Q: Why did the chicken cross the Möbius strip?
A: To get to the other . . . er . . . !
Although we do not actually see a Möbius strip in the episode “Möbius Dick,” the good news is that there are plans to feature one of these odd bits of mathematical tomfoolery in an upcoming Futurama plot. When I visited David X. Cohen at the Futurama offices in the fall of 2012, he told me about an episode in the upcoming season titled “2-D Blacktop,”24 which will star Professor Farnsworth. Cohen explained that the storyline involves the elderly proprietor of Planet Express turning into a speed freak who soups up his spaceship in order to race it on a Möbius drag strip. The interesting feature of such a track—as demonstrated by the felt-tip experiment—is that Farnsworth will need to complete two laps in order to get back to where he started.
Cohen revealed a few plot details: “Leela gets mad at the Professor and they end up racing on the Möbius drag strip. Leela is leading, but the Professor has this big racing move called the dimensional drift. He spins the wheel while pulling the emergency brake, which causes him to drift through one dimension higher than where he is. So, he skids out of the third dimension, then passes briefly through the fourth dimension, so he can reappear back in the third dimension further along the track.”
Unfortunately, shifting up and down through dimensions also leaves Professor Farnsworth traveling in the opposite direction to Leela. Their vehicles collide head-on, thereby crushing them both down into the second dimension! The next scene then takes place against a dimensionally challenged landscape.
In many ways, “2-D Blacktop” is the antidote to “Homer3.” That episode from The Simpsons explored the consequences of being lifted into a higher dimension, drawing upon a Twilight Zone episode for its inspiration. By contrast, “2-D Blacktop” explores what it means to be squashed down to a lower dimension, and it too is inspired by a classic piece of science fiction.
“2-D Blacktop” is an homage to a Victorian sci-fi novella titled Flatland, by Edwin A. Abbott. Subtitled A Romance of Many Dimensions, the story begins in a two-dimensional world known as Flatland. This universe is composed of a single surface populated with various shapes, such as line segments (women), triangles (working-class men), and squares (middle-class men). Essentially, the greater the number of sides, the higher the status, so women have the lowest status, polygons make up the upper echelons of society, and circles are high priests. As a theologian who had studied mathematics at the University of Cambridge, Abbott was keen that readers appreciated his Flatland as both a social satire and an adventure in geometry.
The central character and narrator is a Square, who has a dream in which he visits Lineland, a one-dimensional universe, where a population of points are confined to traveling along a single line. The Square talks to the points and tries to explain the concept of a second dimension and the resulting variety of shapes that occupy Flatland, but the points remain confused. They cannot even appreciate the true nature of the Square, because his shape is inconceivable from their one-dimensional point of view. They see the Square as a line, because that is the cross section that a square makes as it passes through Lineland.
After waking up and realizing that he is back in his Flatland, the Square’s adventures continue when he is visited by a Sphere, an object from the exotic third dimension. Of course, this time it is the Square who is baffled, because he can only perceive the Sphere as a Circle, which is the cross section that the Sphere makes as it passes through Flatland. However, everything begins to make sense when the Sphere drags the Square up into Spaceland. As the Square looks down upon his fellow Flatlanders from the third dimension, he can even speculate about the possibilities of a fourth, a fifth, and even higher dimensions.
When he returns to Flatland, the Square tries to spread the gospel of the third dimension, but nobody wants to listen. Worse still, the authorities clamp down on such blasphemy. In fact, the leaders of Flatland already know of the existence of the Sphere, so they arrest the Square in order to keep the third dimension a secret. The story ends tragically with the Square locked up in prison for telling the truth.
So how does the forthcoming Futurama episode pay tribute to Flatland? When Professor Farnsworth and Leela collide in “2-D Blacktop,” the head-on impact transforms them into flat versions of themselves, sliding around in a flat landscape, which is populated by flat animals, flat plants, and flat clouds.
The animation adheres strictly to the rules of a two-dimensional world, which means that no object can pass over another object, only around it. However, while I watched a rough cut two-dimensional sequence from “2-D Blacktop” with editor Paul Calder, he spotted the fluffy edges of one cloud overlapping slightly with the fluffy edges of another cloud. Overlaps are forbidden in a two-dimensional world, so this will require fixing before the episode is aired.
As they attempt to understand the implications of their new world, Leela and the Professor gradually realize that their digestive canals vanished when they were squashed from three dimensions down to two. This is a necessary part of the transformation process, because a digestive canal in two dimensions is a recipe for disaster. To appreciate the problem, imagine the Professor as a flat cut-out figure facing to the right. Then draw a line from his mouth to his posterior, representing his gastrointestinal canal. Finally cut along this line and slightly separate the two parts of the Professor’s body; the canal is a tunnel in three dimensions, but is simply a gap in two dimensions. Now you can see the problem. With a digestive system in place, the Professor’s body would drift apart in two dimensions. Obviously the same would be true for Leela.
However, without digestive tracts, the Professor and Leela are unable to eat. The other creatures in this two-dimensional world survive by somehow absorbing nutrients, as opposed to eating and excreting food, but the Professor and Leela have not mastered this trick.
In short, for the Professor and Leela, digestive tracts are a case of “can’t live with them, can’t live without them.” Hence, they have to escape their two-dimensional world before they starve to death, and fortunately the writers come to their rescue. Cohen explained: “The Professor and Leela have this realization. They can use the dimensional drift to get out of the second dimension and into the third dimension. We actually have this amazing sequence, because they fly through this huge fractal landscape that represents the area between two dimensions and three dimensions. The scene contains some pretty amazing computer graphics.”
The fractal landscape is particularly appropriate, because fractals actually exhibit a fractional dimensionality. The fractal landscape appears on the journey between the two-dimensional and three-dimensional worlds, which is exactly where one might expect to find a fractional dimension.
If you want to know more about fractals, please refer to appendix 4, where there is a very brief overview of this topic, focusing particularly on how an object can possibly be fractionally dimensional.
The Möbius strip in “2-D Blacktop” resonates with a mathematical concept that appears in “The Route of All Evil” (2002). This episode has a subplot that involves Bender turning himself into a home brewery. He gets the idea after he and his Planet Express colleagues visit a 711 convenience store to buy some alcohol. The store stocks Bender’s usual tipple, Olde Fortran malt liquor, named in honor of FORTRAN (FORmula TRANslation), a computer-programming language developed in the 1950s. The shelves are also stacked with St. Pauli’s Exclusion Principle Girl beer, which combines the name of an existing beer (St. Pauli Girl) with one of the foundations of quantum physics (the Pauli exclusion principle). Most interesting of all is a third brew called Klein’s, which comes in a strange flask. Aficionados of weird geometry will recognize that this is a Klein bottle, which is closely related to the Möbius strip.
The beer is called Klein’s in honor of Felix Klein, one of the greatest German mathematicians of the nineteenth century. His destiny may have been dictated the moment he was born, because each element of his date of bi
rth, April 25, 1849, is the square of a prime number:
April
25
1849
4
25
1,849
22
52
432
Klein’s research ranged across several areas, but he is most famous for the so-called Klein bottle. As with the Möbius strip, it will be easier to understand the shape and structure of a Klein bottle if you construct your own. You will require:
(a) a sheet of rubber,
(b) some sticky tape,
(c) a fourth dimension.
If, like me, you do not have access to a fourth dimension, then you can imagine how we might theoretically build a pseudo–Klein bottle in three dimensions. First, imagine rolling the rubber sheet into a cylinder and taping it along its length as shown here in the first diagram. Then mark the two ends of the cylinder with arrows going in opposite directions. Next, and this is the tricky step, you must introduce a twist in the cylinder so that you can connect the two ends with both arrows heading in the same direction.
This is where the fourth dimension would come in very useful, but instead you will have to make do with a minor fudge. As shown in the middle two diagrams, bend the cylinder back on itself, and then imagine pushing one end of the cylinder through the wall of the selfsame cylinder and up the inside. Finally, after this self-intersection step, roll the penetrating end of the cylinder downward, as in the fourth diagram, in order to connect the two ends of the cylinder. Crucially, when this connection is made, the arrows on each end of the cylinder will be pointing in the same direction.
Both this Klein bottle and the Klein beer bottle in Futurama are self-intersecting, because they both exist in three dimensions. By contrast, a Klein bottle in four dimensions would avoid the necessity for self-intersection. In order to explain how an extra dimension can help avoid self-intersection, let us consider a similar situation involving fewer dimensions.
Imagine a figure eight shape made with a pen on a piece of paper. Inevitably, the ink line intersects itself at the center of the eight, in the same way that that the cylinder intersects itself at the center of the Klein bottle. The inky intersection occurs because the line is trapped within a two-dimensional surface. The problem does not arise, however, if a third dimension is added and the figure eight is created with a piece of rope. One section of the rope can rise up into this third dimension as it overlays another section, so there is no need for the rope to intersect itself. Similarly, if the rubber sheet cylinder could rise up into the fourth dimension, then it would be possible to create a Klein bottle without a self-intersection.
Another way to think about why the Klein bottle intersects itself in three dimensions, but not in four, is to consider how we might view a windmill in three dimensions compared with two dimensions. In three dimensions, we can see that the blades sweep around in front of the vertical tower. However, the situation is different if we look at a shadow of the windmill projected onto the grass. In this two-dimensional representation, the blades appear to sweep through the tower over and over again. The blades intersect the tower in the two-dimensional projection, but not in the three-dimensional world.
The architecture of a Klein bottle is obviously different from that of an ordinary bottle, which in turn leads to a remarkable property. This becomes apparent if we imagine traveling over the surface of the Klein bottle here. In particular, imagine following the path of the black arrow, which is positioned on the outer surface of the Klein bottle.
The arrow moves upward, then loops around the outside of the neck and dives down to the intersection point, where the arrow’s head becomes grey. This indicates that the arrow is now entering the inside of the bottle. As the arrow moves forward, it soon passes its starting position, except that now it is inside the bottle. If the arrow continues its journey up toward the neck and down again to the base, it then returns to the outer surface and eventually arrives back at its original position. Because the arrow is able to journey smoothly between the inner and outer surfaces of the Klein bottle, this indicates that the two surfaces are actually both part of the same surface.
Of course, without a well-defined inside and outside, the Klein bottle fails one of the main criteria required for a fully functioning bottle. After all, how can you put beer in a Klein bottle, when in is the same as out?
In fact, Klein never called his creation a bottle. It was originally called a Kleinsche Fläche, meaning a “Klein surface,” which is appropriate as it consists of a single surface. However, English-speaking mathematicians probably misheard this as Kleinsche Flasche, which translates into English as “Klein bottle,” and the name stuck.
Finally, returning to a point raised earlier, the Klein bottle and the Möbius strip are closely related to each other. The most obvious connection is that both the strip and the bottle share the curious property of having only one surface. A second, and less obvious, connection is that a Klein bottle sliced into two halves creates a pair of Möbius strips.
Unfortunately, you cannot perform this party trick, because it is only possible to slice a Klein bottle if you have access to a fourth dimension. However, you can slice a Möbius strip. Indeed, I would encourage you to cut a Möbius strip along its length in order to find out what happens.
Finally, if you have become hooked on slicing strips, here is one more suggestion for your new hobby of geometry surgery. First, create a strip with a full 360-degree twist (as opposed to the half twist in a Möbius strip). What happens when this strip is cut along its length? It takes a twisted mind to predict the outcome of this twisted dissection.
CHAPTER 17
The Futurama Theorem
Due to his sometimes geriatric-delinquent antics, it is easy to forget that Futurama’s Professor Hubert J. Farnsworth is a mathematical genius. In fact, in the feature-length The Beast with a Billion Backs (2008), we learn that Farnsworth has been awarded a Fields Medal, the highest accolade in mathematics. It is sometimes dubbed the Nobel Prize of Mathematics, but the title of Fields Medallist is arguably even more prestigious than Nobel laureate, because the medals are only awarded every four years.
The Professor regularly discusses his mathematical ideas in a lecture course “The Mathematics of Quantum Neutrino Fields,” which takes place at Mars University, where he is a tenured professor. A tenured position is essentially a job for life, which means that the Professor has to avoid the hazard of tenure-induced mental stagnation. This is a well-known phenomenon in academic circles, and the problem was highlighted by the American philosopher Daniel C. Dennett in his book Consciousness Explained: “The juvenile sea squirt wanders through the sea searching for a suitable rock or hunk of coral to cling to and make its home for life. For this task, it has a rudimentary nervous system. When it finds its spot and takes root, it doesn’t need its brain anymore so it eats it! (It’s rather like getting tenure.)”
Rather than stagnating, Farnsworth has used his tenured position to dabble in other areas of research. So, as well as being a mathematician, he is also an inventor. Indeed, it is no coincidence that Groening and Cohen named the Professor after Philo T. Farnsworth (1906–71), a prolific American inventor who held over one hundred U.S. patents ranging from TV technology to a mini nuclear fusion device.
One of the Professor’s oddest inventions is the Cool-O-Meter, which accurately assesses the level of cool possessed by a person, with the measurement given in units of megafonzies. One fonzie is the quantity of cool associated with Arthur Fonzarelli, the lead character in the 1970s sitcom Happy Days. By choosing a nomenclature based on an iconic figure, Farnsworth was echoing the millihelen, which is a tongue-in-cheek unit of beauty based on the famous reference to Helen of Troy in Christopher Marlowe’s Doctor Faustus: “Was this the face that launch’d a thousand ships / And burnt the topless towers of Ilium?” Therefore the millihelen is technically defined as “a unit of measure of pulchritude, corresponding to the amount of beauty required to launch one ship.”
br /> From a mathematical point of view, the Professor’s most interesting invention is the Mind-switcher, which appears in “The Prisoner of Benda” (2010). As the name suggests, the machine takes two sentient beings and swaps their minds, allowing them to inhabit each other’s bodies. The mathematics is not in the mind-switching per se, but rather is required to help unravel the mess caused by such mental juggling. Before discussing the nature of this mental arithmetic, let us explore the episode in detail and understand exactly how the Mind-switcher works.
“The Prisoner of Benda” begins with an opening caption that reads, “What happens in Cygnus X-1 stays in Cygnus X-1,” echoing the well-known maxim “What happens in Vegas stays in Vegas.” In the case of Cygnus X-1, this is literally true, because it is the name of a black hole in the constellation Cygnus, and whatever happens in a black hole is forever condemned to remain in the black hole. The writers probably picked Cygnus X-1 because it is considered a glamorous black hole, thanks to being the subject of a famous wager. The mathematician and cosmologist Stephen Hawking had initially doubted that the object in question was indeed a black hole, so he placed a bet with his colleague Kip Thorne. When careful observations proved that he was wrong, Hawking had to buy Thorne a one-year subscription to Penthouse magazine.
The episode’s title is a pun based on the Victorian novel The Prisoner of Zenda, by Anthony Hope, in which King Rudolf of Ruritania (a fictional country) is drugged and kidnapped by his evil brother prior to his coronation. In order to save the crown from falling into the wrong hands, Rudolf’s English cousin exploits his resemblance to the king and adopts his identity. In short, the plot of The Prisoner of Zenda revolves around someone taking on a new identity, which is also the central theme of “The Prisoner of Benda.”