by Simon Singh
Therefore, there have always been plenty of people willing to discuss mathematics during script meetings. Yet, despite their fondness for arcane diversions, the writers of The Simpsons realized that a seminar about infinity, Cantor, and Hilbert’s Hotel could be a distraction when it took place in the middle of a scriptwriting session. Fortunately, a solution was found, something that would encourage more mathematical discussion without disrupting the scripting process: Math Club.
The idea for the club was the result of a conversation in a Los Angeles bar between Matt Warburton and Roni Brunn. Warburton, who had studied cognitive neuroscience at Harvard University, had become a writer for The Simpsons soon after the series began and stayed on the team for more than a decade. Brunn had been part of the comedy scene while at Harvard and had been a Harvard Lampoon editor, but her career had focused on fashion and music after graduation.
“Math Club started with my sad realization that I was getting less sharp after graduating college,” recalls Brunn. “I was envious of book clubs. I don’t really like reading novels, but wanted a social setting for intellectual discussions. One night at a bar, I was telling Matt Warburton that it’s not fair there are only book clubs, and that there should be a math club. He gave a noncommittal ‘yeah’ and went on with his beer. We talked about the numerous Simpsons writers who have backgrounds in mathematics, and it was enough encouragement for me to get started.”
Contrary to what Brad Pitt might have advised, the first rule of Math Club was that you do talk about Math Club. In fact, evangelizing was encouraged. The core members were those who wrote for The Simpsons, but Math Club was open to teachers, researchers, and anyone else in Los Angeles who was interested in mathematics.
The first Math Club meeting took place at Brunn’s apartment in September 2002. The inaugural lecture was titled “Surreal Numbers” and was delivered by J. Stewart Burns, who had started work on a PhD in mathematics before joining The Simpsons. One by one, Burns’s colleagues gave their own Math Club lectures, with titles such as “An Introduction to Graph Theory” and “A Random Selection of Problems in Probability.”
Although Math Club was an informal gathering of friends and colleagues with a common interest, the lecturers often had impeccable academic credentials. Ken Keeler, whose lecture was titled “Subdivision of a Square,” is one of the most mathematically gifted writers on The Simpsons. He graduated summa cum laude from Harvard University, recognition that he was one of the most brilliant applied mathematicians to complete a bachelor’s degree in the class of 1983. He then moved to Stanford University and studied for a master’s degree in electrical engineering before returning to Harvard, where he received a PhD in applied mathematics with the snappily titled doctoral thesis “Map Representations and Optimal Encoding for Image Segmentation.” Keeler then joined AT&T Bell Laboratories in New Jersey, whose researchers have won seven Nobel Prizes. During this period, Keeler crossed paths with Jeff Westbrook. They were both active in the same area of research and co-authored a paper titled “Short Encodings of Planar Graphs and Maps.”14 They also co-authored a script for the sci-fi TV series Star Trek: Deep Space Nine, which involved two stand-up comedians starting a war after insulting every alien in the audience during their routines.
Math Club gradually grew in size. Sometimes, in order to accommodate all the members, it was necessary to hold sessions outside and use a suspended bedsheet as a makeshift projector screen. The biggest audiences, roughly one hundred people, turned up to hear the big-name mathematicians, such as Dr. Ronald Graham, the chief scientist at the California Institute for Telecommunications and Information Technology (Cal(IT)2). Incidentally, Graham was well known as having co-authored more than two dozen papers with Paul Erdős, and he was the foremost figure in popularizing the notion of Erdős numbers. One of Graham’s other claims to fame is Graham’s number, which set a record in 1977 for the largest number ever used in a mathematical paper. To get a sense of its size, consider the Planck volume, which is the smallest unit of volume in physics. It is possible to squeeze 1073 such volumes inside a single hydrogen atom. If the digits of Graham’s number were to be inscribed into the fabric of the cosmos so that each digit occupied just one Planck volume, then the entire visible universe would still not be large enough to contain it. It might be comforting to know that its last ten digits are ...2464195387.
One of the most memorable Math Club lectures was given by David S. Cohen, creator of Homer’s last theorem. Cohen’s talk was special because it was dedicated to explaining the research he had conducted prior to becoming a comedy writer. Having graduated with a degree from Harvard University, Cohen then spent a year at the Harvard Robotics Laboratory, later going on to complete a master’s degree in computer science at the University of California, Berkeley. While at Berkeley, Cohen conducted research into the so-called pancake sorting problem, and this topic formed the basis of his Math Club lecture.
The pancake sorting problem had first been posed in 1975 by Jacob E. Goodman, a geometer at the City College of New York, who used the pseudonym Harry Dweighter (harried waiter). He wrote:
The chef in our place is sloppy, and when he prepares a stack of pancakes they come out all different sizes. Therefore, when I deliver them to a customer, on the way to the table I rearrange them (so that the smallest winds up on top, and so on, down to the largest at the bottom) by grabbing several from the top and flipping them over, repeating this (varying the number I flip) as many times as necessary. If there are n pancakes, what is the maximum number of flips (as a function of n) that I will ever have to use to rearrange them?
In other words, if Homer visits Springfield’s Municipal House of Pancakes, as featured in “The Twisted World of Marge Simpson” (1997), and the waiter delivers him n pancakes in a random size order, how many flips will be required to put them into the correct size order in the worst-case scenario? This number of flips is known as the pancake number, Pn. The challenge is to find a formula that predicts Pn.
The pancake sorting problem immediately captured the interest of mathematicians for two reasons. First, it seemed to offer potential insights into solving computer science problems, because rearranging pancakes has parallels with rearranging data. Second, it was a deceptively difficult puzzle, and mathematicians adore problems that are borderline impossible.
Some simple cases illuminate the problem. First, what is the pancake number for just one pancake? The answer is zero, because the pancake cannot arrive in the wrong order. So, P1 = 0.
Next, what is the pancake number for two pancakes? Either the pancakes arrive in the correct order, or the reverse order. The worst case is easy to identify, and it requires only one flip to overturn both pancakes at once to transform them into the correct arrangement of pancakes. So, P2 = 1.
Next, what is the pancake number for three pancakes? This is trickier, because there are six possible starting arrangements. Depending on the starting arrangement, the number of flips required to reach the correct arrangement varies from zero to a worst-case scenario of three, so P3 = 3.
In most cases, you can work out for yourself how to obtain the correct order in the appropriate number of flips. However, for the worst-case scenario, the reordering process is not obvious, so this series of three flips is shown below. Each row indicates the action of one flip, namely where the spatula is inserted and the pancake order after the flip.
As the pile of pancakes grows, the problem becomes increasingly difficult as there are more and more possible starting arrangements, and an increasing number of possible flipping procedures. Worse still, there seems to be no pattern in the series of pancake numbers (Pn). Here are the first nineteen pancake numbers:
n
1
2
3
4
5
6
7
8
9
10
P
0
1
3
4
&nbs
p; 5
6
8
9
10
11
n
11
12
13
14
15
16
17
18
19
20
P
13
14
15
16
17
18
19
20
22
?
The sheer difficulty in running through all the pancake permutations and possible flipping strategies means that even very powerful computers have not yet been able to calculate the twentieth pancake number. And, after more than three decades, nobody has been able to sidestep the brute force computational approach by finding a clever equation that predicts pancake numbers. So far, the only breakthroughs have been in finding equations that set limits on the pancake number. In 1979, the upper limit for the pancake number was shown to be less than (5n + 5)/3 flips. This means that we can take a foolishly large number of pancakes, such as a thousand, and know for a fact that the pancake number (i.e., the number of flips required to rearrange the pancakes into size order in the worst-case scenario) will be less than
Thus, given that you cannot perform a third of a flip, P1,000 is less than or equal to 1,668. This result is famous, because it was published in a paper that was co-authored by William H. Gates and Christos H. Papadimitriou. William H. Gates is better known as Bill Gates, co-founder of Microsoft, and this is thought to be the only research paper that he has ever published.
The Gates paper, based on work he did as an undergraduate at Harvard, also mentions a devious variation of the problem. The burnt pancake problem involves pancakes that are burnt on one side, so the challenge is to flip them into the right orientation (burnt side down), as well as flipping them into the correct size order. This is the problem that was addressed by David S. Cohen while at Berkeley.
Cohen authored a paper15 on the burnt pancake problem in 1995, which set the lower and upper bounds for burnt pancake flipping between 3n/2 and 2n – 2. If we again use the example of 1,000 pancakes, but this time burnt, then we know that the number of flips required to orient and order them in the worst-case scenario is between 1,500 and 1,998.
This is what makes the writers of The Simpsons unique. They not only attend Math Club, but they also deliver rigorous lectures and even author serious mathematical research papers.
David S. Cohen recounted an anecdote that shows how the writers sometimes even astonish themselves when they realize the sheer level of mathematical prowess within the team: “I had written this paper on pancake numbers with help from my adviser, Manuel Blum, who’s a well-known computer scientist, and we submitted it to a journal called Discrete Applied Mathematics. I subsequently left graduate school to come and write for The Simpsons. After the paper was accepted, there was an extremely long lag between it being submitted, revised, and published. So, by the time the paper was published, I had been working at The Simpsons for a while, and Ken Keeler had also been hired at that point. So, finally the research article appeared, and I came in with the reprints of this article and I said, ‘Hey, I’ve got an article in Discrete Applied Mathematics.’ Everyone was quite impressed, except Ken Keeler, who said, ‘Oh yeah, I had a paper in that journal a couple of months ago.’”
With a wry smile on his face, Cohen bemoaned: “What does it mean that I come to write for The Simpsons and I cannot even be the only writer on this show with a paper in Discrete Applied Mathematics?”
CHAPTER 10
The Scarecrow Theorem
Homer Simpson is not usually considered an intellectual powerhouse, instead enjoying a reputation as one of Springfield’s more down-to-earth citizens. In “Homer vs. the Eighteenth Amendment” (1997), he offers a toast that explains his simple philosophy of life: “To alcohol! The cause of, and solution to, all of life’s problems.”
Nevertheless, the writers do occasionally allow Homer off the leash in order to explore the nerdier side of his character. We have already seen this in the 1998 episode “The Wizard of Evergreen Terrace,” and there are several other episodes in which Homer shows that he can be a poster boy for geek pride. For example, the world’s most prestigious scientific journal, Nature, praised him for a comment he makes in the episode “The PTA Disbands” (1995). After catching his daughter trying to build a perpetual motion machine, he puts her firmly in her place: “Lisa, in this house we obey the laws of thermodynamics!”
As well as parroting some of science’s most fundamental laws, Homer also occasionally sets the scientific agenda. In “E-I-E-I-D’oh” (1999), he turns his hand to farming and sprinkles plutonium on his fields to boost his yield. Not surprisingly, the resulting plants are mutants. Homer calls his new crop tomacco, because the plants have the outward appearance of tomatoes and yet contain tobacco inside.
Rob Bauer, a Simpsons fan from Oregon, saw the episode and was inspired to replicate Homer’s achievement. Instead of using radioactive material, he grafted tobacco roots onto a tomato plant and waited to see what would happen. It was not a completely crazy idea, because tomatoes and tobacco both belong to the nightshade family of plants, so grafting such plant relatives might enable the properties of one plant to transfer to the other. Indeed, the leaves of Bauer’s tomato plant did contain nicotine, proving that science fact can be almost as strange as science fiction.
The writers also encouraged Homer’s intellectual side to flourish in “They Saved Lisa’s Brain,” an episode that has already been discussed in Chapter 7. After Stephen Hawking saves Lisa from a baying mob, the story ends with Professor Hawking chatting to Lisa’s father in Moe’s Tavern, where he is impressed with Homer’s ideas about cosmology: “Your theory of a doughnut-shaped universe is intriguing . . . I may have to steal it.”
This sounds ridiculous, but mathematically minded cosmologists claim that the universe might actually be structured like a doughnut. In order to explain how this geometry is possible, let us simplify the universe by imagining that the entire cosmos is flattened from three dimensions into two dimensions, so that everything exists on a sheet. Common sense might suggest that this universal sheet would be flat and extend to infinity in all directions. But cosmology is rarely a matter of common sense. Einstein taught us that space can bend, which leads to all sorts of other potential scenarios. For example, imagine that the universal sheet is not infinite, but instead has four edges, so that it looks rather like a large rectangular sheet of rubber. Next, imagine joining the two long edges of the sheet so it forms a cylinder, then connecting the two ends of the cylinder so that the whole sheet has been transformed into a hollow doughnut. This is exactly the sort of universe that Hawking and Homer were discussing.
If you lived on the surface of this doughnut universe, you could follow the grey arrow and eventually return to your original position. Alternatively, you could follow the black arrow and, again, you would end up back where you started. The doughnut universe behaves rather like the spacescape of Asteroids, Atari’s best selling video game of all time. If the player’s ship flies eastward, then it leaves the screen on the right and returns on the left, eventually returning to its original position. Similarly, if the ship heads northward, then it leaves the top of the screen and reenters at the bottom, eventually returning once again to where it started.
Of course, we have discussed the theory only in terms of two dimensions, but within the laws of physics it is permissible for a three-dimensional universe to be rolled into a cylinder and formed into a doughnut. For nonmathematicians, it is almost impossible to visualize manipulating three-dimensional space in this manner, but Hawking and Homer understand that the doughnut is a perfectly viable reality for the shape of our universe. As the British scientist J. B. S. Haldane (1892–1964) once said: “My suspicion is that the Universe is not only queerer than we suppose, but queerer than we can suppose.”
In other episodes, the writers create a trigger event that galvanizes Homer’s brain, which in turn allows him to excel in mathematics. In “HOMЯ” (2001), Homer removes a crayon that has been lodged in his brain and suddenly realizes that he can use calculus to prove that God does not exist. He shows the proof to Ned Flanders, his God-fearing neighbor, who is initially suspicious of Homer’s claim to have made God vanish in a puff of logic. Flanders examines the proof and mutters: “We’ll just see about that . . .uh-oh. Well, maybe he made a mistake . . .Nope. It’s airtight. Can’t let this little doozy get out.” Unable to find any flaw that will undermine Homer’s logic, Flanders sets fire to the proof.
This scene pays homage to one of the most famous episodes in the history of mathematics, when the greatest mathematician of the eighteenth century, Leonhard Euler, pretended to prove the opposite of Homer’s conclusion, namely that God does exist. The incident took place while he was at the court of Catherine the Great in St. Petersburg. Catherine and her courtiers were becoming increasingly concerned about the influence of the visiting French philosopher Denis Diderot, who was an outspoken atheist. He was also supposedly terrified of mathematics. Hence, Euler was asked to construct a fake equation that would apparently prove the existence of God and put an end to Diderot’s heresies. When he was publicly confronted with Euler’s complicated equation, Diderot was left speechless. Diderot became the laughingstock of St. Petersburg after this humiliating encounter, and he soon asked for permission to return to Paris.
