A hexagon has more sides than a square, which makes it a better approximation to a circle. This explains why it delivers tighter limits for the value of π. Nonetheless, there is still a large margin of error here. So, Archimedes persisted, repeating his method with increasingly multisided polygons, using shapes that approximated ever closer to a circle.
Indeed, Archimedes persevered to the extent that he eventually trapped a circle between two 96-sided polygons, and he calculated the perimeters of both shapes. This was an impressive feat, particularly bearing in mind that Archimedes did not have modern algebraic notation, he had no knowledge of decimals, and he had to do all his lengthy calculations by hand. But it was worth the effort, because he was able to trap the true value of π between 3.141 and 3.143.
Fast-forwarding eight centuries to the 5th century A.D., the Chinese mathematician Zu Chongzhi took the Archimedean approach another step—or another 12,192 steps to be exact—and used two 12,288-sided polygons to prove that the value of π lay between 3.1415926 and 3.1415927.
The polygonal approach reached its zenith in the seventeenth century with mathematicians such as the Dutchman Ludolph van Ceulen, who employed polygons with more than 4 billion billion sides to measure π to 35 decimal places. After he died in 1610, the engraving on his tombstone explained that π was more than 3.14159265358979323846264338327950288 and less than 3.14159265358979323846264338327950289.
As you may have deduced by this point, measuring π is a tough job, and one that would carry on for eternity. This is because π is an irrational number. So, is there any point in calculating π to any greater accuracy? We will return to this question later, but for the time being we have covered enough essential π information to provide the context for the mathematical joke in the episode “Bye, Bye, Nerdie.”
The plot of the episode centers on the bullying of nerds, which continues to be a global problem despite the wise words of the American educationalist Charles J. Sykes, who wrote in 1995: “Be nice to nerds. Chances are you’ll end up working for one.” When Lisa endeavors to explain why bullies cannot resist picking on nerds, she suspects that nerds are emitting a scent that marks them out as victims. She persuades some of her nerdiest school friends to work up a sweat, which she collects and analyzes. After a great deal of research, she finally isolates a pheromone emitted by every “geek, dork, and four-eyes” that could be responsible for attracting bullies. She names this pheromone poindextrose, in honor of Poindexter, the boy genius character created for the 1959 cartoon series Felix the Cat.
In order to test her hypothesis, she rubs some poindextrose on the jacket of the formidable ex-boxer Drederick Tatum, who is visiting her school. Sure enough, the pheromone attracts Nelson Muntz, the school bully. Even though Nelson knows it is preposterous and inappropriate for a schoolyard bully to attack an ex-boxer, he cannot resist the allure of poindextrose and even gives Tatum a wedgie. Lisa has the proof she needs.
Excited by her discovery, Lisa decides to deliver a paper (“Airborne Pheromones and Aggression in Bullies”) at the 12th Annual Big Science Thing. The conference is hosted by John Nerdelbaum Frink Jr., Springfield’s favorite absentminded professor. It is Frink’s responsibility to introduce Lisa, but the atmosphere is so intense and the audience so excitable that he struggles to bring the conference to order. Frustrated and desperate, Frink eventually calls out: “Scientists . . . Scientists, please! I’m looking for some order. Some order, please, with the eyes forward . . . and the hands neatly folded . . . and the paying of attention . . . Pi is exactly three!”
Suddenly, the noise stops. Professor Frink’s idea worked, because he correctly realized that declaring an exact value for π would stun an audience of geeks into silence. After thousands of years of struggling to measure π to incredible accuracy, how dare anybody replace 3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117067982148086513... with 3!
The scene echoes a limerick written by Professor Harvey L. Carter (1904–94), a historian at Colorado College:
’Tis a favorite project of mine,
A new value of pi to assign.
I would fix it at 3
For it’s simpler, you see,
Than 3 point 1 4 1 5 9.
However, Frink’s outrageous statement was not based on Carter’s whimsical limerick. Instead, Al Jean explained that he had suggested the “Pi is exactly three!” line because he had recently read about an incident that took place in Indiana in 1897, when politicians attempted to legislate an official (and wildly incorrect) value for π.
The Indiana Pi Bill, officially known as House Bill No. 246, of the 1897 sitting of the Indiana General Assembly was the brainchild of Edwin J. Goodwin, a physician from the town of Solitude in the southwest corner of the state. He had approached the assembly and proposed a bill that focused on his solution to a problem known as “squaring the circle,” an ancient problem that had already been proven impossible in 1882. Goodwin’s complicated and contradictory explanation contained the following line relating to the diameter of a circle:
“. . . the fourth important fact, that the ratio of the diameter and circumference is as five-fourths to four.”
The ratio of the diameter to the circumference is equal to π, so Goodwin was effectively dictating a value for π according to the following recipe:
Goodwin said that Indiana schools could use his discovery without charge, but that the state and he would share the profits from royalties charged to other schools who wished to adopt a value of 3.2 for π. Initially, the technical nature of the bill meant that it baffled the politicians, who bounced it from the House of Representatives to the Finance Committee to the Committee on Swamplands and finally to the Committee on Education, where an atmosphere of confusion led to it being passed without any objection.
It was then up to the state senate to ratify the bill. Fortunately, a certain Professor C. A. Waldo, who was then head of the Mathematics Department at Purdue University in West Lafayette, Indiana, happened to be visiting the statehouse during this period to discuss funding for the Indiana Academy of Science. By chance, someone on the funding committee showed him the bill and offered to introduce him to Dr. Goodwin, but Waldo replied that this would not be necessary, as he already knew enough crazy people.
Instead, Professor Waldo worked hard to raise concerns with the senators, who began to ridicule Goodwin and his bill. The Indianapolis Journal quoted Senator Orrin Hubbell: “The Senate might as well try to legislate water to run up hill as to establish mathematical truth by law.” Consequently, when the bill was debated a second time, there was a successful motion to postpone it indefinitely.
Professor Frink’s absurd declaration that π equals 3 is a neat reminder that Goodwin’s postponed bill still exists in a filing cabinet in the basement of the Indiana statehouse, waiting for a gullible politician to resuscitate it.
CHAPTER 3
Homer’s Last Theorem
Every so often, Homer Simpson explores his inventing talents. In “Pokey Mom” (2001), for instance, he creates Dr. Homer’s Miracle Spine-O-Cylinder, which is essentially a battered trash can with random dents that “perfectly match the contours of the human verti-brains.” He promotes his invention as a treatment for back pain, even though there is not a jot of evidence to support his claim. Springfield’s chiropractors, who are outraged that Homer might steal their patients, threaten to destroy Homer’s invention. This would allow them once again to corner the market in back problems and happily promote their own bogus treatments.
Homer’s inventing exploits reach a peak in “The Wizard of Evergreen Terrace” (1998). The title is a play on the Wizard of Menlo Park, the nickname given to Thomas Edison by a newspaper reporter after he established his main laboratory in Menlo Park, New Jersey. By the time he died in 1931, Edison had 1,093 U.S. patents in his name and had become an inventing legend.
The episode focuses on Hom
er’s determination to follow in Edison’s footsteps. He constructs various gadgets, ranging from an alarm that beeps every three seconds just to let you know that everything is all right to a shotgun that applies makeup by shooting it directly onto the face. It is during this intense research and development phase that we glimpse Homer standing at a blackboard and scribbling down several mathematical equations. This should not be a surprise, because many amateur inventors have been keen mathematicians, and vice versa.
Consider Sir Isaac Newton, who incidentally made a cameo appearance on The Simpsons in an episode titled “The Last Temptation of Homer” (1993). Newton is one of the fathers of modern mathematics, but he was also a part-time inventor. Some have credited him with installing the first rudimentary flapless cat flap—a hole in the base of his door to allow his cat to wander in and out at will. Bizarrely, there was a second smaller hole made for kittens! Could Newton really have been so eccentric and absentminded? There is debate about the veracity of this story, but according to an account by J. M. F. Wright in 1827: “Whether this account be true or false, indisputably true is it that there are in the door to this day two plugged holes of the proper dimensions for the respective egresses of cat and kitten.”
The bits of mathematical scribbling on Homer’s blackboard in “The Wizard of Evergreen Terrace” were introduced into the script by David S. Cohen, who was part of a new generation of mathematically minded writers who joined The Simpsons in the mid-1990s. Like Al Jean and Mike Reiss, Cohen had exhibited a genuine talent for mathematics at a young age. At home, he regularly read his father’s copy of Scientific American and toyed with the mathematical puzzles in Martin Gardner’s monthly column. Moreover, at Dwight Morrow High School in Englewood, New Jersey, he was co-captain of the mathematics team that became state champions in 1984.
David S. Cohen pictured in the Dwight Morrow High School yearbook of 1984. The running joke was that everyone on the Math Team was co-captain, so that they all could put it on their college applications.
Along with high school friends David Schiminovich and David Borden, he formed a teenage gang of computer programmers called the Glitchmasters, and together they created FLEET, their very own computer language, designed for high speed graphics and gaming on the Apple II Plus. At the same time, Cohen maintained an interest in comedy writing and comic books. He pinpoints the start of his professional career to cartoons he drew while in high school that he sold to his sister for a penny.
Even when he went on to study physics at Harvard University, he maintained his interest in writing and joined the Harvard Lampoon, eventually becoming president. Over time, like Al Jean, Cohen’s passion for comedy and writing overtook his love of mathematics and physics, and he rejected a career in academia in favor of becoming a writer for The Simpsons. Every so often, however, Cohen returns to his roots by smuggling mathematics into the TV series. The symbols and diagrams on Homer’s blackboard provide a good example of this.
Cohen was keen in this instance to include scientific equations alongside the mathematics, so he contacted one of his high school friends, David Schiminovich, who had stayed on the academic path to become an astronomer at Columbia University.
The first equation on the board is largely Schiminovich’s work, and it predicts the mass of the Higgs boson, M(H0), an elementary particle that that was first proposed in 1964. The equation is a playful combination of various fundamental parameters, namely the Planck constant, the gravitational constant, and the speed of light. If you look up these numbers and plug them into the equation,4 it predicts a mass of 775 giga-electron-volts (GeV), which is substantially higher than the 125 GeV estimate that emerged when the Higgs boson was discovered in 2012. Nevertheless, 775 GeV was not a bad guess, particularly bearing in mind that Homer is an amateur inventor and he performed this calculation fourteen years before the physicists at CERN, the European Organization for Nuclear Research, tracked down the elusive particle.
The second equation is . . . going to be set aside for a moment. It is the most mathematically intriguing line on the board and worth the wait.
The third equation concerns the density of the universe, which has implications for the fate of the universe. If Ω(t0) is bigger than 1, as initially written by Homer, then this implies that the universe will eventually implode under its own weight. In an effort to reflect this cosmic consequence at a local level, there appears to be a minor implosion in Homer’s basement soon after viewers see this equation.
Homer then alters the inequality sign, so the equation changes from Ω(t0) > 1 to Ω(t0)
The fourth line on the blackboard is a series of four mathematical diagrams that show a doughnut transforming into a sphere. This line relates to an area of mathematics called topology. In order to understand these diagrams, it is necessary to know that a square and a circle are identical to each other according to the rules of topology. They are considered to be homeomorphic, or topological twins, because a square drawn on a rubber sheet can be transformed into a circle by careful stretching. Indeed, topology is sometimes referred to as “rubber sheet geometry.”
Topologists are not concerned with angles and lengths, which are clearly altered by stretching the rubber sheet, but they do care about more fundamental properties. For example, the fundamental property of a letter A is that it is essentially a loop with two legs. The letter R is also just a loop with two legs. Hence, the letters A and R are homeomorphic, because an A drawn on a rubber sheet can be transformed into an R by careful stretching.
However, no amount of stretching can transform a letter A into a letter H, because these letters are fundamentally different from each other by virtue of A consisting of one loop and two legs and H consisting of zero loops. The only way to turn an A into an H is to cut the rubber sheet at the peak of the A, which destroys the loop. However, cutting is forbidden in topology.
The principles of rubber sheet geometry can be extended into three dimensions, which explains the quip that a topologist is someone who cannot tell the difference between a doughnut and a coffee cup. In other words, a coffee cup has just one hole, created by the handle, and a doughnut has just one hole, in its middle. Hence, a coffee cup made of a rubbery clay could be stretched and twisted into the shape of a doughnut. This makes them homeomorphic.
By contrast, a doughnut cannot be transformed into a sphere, because a sphere lacks any holes, and no amount of stretching, squeezing, and twisting can remove the hole that is integral to a doughnut. Indeed, it is a proven mathematical theorem that a doughnut is topologically distinct from a sphere. Nevertheless, Homer’s blackboard scribbling seems to achieve the impossible, because the diagrams show the successful transformation of a doughnut into a sphere. How?
Although cutting is forbidden in topology, Homer has decided that nibbling and biting are acceptable. After all, the initial object is a doughnut, so who could resist nibbling? Taking enough nibbles out of the doughnut turns it into a banana shape, which can then be reshaped into a sphere by standard stretching, squeezing, and twisting. Mainstream topologists might not be thrilled to see one of their cherished theorems going up in smoke, but a doughnut and a sphere are identical according to Homer’s personal rules of topology. Perhaps the correct term is not homeomorphic, but rather Homermorphic.
The second line on Homer’s blackboard is perhaps the most interesting, as it contains the following equation:
3,98712 + 4,36512 = 4,47212
The equation appears to be innocuous at first sight, unless you know something about the history of mathematics, in which case you are about to smash up your slide rule in disgust. For Homer seems to have achieved the impossible and found a solution to the notorious mystery of Fermat’s last theorem!
Pierre de Fermat fi
rst proposed this theorem in about 1637. Despite being an amateur who only solved problems in his spare time, Fermat was one of the greatest mathematicians in history. Working in isolation at his home in southern France, his only mathematical companion was a book called Arithmetica, written by Diophantus of Alexandria in the third century A.D. While reading this ancient Greek text, Fermat spotted a section on the following equation:
x 2 + y 2 = z 2
This equation is closely related to the Pythagorean theorem, but Diophantus was not interested in triangles and the lengths of their sides. Instead, he challenged his readers to find whole number solutions to the equation. Fermat was already familiar with the techniques required to find such solutions, and he also knew that the equation has an infinite number of solutions. These so-called Pythagorean triple solutions include
32 + 42
=
52
52 + 122
=
132
1332 + 1562
=
2052
So, bored with Diophantus’ puzzle, Fermat decided to look at a variant. He wanted to find whole number solutions to this equation:
x 3 + y 3 = z 3
Despite his best efforts, Fermat could only find trivial solutions involving a zero, such as 03 + 73 = 73. When he tried to find more meaningful solutions, the best he could offer was an equation that was out of kilter by just one, such as 63 + 83 = 93 – 1.