Prove me wrong, I dare you.) Nonetheless, this is a true statement. (The general theorem is “For any positive integer n, there is a power of two whose first digits make up the number n.”) Maybe I’ll give the proof in a future post. In the meantime:

A biologist, a physicist, and a mathematician are having lunch at a sidewalk cafe. They see two people enter an apparently-empty building across the street; a few minutes later, the same two people leave with a third person. The physicist says “Our initial observation must have been incorrect.” The biologist says “They must have multiplied.” The mathematician says, “If another person enters the building, it will be empty again.”

Math gets some ridicule for being a “head in the clouds” sort of subject, where decades-long research programs could have no immediate applications to the real world. This isn’t all that fair—there are lots of applications of mathematics! More to the point, though: one of the best things about mathematics is that it has real-world applications regardless of whether a mathematician is looking for them or not. A famous example is Riemannian geometry, which was a paragon of “head in the clouds” mathematics, until it became the framework for general relativity theory. It’s a similar story for linear algebra and quantum mechanics. And here are some other theory/application pairings you might not know about:  group theory & quarksfield theory & cryptography, and even Graeco-Latin squares & experimental design (see Klyve & Stemkoski and then go find Ronald Fisher’s book, Design of Experiments).

This last one’s just for fun:

Professor Wilson and his family recently moved to a new neighborhood. Since the professor was known to be absent-minded, his wife wrote down their new address on a slip of paper and gave it to him as he left the house for work that morning. At some point during the day, he needed to write down an important equation for a student, and took out the slip of paper to write it on the back, and gave it to the student. Forgetting all about the paper, and his new house, he went back to his old house. Realizing his mistake when he arrived, he said to the young girl on the front step, “Hello, little girl, do you know where the Wilsons live?” The girl said, “It’s okay, daddy, mommy sent me here to get you.”

I can be absent-minded sometimes, but I’m very glad it’s not this bad.

Finally, if you’re yearning for some more math geek humor, go here next.

Mini-Update: Math(s) on Twitter

In lieu of my next full post, here are some cool mathematics links from around the interwebs!

1. Maths History: This feed is managed by the British Society for the History of Mathematics, and posts births and deaths of famous mathematicians & scientists throughout history. It’s like your morning coffee, but for math & history nerds. Today’s roast? John Flamsteed:

John Flamsteed (1646-1719), first Astronomer Royal, published accurate astronomical observations, was born 19 Aug http://t.co/GcFwBvwhuf
— Maths History (@mathshistory) August 19, 2013

2. Amazing Maps: This is truly the Internet sensation of the week, which I’ve seen shared all over Twitter and Facebook. The name tells you all you need to know about this feed, so let me just share a few of the highlights:

3. Amazing Data: A relatively-recent spinoff of the previous account, this one focuses on creative data display. Here’s a small sampling:

And, just for fun, here’s another cool map.

Make Your Own Eternally-Sunlit Empire

You may have heard that famous phrase “The sun never sets on the British Empire.” While the statement is still true (as Randall Munroe explains in his excellent What If? blog), it originally referred to the Spanish Empire of the 16th and 17th centuries. In one of his theological tracts, An Advertisement Touching a Holy War, the estimable Francis Bacon (everybody loves Bacon, amiright?) says

“…both the East and the West Indies being met in the crown of Spain, it is come to pass, that, as one saith in a brave kind of expression, the sun never sets in the Spanish dominions, but ever shines upon one part or other of them…”

Regardless of which country the phrase originally referred to, there’s an interesting geography question here—how would you know if the sun never did set on your empire? It turns out that high school geometry is enough to solve the problem, at least approximately.

Let’s lay down some basic observations first:

  • Your empire would need to have nonzero land area: Let’s ignore undersea kingdoms (sorry, but Doggerland doesn’t count).
  • Your empire would need to have at least 3 territories: Theoretically, if your empire consisted of two antipodal points, the sun would set in one territory at the same time that it rises in the other. But that’s boring.
  • The Earth is (nearly) a sphere: For the purposes of this post, it will be a sphere. Otherwise, the math gets really messy).
  • On the Equinox, it doesn’t matter what your territories’ latitudes are: If the sun’s shining on 40° N, 88° W, then it’s also shining on 0° N, 88° W and 89° N, 88° W. And if your points don’t lie beyond the Arctic or Antarctic Circles, then you’re safe for most of the year.

Given these restrictions, we can make a key simplification: all three territories lie on the Equator, and consist of single points. Now, we can examine the problem two-dimensionally—imagine you’re looking down on the earth from above the North Pole, and that your three points are distributed around the Equator. It’d look something like this:

Screen shot 2013-08-02 at 9.14.41 AM

If we view the sun as being infinitely distant (apologies to Calvin’s dad), the sunlit side will be a semicircle that rotates around the disk of our 2-dimensional Earth. Looking at the picture above, it should be clear that the only way for your empire to be completely dark is for there to be a territory-free arc of more than 180°. Now here’s a cool fact: your empire will be eternally-sunlit exactly when the circle’s center lies within that red triangle. (Think about it: can you place a sunlit semicircle on the edge if the center isn’t inside the triangle?)

It turns out this fact relies on a 2,600-year-old geometry fact called Thales’ theorem. Here it is:

Thales’ Theorem: Given three points A, B and C on a circle with center O, if the line AC passes through O then ∠ABC is a right angle.

Legend has it that Thales celebrated the discovery of this theorem by sacrificing an ox (good times!). A generalization of the theorem appeared in Eulcid’s Elements as Proposition 33 of Book III. Now, here’s an easy generalization of Thales’ theorem and Proposition 33:

Theorem: Given three points A, B, C on a circle with center O, the triangle ABC contains O exactly when ∠AOC is less than 180°.

It gets even easier if we notice that the diagram above makes some unstated assumptions about the points: specifically, A, B, and C are given in order of increasing longitude, and the International Date Line lies on the arc AC. So, using some common sense, we get the following simple rule for our empire:

A nation is perpetually-sunlit if, and only if, it has three territories A, B, C (listed in increasing longitude) for which the longitude of C minus the longitude of A is greater than 180°.

Based on this formula, only two nations are perpetually-sunlit today: the United Kingdom (take Pitcairn Island, London, and Diego Garcia) and France (Martinique, Paris, and New Caledonia). The Dutch qualified until they gave up Netherlands New Guinea in 1962.

So, how could you go about building an eternally-sunlit empire of your own? Unfortunately, you don’t have many options—most of the world’s land has been claimed already. But there are a few scraps of territory you could snap up:

  1. The Bir Tawil (21.871° N, 33.737° E): A trapezoid of sand in the Sahara, the Bir Tawil is on the Egypt-Sudan border but is claimed by neither country. Go plant a flag there, and it’s yours.
  2. Russkaya Station (74.766° S, 136.803° W): This Antarctic station, closed in 1990, is located in Marie Byrd Land, which is not claimed by any country. You could probably buy it for the right price, and, if you felt like it, use it to film your own fanfic episodes of the X-Files. (And yes, there are certainly times of the year when it’d be in darkness, but beggars can’t be choosers.)
  3. After that, you need only rustle up some sovereign territory east of 33.73° E and west of 136.87° W. I’d suggest looking at Private Islands Magazine (my local library doesn’t carry it, I have no idea why) for ideas. Of course, you’ll still have to lead a revolt against the country that your new island is situated in.

And stay away from Yadua Island—I call dibs.