There’s more science than meets the eye on The Big Bang Theory. Literally. Tonight’s viewers were undoubtedly puzzled by the presence of a little non-Euclidean geometry on the white boards. Why was it there? What is it?
First let us recall that Euclid himself developed his non-non-Euclidean geometry in 300 B.C. He started by assuming as little as possible and left it to generations of junior high school students to prove all the rest, that triangles have 180 degrees and so forth. As is typical in mathematics, the game is to use as few, most elegant, postulates as possible and leave the rest to clever derivation.
Every junior-high school student learns Euclid’s five postulates (briefly listed, with some legalities dropped):
1. Any two points can be joined by exactly one line segment.
2. A line segment can be extended to an infinite line.
3. For any line segment, a circle can be drawn.
4. All right angles are the same.
Now these four sound pretty elementary. It is hard to imagine ever being able to prove or disprove them and so they serve “obvious” as starting points. But there is a pesky “fifth postulate” Euclid added:
5. For a point outside a line, there is exactly one line through the point that never meets the original line.
Now Euclid was a smart guy. Why not just prove “the fifth postulate” from the first four? He tried. For 2000 years, mathematicians tried. Even Karl Friedrich Gauss. No dice. It was not proved. Modern mathematicians know it could never have been proved from the first four.
So mathematicians made lemons out of lemonade. In 1826, Nicolai Lobachevsky said, let me assume there are not one, but many, such parallel lines. In the 1850’s Bernhard Riemann said let’s assume there are none. Chaos ensued. Without the fifth postulate you cannot even prove all triangles have 180 degrees. Crazier still, assuming Lobachevsky’s version of the fifth postulate gives you less than 180 degrees in all triangles. Riemann’s gives you more than 180. These are some strange triangles.
(As it happens, it was all worked out decades earlier by Gauss and he stuck it in his desk without publishing. Note to aspiring mathematicians…whatever you work on, there’s a good chance Gauss already tried it.)
A few decades later, at the turn of the 20th century, Albert Einstein and others realized there was more than abstraction to the fifth postulate, but that it relates to an actual uncertainty we have about our own Universe. Being unable to prove Euclid’s fifth postulate is equivalent to not knowing if we live in curved space or a flat one. We don’t really know if extremely large triangles in our Universe have exactly 180 degrees or a little more or a little less. How can a triangle have more than 180 degrees? Let’s take a trip:
Let’s walk (and dogsled, and swim) from the North Pole along the Greenwich meridian (0 deg. longitude) through England and down to the Equator. Here make a right turn. That’s a 90 degree angle. Go until you hit San Salvador Island, west of Ecuador, located on the Equator at the 90th meridian (+90 degrees) . Now take another right turn (another 90 degrees) and head back north through St. Louis, back to the North Pole. So far we’ve taken two right turns, 90+90=180 degrees. But then we hit the North pole again, coming up through Canada. We find we arrive having completed a triangle. We took three turns. But you are coming in at right angles to where we left. The last angle has 90 degrees. So we’ve made a triangle containing 90+90+90= 270 degrees, not 180. Such is life on a curved surface. Even if you can’t see the curvature while moving around, it is there.
You can do the same in a Lobachevsky space. Imagine walking around on a Pringle’s potato chip.
Often it is the case that a piece of mathematical abstraction has a direct bearing on our physical world. In this case, it just took over 2000 years to realize it.
In his fascinating 1965 sequel to Flatland, Dionys Burger’s Sphereland chronicle’s the life of the grandson of A. Square, named A. Hexagon. A surveyor in Flatland is distressed to find the angles of triangles do not add to 180 degrees. A. Hexagon, being the progeny of the insightful hero of Flatand, awakens his surveyor friend to the fact that Flatlanders really inhabit a curved space. Like our trip, triangles their angles do not have angles that add up to 180 degrees.
Now even our own, non-fictional, 3-dimensional space may be curved. We just don’t know. Astrophysicists measure the largest triangles they can find looking for small deviations. They use the rays of the oldest light in the universe, the microwaves left over from just 380,000 years after the Big Bang. So far, all triangles add up to 180 degrees, but with more precision we may at any time find we live in a curved space. And that is not a problem. Everywhere junior high school geometry books would just have to revise Euclid’s fifth postulate. (That’s good news for the textbook publishers, who are always looking to come out with a new edition to sell. Even if there is nothing new, I find for my own classes, they will just re-number the questions every 3 years and call it a new edition.)
Fun stuff, but what did this have to do with the show? Why did we put it on the whiteboard? Actually it directly references a speech Sheldon had in the script. Had. It was an earlier version script.
The scripts go through many revisions throughout the production week. Every day the actors rehearse and the writers improve the script. Comedy seems to work like an experimental science. The writers sometimes find something better. And for this week, after I already sent in the equations and diagrams to the set dressers, the writers rewrote Sheldon’s lines about retreating into a Riemannian space to relax.
So viewers never saw this little piece of science. The writers replaced it with something funnier: Riemannian space was replaced by Sheldon’s favorite location in Sim City. But maybe it still relates. After all, how well has anyone measured the triangles of Sheldonopolis anyway?