Another pioneering aspect of The Simpsons, according to writer Patric Verrone, has been the development of the freeze-frame gag: “If it wasn’t invented at The Simpsons, it’s been perfected here. It’s a joke that just goes by unnoticed in the normal course of viewing, so you have to freeze the frame to see it. A lot of them are typically book titles or signs. It’s harder to put that sort of thing in a live-action show.”
Freeze-frame gags—which can last literally for just a single frame, or sometimes for a little longer—were included in The Simpsons from the beginning. In “Bart the Genius,” the first proper episode of The Simpsons, we see a library that contains both The Iliad and The Odyssey. Blink and you would have missed them. The joke, of course, is that these ancient Greek texts were written by Homer.
Freeze-frame gags were an opportunity to increase the comedic density of the show, but they also enabled the writers to introduce obscure references that rewarded viewers with niche knowledge. In that same episode, one of the students momentarily flashes his Anatoly Karpov lunchbox. Karpov was world chess champion from 1975 to 1985. His other claim to fame is that he holds the record for being the seller of the most valuable stamp from the Belgian Congo, auctioned off at $80,000 in 2011. If a viewer did not see the gag, then nothing was lost. However, if just one viewer noticed and appreciated the reference, then the writers considered it to be worth the effort.
To a large extent, the freeze-frame gag was a product of technological developments. Roughly 65 percent of American households owned a video cassette recorder by 1989, when The Simpsons was launched. This meant that fans could watch episodes several times and pause a scene when they had spotted something curious. At the same time, more than 10 percent of households had a home computer and a few people even had access to the Internet. The following year saw the birth of alt.tv.simpsons, a Usenet newsgroup that allowed fans to share, among other things, their freeze-frame discoveries.
According to Chris Turner, author of Planet Simpson, the most extreme version of freeze-frame humor appears in “Homer Badman” (1994), an episode in which a sensationalist investigative show called Rock Bottom falsely accuses Homer of lecherous behavior. The host, Godfrey Jones, is forced to make an apology on air and issue a correction, which takes the form of text rapidly scrolling down the screen. The average viewer sees nothing more than a blur, but there were thirty-four freeze-frame gags in four seconds, all perfectly legible for anybody willing to pause the episode and step through the corrections frame by frame.
Crucially, freeze-frame gags provided opportunities for the mathematical writers on The Simpsons to throw in some references that would appeal to hard-core number nerds. For example, “Colonel Homer” (1992) features the first appearance of the local movie theater, and eagle-eyed viewers would have noticed that it is called the Springfield Googolplex. In order to appreciate this reference it is necessary to go back to 1938, when the American mathematician Edward Kasner was in conversation with his nephew Milton Sirotta. Kasner casually mentioned that it would be useful to have a label to describe the number 10100 (or 10,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000). The nine-year-old Milton suggested the word googol.
Freeze-Frame Gags from The Simpsons
“HOMER BADMAN” (1994)
Lines on Rock Bottom correction list
If you are reading this, you have no life.
Our viewers are not pathetic sexless food tubes.
Quayle is familiar with common bathroom procedures.
The people who are writing this have no life.
“DUMBBELL INDEMNITY” (1998)
Sign outside Stu’s Disco
You Must Be at Least This Swarthy to Enter
“LARD OF THE DANCE” (1998)
Name of shop offering “Winter Madness Sale”
Donner’s Party Supplies
“BART VS. LISA VS. THE THIRD GRADE” (2002)
Title of Lisa’s book
Love in the Time of Coloring Books
“CO-DEPENDENT’S DAY” (2004)
Sign outside First Church of Springfield
We Welcome Other Faiths (Just Kidding)
“BART HAS TWO MOMMIES” (2006)
Sign at Left-Handers Convention
Today’s Seminar—Ambidextrous: Lefties in Denial?
In his book Mathematics and the Imagination, Kasner recalled how the conversation with his nephew continued: “At the same time that he suggested ‘googol’ he gave a name for a still larger number: ‘Googolplex.’ A googolplex is much larger than a googol, but is still finite, as the inventor of the name was quick to point out. It was first suggested that a googolplex should be 1, followed by writing zeros until you get tired.”
The uncle rightly felt that the googolplex would then be a somewhat arbitrary and subjective number, so he suggested that the googolplex should be redefined as 10googol. That is 1 followed by a googol zeroes, which is far more zeroes than you could fit on a piece of paper the size of the observable universe, even if you used the smallest font imaginable.
These terms—googol and googolplex—have become moderately well known today, even among members of the general public, because the term googol was adopted by Larry Page and Sergey Brin as the name of their search engine. However, they preferred a common misspelling, so the company is called Google, not Googol. The name implies that the search engine provides access to vast amounts of information. Google headquarters is, not surprisingly, called the Googleplex.
Simpsons writer Al Jean recalls that the Springfield Googolplex freeze-frame gag was not in the original draft of the script for “Colonel Homer.” Instead, he is confident that it was inserted in one of the collaborative rewrites, when the mathematical members of the team tend to exert their influence: “Yeah, I was definitely in the room for that. My recollection is that I didn’t pitch Googolplex, but I definitely laughed at it. It was based on theaters that are called octoplexes and multiplexes. I remember when I was in elementary school, the smart-ass kids were always talking about googols. That was definitely a joke by the rewrite room on that episode.”
Mike Reiss, who had worked with Jean on The Simpsons since the first season, thinks that the Springfield Googolplex was possibly his freeze-frame gag. When a fellow writer raised a concern that the joke was too obscure, Reiss remembers being very protective: “Someone made some remark about me giving him a joke that nobody was ever going to get, but it stayed in . . . It was harmless; how funny can the name of a multiplex theater be?”
Another mathematical freeze-frame appears in “MoneyBART.” In fact, you may already have glimpsed it in the frame presented in Chapter 6. Here is a close-up in order to help identify the freeze-frame reference.
When Lisa is studying to become a first-rate baseball coach, we see her surrounded by books, and one of the spines displays the title “eiπ + 1 = 0.” If you have studied mathematics beyond high school, then you may recognize this as Euler’s equation, sometimes referred to as Euler’s identity. An explanation of Euler’s equation would involve a degree of complexity that is beyond the scope of this chapter, but there is a partial and moderately technical explanation in appendix 2. In the meantime, we will focus on the initial component of the equation, which is a peculiar little number known as e.
The number e was discovered when mathematicians began to study a fascinating question about the usually tedious subject of bank interest. Imagine a simple investment scenario, in which one invests $1.00 in an extraordinarily convenient and generous bank account that offers 100 percent interest per year. At the end of the year, that $1.00 would have accrued $1.00 interest, giving a total of $2.00.
Now, instead of 100 percent interest after one year, consider a scenario in which the interest is halved, but calculated twice. In other words, the investor receives 50 percent interest after both six and twelve months. Thus, after the first six
months, the $1.00 would have accrued $0.50 interest, giving a total of $1.50. During the second six months, interest is gathered on both the $1.00 and the additional $0.50 interest that has already accrued. Therefore the additional interest added after twelve months is 50 percent of $1.50, which equals $0.75, resulting in an overall total of $2.25 at the end of the year. This is known as compound interest.
As you can see, the good news is that this half-year compound interest is more profitable than simple annual interest. The bank balance could have been even higher if the compound interest had been calculated more frequently. For instance, if it had been calculated quarterly (25 percent every three months), then the total would have been $1.25 at the end of March, $1.56 at the end of June, $1.95 at the end of September, and $2.44 at the end of the year.
Al Jean (who is holding an iron) was in the room when Mike Reiss (seated, left) suggested Googolplex as the name of Springfield’s movie theater. This 1981 photograph shows them while at Harvard in the “Lampoon Castle.” Patric Verrone, who is seen here juggling pool balls, is also a successful TV comedy writer, with a list of credits that includes a 2005 episode of The Simpsons entitled “Milhouse of Sand and Fog.” The fourth member of the group is Ted Phillips, who passed away in 2005. Although he had a talent for writing, he went on to pursue a career in law in South Carolina and was a respected local historian. He is name-checked in the episode “Radio Bart” (1992) and also has a character (Duke Phillips) named after him in The Critic, an animated series created by Jean and Reiss.
If n is the number of increments (i.e., the number of times per year that interest is calculated and added), then the following formula can be used to calculate the final sum (F ) when the compound interest is also calculated at monthly, weekly, daily, and even hourly intervals:
F = $(1 + ⅟n)n
Initial sum
Annual interest
Time increment
Number of increments (n)
Incremental interest
Final sum (F)
$1.00
100%
1 year
1
100.00%
$2.00
$1.00
100%
½ year
2
50.00%
$2.25
$1.00
100%
¼ year
4
25.00%
$2.4414...
$1.00
100%
1 month
12
8.33%
$2.6130...
$1.00
100%
1 week
52
1.92%
$2.6925...
$1.00
100%
1 day
365
0.27%
$2.7145...
$1.00
100%
1 hour
8,760
0.01%
$2.7181...
By the time compound interest is calculated on a weekly basis, we are almost $0.70 better off than if we had been earning only simple annual interest. However, after this point, calculating the compound interest even more frequently achieves only one or two more pennies. This leads us to the fascinating question that began to obsess mathematicians: If the compound interest could be calculated not just every hour, not just every second, not just every microsecond, but at every moment, what would be the final sum at the end of the year?
The answer turns out to be $2.718281828459045235360287471352662497757247093699959574966967627724076630353547594571382178525166427.... As you can probably guess, the decimal places continue to infinity, it is an irrational number, and it is the number that we call e.
2.718... was named e because it relates to exponential growth, which describes the surprising rate of growth experienced when money gathers interest year after year, or when anything repeatedly grows by a fixed rate again and again. For example, if the investment did increase in value by a factor of 2.718... year after year, then $1.00 becomes $2.72 after year one, then $7.39 after year two, then $20.09, then $54.60, then $148.41, then $403.43, then $1,096.63, then $2,980.96, then $8,102.08, and finally $22,026.47, in just ten years.
Such staggering rates of sustained exponential growth are rare within the world of financial investment, but there are concrete examples elsewhere. The most famous illustration of exponential growth has taken place in the world of technology and is known as Moore’s law, named after Gordon Moore, co-founder of Intel. In 1965, he observed that the number of transistors on a microprocessor chip doubles approximately every two years, and he predicted that this trend would continue. Sure enough, Moore’s law has held true decade after decade. The forty years between 1971 and 2011 have resulted in twenty doublings in the number of transistors. In other words, there has been an improvement by a factor of 220, or roughly one million, in the number of transistors on a chip over four decades. This is why we now have microprocessors with vastly improved performance at hugely reduced costs compared with the 1970s.
By way of analogy, it is sometimes said that if cars had achieved the same rapid improvement as computers, then a Ferrari would cost just $100 today and would manage a million miles per gallon . . . but it would also crash once a week.
Being linked to compound interest and exponential growth is interesting, but e has much more to offer the world. Just like π, the number e crops up in all sorts of unexpected situations.
For example, e is at the heart of the so-called problem of derangements, more commonly known as the hat check problem. Imagine that you are running the cloakroom at a restaurant, collecting hats from customers and putting them in hat boxes. Unfortunately, you do not make a note of which hat belongs to which person. As the diners return later in the evening, you hand back the hat boxes at random and wave good-bye to the customers before they have a chance to open the boxes. What is the probability that none of the boxes contains the right hat for the right person? The answer depends on the number of customers (n), and the probability for zero matches, labeled P(n), can be found according to the following formula18:
So for one guest the probability of zero matches is 0, because the one hat will inevitably reach the right person:
For two guests, the probability of zero matches is 0.5:
For three customers, the probability of zero matches is 0.333:
For four customers, the probability is roughly 0.375, and for ten customers it is approximately 0.369. As the number of customers tends to infinity, the probability settles down to 0.367879..., which is 1/2.718..., or 1/e.
You can test this for yourself by taking two decks of cards and shuffling each of them separately, so that the two decks are randomized. One deck represents the random way that hats were put in boxes, while the other deck represents the random order in which customers will return to collect their hats. Place the two decks side by side and turn over cards one at a time from the top of each deck. If both cards have the same suit and rank, then this counts as a match. The probability of zero matches after going through both decks will be close to 1/e, which is roughly 0.37, or 37 percent. In other words, if you repeat this entire process one hundred times, then you can expect a very poor social life and roughly thirty-seven pairs of decks with zero matches. The hat check problem might seem trivial, but it is a fundamental question in an area known as combinatorial mathematics.
The number e also crops up in the study of a type of curve known as a catenary, which is the shape formed by a chain hammocked between two points. The term was coined by Thomas Jefferson and is based on the Latin word catena, meaning “chain.” The shape of a catenary curve is described by the following equation, which has e at its heart, twice:
The silk in a spider’s web forms a series of catenaries between the spokes, which prompted French entomologist Jean-Henri Fabre to write in La Vie des Araignées (The Life of the Spider): “Here we have the abracadabric number e rea
ppearing, inscribed on a spider’s thread. Let us examine, on a misty morning, the meshwork that has been constructed during the night. Owing to their hygrometrical nature, the sticky threads are laden with tiny drops, and, bending under the burden, have become so many catenaries, so many chaplets of limpid gems, graceful chaplets arranged in exquisite order and following the curve of a swing. If the sun pierce the mist, the whole lights up with iridescent fires and becomes a resplendent cluster of diamonds. The number e is in its glory.”
We can also find e popping up in a completely different area of mathematics. Imagine using the randomization button on a calculator to generate random numbers between 0 and 1, and then continuing to add them together until the total exceeds 1. Sometimes it will require two random numbers, usually three, and occasionally four or more numbers to reach a total bigger than 1. However, on average, the number of random numbers required to exceed 1 is 2.71828..., which, of course, is e.
There are numerous other examples demonstrating that e plays a diverse and fundamental role in several areas of mathematics. This explains why so many number lovers have a particularly emotional attachment to it.
For example, Donald Knuth, professor emeritus at Stanford University and a godlike figure in the world of computing, is an e enthusiast. After authoring Metafont, his font-creation software, he decided to release updates with version numbers that relate to e. This means that the first version was Metafont 2, then Metafont 2.7, then Metafont 2.71, and so on up to the current Metafont 2.718281. Each new version number is a closer approximation to the true value of e. This is only one of several ways in which Knuth has expressed his quirky approach to his work. Another example is the index of his seminal work The Art of Computer Programming, volume 1, in which the entry for “Circular definition” points to “Definition, circular,” and vice versa.