Of course, the success of this franchise should not be viewed in isolation, but as just a part of the larger vampire pop culture renaissance.  HBO’s True Blood, also based on a book series involving a girl who knocks boots with the undead, is going strong into its third season this summer, and the CW’s Vampire Diaries will return for a second season this fall.  And just when I thought the market for vampire-themed programming had become saturated, ABC premiered its own summer show featuring blood suckers called The Gates.  Clearly there is a trend here, with the ever-growing popularity of the vampire at its center.  No doubt Eddie Murphy is rolling in his undead grave for not releasing Vampire in Brooklyn 10 years later.

While there are many words that could be used to describe these shows and movies that place supernatural love triangles at their center, “realistic” is not one of them.  Nevertheless, there are a handful of people who have taken a critical eye to the vampire phenomenon and have used mathematical models to gain insight into how the populations of such creatures might behave in real life.  Just like the fights between Team Edward and Team Jacob, however, the debate over whether vampires could actually exist rages on.

Not long ago, an article went around the web purporting that a team of physicists had proven that vampires could not exist.  The physicists, Costas Efthimiou and Sohang Gandhi, posted a paper to the arXiv in which they purport to use physics to dispel pop culture portrayals of ghosts and zombies, in addition to vampires.  Their argument for debunking vampires rests on the following assumptions:

1. When a vampire bites a human, that human becomes a vampire (we will return to this assumption later).
2. Vampires need to feed on a human once every month (a conservative estimate when compared to what popular culture would have us believe).
3. Assume the first vampire came into existence in 1600, when the human population was roughly 500 million.
4. Ignore human mortality rates due to other factors, and ignore human birth rates as well.

With these hypotheses, they show that vampires would wipe out humanity in just 2 1/2 years.  In fact, no matter the size of the initial human population, their model will lead to humanity’s extinction in a short amount of time.  This is true even if we assume a more conservative estimate on the length of time vampires can go between feedings.

The reason is simple.  Using their model, the first vampire will feed after one month, creating a new vampire (by assumption 1).  After 2 months, those 2 vampires will each feed, giving us a total of 4 vampires.  After 3 months, those 4 vampires will feed, giving us a total of 8 vampires.  The pattern continues – after n months, the vampire population will be 2ⁿ. In other words, the population of vampires will grow exponentially.  Moreover, because of the assumption on the birth and mortality rate of mankind, we see that as the population of vampires grows exponentially, so too must the population of humans shrink exponentially.  This means that at some point (sooner than you might think), humans would be wiped out.

The careful reader, however, will note a number of problems with this analysis.  For one, ignoring the birth rate of humans means that the model’s date of extinction is premature.  However, Efthimiou and Gandhi point out that even if we include the birth rate, that rate would not be high enough to counteract the explosion in the vampire population.  A more serious flaw, however, is in not considering the mortality rate of the vampires themselves.  After all, once people realize there are vampires in their midst, wouldn’t they fight back, or at least defend themselves so that not all of the vampires could feed?  Assuming that every vampire would be able to feed whenever necessary seems unrealistic.

What’s more, assuming that vampires can only satisfy themselves with human blood, it seems unreasonable to assume that vampires would feast so carelessly, without regard to the diminishing supply of their food.  If vampires killed all humans, they in turn would die (again), and so it seems reasonable to expect that vampires would apply a better strategy, one in which they kept the human species afloat so that they could themselves continue to exist.  Just ask Ethan Hawke.

In a brief article for Math Horizons, mathematician Dino Sejdinovic addresses these issues and highlights an article from 1982 that modeled the vampire outbreak more realistically, by including the human birth rate and vampire mortality rate.  In doing so, the mathematics becomes less fit for a general audience, but it also gives us a more interesting picture – regardless of the collective desire for human blood, vampires can act in a way that the ratio of vampires to humans reaches an eventual equilibrium.  In other words, it doesn’t seem right to throw out the idea of vampires based on purely mathematical arguments.

Interestingly, though, all of these analyses rest upon assumption (1), which states that humans always become vampires once bitten.  In the modern incarnation of these creatures, however, this assumption no longer appears to be valid.  For example, in both True Blood and The Vampire Diaries, the process of turning into a vampire requires consent (I guess it’s more romantic that way); not only must the vampire drink the human’s blood, but the human must also drink the vampire’s blood.  In this case, it is possible for vampires to satiate themselves without killing humans (provided the vampires can show enough restraint) or increasing their own population.

There also appear to be rules governing population control in vampire communities.  For example, in an episode of True Blood, one vampire is tasked with creating a new vampire as penance for murdering one of his own kind.  Are such rules keeping the population stable widespread?  How might such rules, in conjunction with a weakening of assumption (1), alter the vampires’ optimal strategy?  I will leave it to the curious reader to discover the answer.

Big Poppa, or Big Daddy?

For purposes of this discussion, however, I will try not to get pulled into the ocean of Nicolas Cage adulation, and will instead try to focus on the mathematical content in his 2009 film, Knowing.  This turns out to be a fairly simple task, as you will soon see.

In Knowing, Cage stars as John Koestler, a professor of astrophysics at MIT (already, the potential for mathematics is promising).  He is a widower with a young son, named Caleb, who goes to a nearby elementary school.  Near the beginning of the film, Caleb’s school holds a celebration in honor of its fiftieth birthday, and as part of the festivities they unearth a time capsule that was buried 50 years prior.  The time capsule is filled with drawings from former students, but Caleb gets a little short-changed: instead of receiving paper with a drawing, his paper just has a long sequence of numbers.

As might be expected by the presence of a long list of numbers, Knowing is less concerned with math than it is with numerology.  Don’t let John Koestler’s profession fool you – aside from one scene in the classroom where you can spot Maxwell’s equations if you know where to look, there is very little in the way of mathematics in this film.  But the more important question to ask is whether this is a bad thing.

While the lack of mathematics may make the trailer seem a bit disingenuous, Knowing is aptly named in the sense that it knows the film’s conceit has nothing to do with mathematics.  Unlike a film such as Pi, which attempts to pass off numerological voodoo as actual mathematics, Knowing is very much aware that Koestler’s analysis of the numbers from the time capsule does not constitute research in mathematics.  At one point his colleague even points this out to him, saying “Whoa. Just step back. Have another look at it! Systems that find meaning in numbers are a dime a dozen. Why? Because people see what they want to see.”

Another difference between this film and Pi is that here, the pattern of the numbers is discovered fairly early on, and it becomes quite predictable.  There is less mystery in the numbers themselves than there is in how a schoolgirl 50 years ago came to write the numbers down.  Thankfully, the answer to the latter question is resolved as well (spoiler alert: it all has to do with angel aliens).

Does this film pertain to mathematics?  Not really.  But nor does it aspire to be.  In that sense, then, I would certainly consider this a better film than Pi.  Should they remake Pi with Nicolas Cage, however, all bets are off.

Last month, Jennie Yabroff wrote an article for Newsweek discussing the new film Precious. I haven’t seen the film, but this trailer makes a fairly strong impression:

The theme of finding redemption through writing is certainly not new to this genre of film, as Yabroff points out. Films such as Dangerous Minds and Freedom Writers have explored this territory before, although perhaps with less success than Precious. However, Yabroff wonders if all this time spent journaling wouldn’t be better spent learning some mathematics.

In her article, Yabroff writes:

The idea that every underprivileged young adult harbors the soul of a Rimbaud is a favorite trope of popular culture. We don’t expect our bankers and lawyers to be secret Baudelaires, but we eagerly accept the idea that every poor person has a “story,” and just needs the right teacher or mentor to give it voice…

…But, contrary to what Hollywood would have us believe, the world does not reward self-expression as readily or consistently as it rewards a good head for numbers. It’s hard for any writer to support herself writing. Precious’s teacher should have known that, and given her a calculator along with that journal.

Stand and Deliver is one film that shows how mathematics can improve the lives of inner city youths, but as a general rule it does seem much more likely that a film will attempt to highlight the potential of students through writing rather than through math. Part of this bias is to be expected, since, as Yabroff points out, movies are written by writers and not mathematicians. Also, given Precious’s illiteracy, it would make more sense to build up her fundamental reading and writing skills before asking her to pick up a math book. Even so, should we be concerned that nearly every film that shows inner city youth turning their lives around does it by emphasizing words over equations?

One may be tempted to argue that it’s just harder to write a film that portrays math honestly and in a compelling way, but I think that’s only true if you’re a lazy writer or you don’t know anything about math. Especially at the high school level, the fundamental concepts could be explained in a way that’s understandable to general audience, and, dare I say it, even interesting. The end result may not seem to be as glamorous if one’s education is focused on math – given the choice, I’m guessing that kids would rather be like Kanye instead of Urkel – but this just means we need to do a better job showing kids why mathematical skills are so attractive.

I for one would love to see more movies like Stand and Deliver, especially ones that follow through and show what can be done with a background in science and math. Who knows when that will happen, though. I guess we need a character with the technical know-how of Steve Urkel, but the charm and sophistication of is suave doppelganger, Stephan Urquelle. I am confident that character is out there, and am hopeful that his (or her) time will soon come.

Sorry Steve – lose the suspenders, and then we can talk.

Recently, I decided to watch this film for a third time, to see how this film compares with other films that involve mathematics. Lou Diamond Phillips was as charming as ever – but how did the math stack up?

For the uninitiated, Stand and Deliver aims to tell the true story of Jaime Escalante, a man who gained a fair amount of press in the 1980′s for developing an extremely successful advanced placement math program at an inner city school in Los Angeles. The film tells the story of the first batch of kids to study Calculus under Escalante’s tutelage, and aims to show that regardless of your background, an understanding of mathematics is not beyond your reach.

Most of the time when I discuss the representation of math in films, there are two main things to consider: the portrayal of mathematicians, and the portrayal of mathematics itself. For this film, Mr. Escalante is not a mathematician, however – he is instead a very good math teacher. Nevertheless, being a good math teacher means he must be good at math, and whenever someone who is good at math is presented on film, there is a danger that the character will have certain stereotypical attributes.

Thankfully, this doesn’t seem to be the case here. Let’s take a closer look at some stereotypes of mathematics.

- People who are good at math are socially awkward.

Jaime Escalante is many things in this film, but socially awkward is not one of them. He’s a charismatic dude with a comb over to match. Sure, some of the things he says may not be entirely appropriate for the classroom, but everyone seems to enjoy it. Plus, he has a stable home life with a loving wife and a son – it’s not often that people who are good at math are shown in such drama free households. +1.

- To be good at math, you must be insane.

This movie shows all types of students, from the nerdy girls who work at their dad’s restaurant, to the wannabe gangsters with no aspirations for higher education. Many of the students come from less than ideal family situations. Nevertheless, there is one thing that binds them all together: their ability to learn math.

While some students are stronger than others, there isn’t one among them who is completely lost. They all learn Calculus, despite the skepticism that surrounds them. At one point Mr. Escalante opines that “students will rise to the level of expectation,” and in this case, he is correct. You don’t need to be crazy to be good at math. Having a good teacher, however, certainly helps. +1.

Real Jaime Escalante versus movie Jaime Escalante.

- Mathematics is inherently difficult and complicated, and only gifted people have a hope of doing well.

This is arguably the most harmful stereotype about mathematics. While it’s certainly true that math is difficult, and that there are those who seem to have an innate mathematical ability, it is certainly not the case that every professional mathematician (or those who use math in a technical career) are math savants. More often, they are simply people who had a few good teachers and were motivated to really understand mathematics.

Usually a film’s mathematics perspective is weighted heavily towards the savant end of the scale, which only reinforces stereotypes about people who study mathematics. Thankfully, this film really emphases the latter standard – that even if you aren’t the most naturally gifted when it comes to math, you can still succeed with enough hard work. All the students in the film work extremely hard, even the ones who may be better at math than the others. And aside from a bit involving Andy Garcia who plays a total tool, the students are all rewarded for their hard work, not only with good grades, but with a deeper understanding of math, and greater confidence about their abilities as students. As Mr. Escalante says, “Calculus is not made to be easy – it already is.” +1.

This film is very different from most films that involve math. Math isn’t presented as some mystical oracle that can only be deciphered by the borderline insane. Instead, it is presented as a difficult subject, but one that can be mastered with dedication and practice. It’s no wonder, then, that this film has secured such an enduring spot in the hearts of math teachers nationwide.

For those of you who may not find the film appealing, there’s always the counterpoint offered by South Park. Mr. Cartmanez may not have the heart of Mr. Escalante, but at least he has the comb over. Their teaching styles couldn’t be more different, and yet in their own way, both are successful. Somehow, though, I don’t think math teachers will find the story of Mr. Cartmanez as appropriate for their students.

The story of Pi centers on a mathematician named Max Cohen, a self professed number theorist – although he never specifies what qualifies him for this title – who spends his days analyzing the stock market and wiping the blood off of his upper lip (I know what you’re thinking, and no, he’s not a cage fighter – that would’ve made the film way better). As he comes closer to “unlocking the secrets” of the stock market (whatever that means), several interested parties begin to come out of the woodwork, all with their own self-interest at heart.

From a math perspective, how does this film stack up? Unfortunately, the answer is poorly. Those wishing to learn some math from their pop culture would be better off with an episode of Sesame Street, or perhaps some School House Rock. In fact, pretty much any math pop culture reference you can think of would probably fare better.

- Mathematicians are really good at calculating things in their heads.

One of our first introductions to Max comes early in the film, when he is leaving his apartment and a young Asian girl approaches him with a calculator. She proceeds to ask him to compute products and quotients of large numbers in his head – things like 421 x 121. Of course, since Max is a mathematician, he has no trouble computing these products. He does it just as quickly as she can type the numbers into her calculator!

As I’ve said before, this isn’t at all a realistic depiction of mathematicians. While there are certainly computational savants amongst us, it is just as common, if not more so, to encounter a mathematician who will willingly admit that he or she is no good at computation. Many mathematicians even take pride in such assertions. Mathematics is as much about doing multiplication in your head as cooking is about opening jars with your bare hands. -1.

- To be successful in math, you have to complete your Ph.D. at an extremely young age.

Max Cohen published his first paper at age 16, and completed his Ph.D. by the time he was 20. While this is certainly not unheard of in academia, the idea that you must be young to be successful is one that is especially pervasive in mathematics.

The problem with this stereotype is that it tends to discourage people from studying mathematics if they are firm in their beliefs that math is solely a young man’s game. In pop culture, most people who are good at math are portrayed as having completed their degrees at a very young age, but in reality these people are the exceptions, not the rules. In mathematics, as Aaliyah will tell you, it is becoming more and more common that age ain’t nothing but a number. -1.

Don’t subscribe to this propaganda: mathematicians enjoy a good samosa just as much as everyone else.

- People who are good at math are socially awkward.

In an early scene, Max is busy looking at numbers on his computer screen (because he’s a number theorist, remember?!) when someone comes knocking on his door. On the other side is his attractive neighbor, who not only sports a British accent, but also has brought him samosas.

Any normal person would welcome such an act with kindness and gratitude. Of course, Max is not a normal person – he is a mathematician. This must explain why he is rude to this woman, both in this scene and later on in the film, despite the fact that she is not only kind to him, but is also cute. That she puts up with his abuse is simply a testament to the power of seduction that comes with studying mathematics, whether intentional or not. -1.

- Number Theory is synonymous with numerology.

Throughout the film, Max stares at numbers. He watches stock prices fluctuate. He prints out random strings of integers from his super old computer and stares at them for long periods of time. Sometimes he even draws circles on the newspaper and shows that he is a genius at math because he can recall the formulas for the circle’s area and circumference.

One notable omission in all of this is that at no point does Max do anything even remotely resembling number theory. If anything, Max’s research would more aptly fall into the realm of financial mathematics, with maybe a splash of ergodic theory thrown into the mix.

At one point Max is so fixated on trying to find a pattern in his work that his mentor, Sol Robeson, tells him, “As soon as you discard scientific rigor, you’re no longer a mathematician; you’re a numerologist.” Coming from a film that depicts nothing BUT numerology, this is somewhat of a surprising statement. -1.

- To be good at math, you must be insane.

Let’s be honest – Max is a total whack job. Not only does he hate it when people deliver him fresh samosas, but he’s obsessive, paranoid (perhaps justifiably so), and irritable. He fantasizes about poking his brains, and at one point he shaves his head and begins drawing on his skull with indelible marker, or so we are led to believe. At least the connotations with phrenology pair nicely with the numerological gobbledygook that permeates the rest of the film.

As I’ve said before (and will no doubt say again), you can be good at math without being crazy. Being crazy is just the frosting on the cake. I kid, I kid. -1.

Wow, look at all the math in that noggin.

- The number π provides an active area of research for mathematicians.

Max’s aforementioned mentor guides Max throughout the film, and often during their discussions, Sol hearkens back to his own youth, and the amazing math he did while researching π.

Sadly, Sol is a century or two late if he expects us to believe he could make a career studying a single number. The whole idea is absurd. I have not heard of a single mathematician who has made a lucrative career studying π, and I think you’d be just as hard pressed to find one. Saying that a mathematician’s research concerns π would be like saying an English literature professor’s research concerns the word “banana.”

The only reasonable conclusion one can draw is that Sol is a fraud. This is further evidenced by the fact that he doesn’t know what density is (weight over volume? Come on, dude). Perhaps Sol was only hanging out with Max to bask in his numerological genius. No doubt his time would’ve been better spent elsewhere. -1.

In summary, this movie scores a whopping -6. This is a pretty poor showing. Sorry, Mr. Aronofsky – if it’s any consolation, I do want to see your new film. Hopefully you are more of an authority on washed up wrestlers than you are on mathematicians.