With a payroll roughly a third the size of the Yankees’, Beane understood that the playing field was not a level one from an economic standpoint.  What’s more, at the end of the 2001 season, three of the A’s star players left Oakland for bigger paychecks.  To fill the void, the film (and book) show how Beane took a more analytic approach, and used statistical analysis to uncover players who were undervalued and could be purchased for much less than they were worth.  Beane, together with Paul DePodesta (Peter Brand in the film, and played by Jonah Hill), used a sabermetric approach to lead the A’s to a league-leading 103 wins for the season. While their first-place ranking for number of wins that year was shared with the Yankees, they spent much less per win than their New York counterparts (the A’s spent the least per win, while the Yankees spent the third most).  Here’s a table comparing the teams; the payroll numbers are taken from here, and differ slightly from the numbers that appear in the book.

Team	Wins	Losses	Payroll	Cost Per Win (millions)
Oakland Athletics	103	59	$40,004,167	$0.388
Minnesota Twins	94	67	$40,225,000	$0.428
Montreal Expos	83	79	$38,670,500	$0.466
Florida Marlins	79	83	$41,979,917	$0.531
Cincinnati Reds	78	84	$45,050,390	$0.578
Pittsburgh Pirates	72	89	$42,323,599	$0.588
Los Angeles Angels	99	63	$61,721,667	$0.624
Tampa Bay Rays	55	106	$34,380,000	$0.625
San Diego Padres	66	96	$41,425,000	$0.628
Chicago White Sox	81	81	$57,052,833	$0.704
Philadelphia Phillies	80	81	$57,954,999	$0.724
Houston Astros	84	78	$63,448,417	$0.755
Kansas City Royals	62	100	$47,257,000	$0.762
St. Louis Cardinals	97	65	$74,660,875	$0.770
Colorado Rockies	73	89	$56,851,043	$0.779
San Francisco Giants	95	66	$78,299,835	$0.824
Seattle Mariners	93	69	$80,282,668	$0.863
Milwaukee Brewers	56	106	$50,287,833	$0.898
Baltimore Orioles	67	95	$60,493,487	$0.903
Atlanta Braves	101	59	$93,470,367	$0.925
Toronto Blue Jays	78	84	$76,864,333	$0.985
Detroit Tigers	55	106	$55,048,000	$1.001
Los Angeles Dodgers	92	70	$94,850,953	$1.031
Arizona Diamondbacks	98	64	$102,819,999	$1.049
Cleveland Indians	74	88	$78,909,449	$1.066
Chicago Cubs	67	95	$75,690,833	$1.130
Boston Red Sox	93	69	$108,366,060	$1.165
New York Yankees	103	58	$125,928,583	$1.223
New York Mets	75	86	$94,633,593	$1.262
Texas Rangers	72	90	$105,726,122	$1.468

Their new approach threw out many pieces of conventional baseball wisdom: stealing bases and bunting were strict no-no’s, for example.  Naturally, these changes brought about some tension, and it’s this tension that makes for the dramatic thrust of the film.  In particular, mathematics takes a backseat, though there are some little cameos for those who are paying attention.

The most significant piece of mathematics making an appearance in the film is the Pythagorean Expectation, a formula discovered by Bill James that estimates a team’s win percentage in terms of its runs scored and runs allowed.  More specifically, the formula asserts that a team’s win percentage is approximately equal to

For example, the 2002 A’s scored a total of 800 runs, and allowed a total of 654 runs, for a Pythagorean Expectation of

(relevant stats can be found here). This compares to the team’s actual win percentage of 103/162, which is around 0.636.

In the film, Peter Brand applies this formula in order to estimate the number of runs the team needs to score, along with the maximum number of runs it can allow, in order to secure a playoff spot.  In one scene, he tells Billy Beane that he thinks the A’s will need to win at least 99 games to guarantee a playoff spot.  In a 162 game season, this equates to a win percentage of around 0.611.  In order to ensure that the Pythagorean Expectation is at least this large, we set

With a little algebra, this is the same as

Brand then informs Beane that in order for this to happen, the team needs to score at least 814 runs, and can allow no more than 645 runs.  This gives a runs allowed to runs scored ratio of 645/814, or around 0.793 < 0.798 (though, if I were being anal, I would point out that with 814 runs scored, the team could allow as many as 649 runs and still have a runs scored to runs allowed to runs scored ratio that is less than 0.798).

While the math formulas on display in the film are accurate, I would be remiss if I did not briefly discuss Hill’s portrayal of Peter Brand.  Overall, Hill does a good job; though Brand is clearly a nerd, Hill’s portrayal usually avoids caricature.

Like every other film featuring characters who are good at math, though, Moneyball can’t help itself from including a scene where we see how good Brand is at math because he can do mental calculations quickly.  This particular scene takes place when Brand is sitting in his first meeting with Beane and the rest of the baseball scouts, and though it serves to highlight the tension that exists between Brand’s new school of thought and more traditional baseball thinking, I think the scene could have been just as effective without the clichéd math exercise.

Also, in the interest of full disclosure, I should point out that there are some who feel the story told in Moneyball (both the film and the book) is an exaggeration.  More specifically, as this Slate article discusses, many people believe that the reason for the A’s success during the early aughts had less to do with sabermetrics, and more to do with the fact that they had awesome pitchers in Tim Hudson, Mark Mulder, and Barry Zito, none of whom feature prominently in the book or film.  While I don’t feel knowledgeable enough to weigh in decisively on this issue, the role of the defense certainly appears to be underrepresented here.

To try and convince you of this, recall that the A’s made it to the playoffs in four consecutive years, from 2000-2003.  Here is some data on how many runs they scored and how many runs they allowed during each of those years, and in 2004, when they did not make the playoffs:

Observe that especially from 2001-2003, while the A’s offense declined, their defense remained consistent in allowing relatively few runs.  Of course, this should not be viewed in a vacuum, but rather in relation to how baseball as a whole performed.  Therefore, it is better to consider not runs scored and runs allowed, but runs scored and runs allowed as a proportion of runs scored and runs allowed in the American League.  With this slight adjustment, we get the following picture:

Note in the above that a proportion of 1 means that the A’s were performing at an average rate, while a proportion greater than 1 indicates above-average performance, and a proportion less than 1 indicates below-average performance.  As we can see from the data, in 2001-2003, the A’s defense was allowing runs at a rate well below the average; in other words, the defense was relatively strong.  On the other hand, during the same period, the offense consistently weakened year-over-year, so that the number of runs the A’s scored was below the league average in 2003-2004.  In particular, during the 2002 season profiled in Moneyball, the number of runs scored took a sharp downturn relative to the league average, while the number of runs allowed still remained well below average.  This indicates, to me at least, that the role of the defense was certainly an important factor in the A’s playoff runs during the 2002 and 2003 seasons.  Note also that in the 2004 season the number of runs allowed rose sharply relative to the league average; without a corresponding uptick in runs scored, the A’s didn’t make it to the playoffs.

Nevertheless, I don’t think the issue is binary; excellent pitching and a sabermetric approach probably combined to help the A’s.  Even though Moneyball only explores one of these issues, it’s still a film well worth seeing.  If you’re no fan of mathematics, don’t worry, there isn’t much on display.  And if you’re no fan of baseball, surprisingly, I think you might enjoy the movie anyway.

As a shining beacon of what is hip, MTV has a responsibility during its movie awards to highlight the most popular films of the year. This is in stark contrast to the priorities of higher brow award shows such as the Oscars, for which artistic achievement is placed on the highest pedestal. This is not to say that these two goals need be mutually exclusive; indeed, since the first MTV Movie Awards was broadcast in 1992, the “Best Film” has agreed with the Academy Award winning best film three times (1997′s Titanic, 2000′s Gladiator, and 2003′s Lord of the Rings: The Return of the King). Even so, a quick glance at the nominated films from these two awards shows each year reveals a fairly small overlap, in general. But if MTV’s goal is to prop up films from an angle focused more on pop culture, it is natural to ask how good of a job they do.

Delicious metal popcorn. (Photo: Jason McDonald/MTV)

This question begets another one: how can we best measure a film’s popularity? My first thought was to consider the rankings on IMDB. There, users can give any film a score from 1 to 10; as a prime example of a range voting system, this seemed like a good place to measure the public’s reception of a film.

The results were mixed. With this metric, comparing the 20 years that both awards shows have been around, the MTV best film scored higher than the Oscar winning best film only 5 times. Oscar trumped 12 times, and the two awards tied three times. The trend is also worth mentioning – after 5 consecutive years of beating or tying the Oscars in IMDB score from 1999-2003, the Oscar winning film has bested the MTV winning film ever since. The disparity has become even larger in recent years (I call this the Twilight effect, as the Twilight films have won best film at the MTV awards for three years running). Here’s a graph of the scores over time (the list of MTV Best film winners is here; Oscar winners can be found here):

While you may argue there isn’t enough data here to draw much of a strong conclusion, the recent trend is fairly convincing. By this metric, it seems like Oscar winning films, at least over the past few years, seem to have been more popular.

Rather than looking at only the winner, though, you might expect to get a better sense of the popularity of films on display by looking at all nominated films, rather than just the winners. If we take the average IMDB score for all nominated best films at each awards show, rather than just the winning film, we get the following picture:

It’s come close, but the average IMDB score of MTV nominated films has never been greater than the average IMDB score for Oscar nominated films. We see the Twilight effect among the averages as well, though it was dampened somewhat this year due to the inclusion of critical darlings Inception, The Social Network, and Black Swan on the MTV nominee list.

“Hold up,” you may be thinking to yourself, “this is all a bunch of hooey.” You may think that IMDB scores are not a very good measure of a film’s popularity. It may be quite likely that IMDB scores are biased towards those same features of a film that make it more likely for Oscar consideration. Perhaps the type of person who goes onto the website to rate films is more likely to be somewhat of a connoisseur, and therefore the tastes reflected by the IMDB community are more likely to reflect the tastes of the Oscars. At the very least, it seems likely that teenage girls are not voting on the website in droves; how else can one explain the Twilight series’ limp average of only 4.9?

What else can we use to measure a film’s popularity? Well, to return to teenage girls, they don’t show their support for the Twilight series by rating it highly on the internet. They show their support by going out and seeing the movie multiple times. So perhaps we should look at box office receipts rather than IMDB score (and, of course, by picking sides in the bitter feud between Edward and Jacob). What sort of picture do we see in this case?

If we only consider the winning film from each awards show, the data looks like this (I’m only considering US box office numbers here):

(I’ve cut the graph so that you can’t see the leap in 1997, when Titanic took top prize at both shows.) Things look a little more erratic, but if you look closely, you’ll see that the MTV award winner has taken in more money than its Oscar winning counterpart 14 times out of 20. The Oscar has favored the larger cash cow only three times.

As with the IMDB rankings, we can try to smooth things out by looking at the average box office returns among nominees, rather than just returns for the winner. This yields the following graph:

These numbers aren’t adjusted for inflation, which may explain in part the growth trend (2009 numbers are bumped up because of Avatar, as well). I’m less interested in the actual numbers than I am the difference between the two graphs. And here we see, in contrast to the IMDB case, that the roles of the two awards shows have flipped. While the average IMDB score of Oscar nominees has always been higher than the average IMDB score of MTV nominees, the average box office return of MTV nominees has always been higher than the average box office return of Oscar nominees.

To return to the original question: do the MTV movie awards highlight more popular films than the Oscars? Well, it depends on how you define “popular.” If popular means highly rated, the claim is somewhat dubious. But if popularity is measured by the almighty dollar, then this seems like a fair conclusion to draw. Whatever your line of thinking, though, I’m fairly confident that current trends will continue, at least until the last of the Twilight films has exited theaters.

As you can see, the story centers around a mathematician who claims to have discovered an integer between 3 and 4.  Forgetting the mathematical particulars for a moment, the source material worries me, mostly because the mathematician featured in the story has been hospitalized following a nervous breakdown brought on by his work (if you have a chance, I’d encourage you to read the story for yourself – the whole thing comes in at under 2,000 words).  Do we really need another film exploring the psyche of the brilliant but frail mathematical mind?

Regarding the mathematical content itself, that frustrates me a little bit in the same way that a film like Pi frustrates me.  Often times, the lack of mathematical understanding is used to create a false illusion of mystery.  There isn’t really anything interesting about the question “What if there was an integer between 3 and 4?” in the same way that there isn’t really anything interesting about the question “What if bananas tasted like apples?”  It’s not as if there is a lack of open mathematical questions that even the layperson could understand (think of the twin prime conjecture, for example, or Goldbach’s conjecture) and which might supply the necessary mystique without sacrificing mathematical authenticity.  Of course I realize this is not a problem unique to mathematics – any specialist will likely find fault with a film focusing on that specialty.  Some faults are more apparent than others, however, and the potential for a misstep when working with this source material seems quite large.

Hopefully this film will eschew the use of the slinky.

Nevertheless, kudos to the filmmakers for exceeding their fundraising goal.  To have a student film budget in excess of $10,000 is quite a feat, and the publicity from their campaign can’t hurt either.  I look forward to seeing their final product, and I hope they can avoid the cliches that so frequently arise when telling these types of stories.  Only time will tell.  I will admit, though, that some of the production art is pretty slick.

(Hat tip to Meg for sending a link to this project my way.)

Consider, for example, the following passage:

How much math do you really need in everyday life? Ask yourself that — and also the next 10 people you meet, say, your plumber, your lawyer, your grocer, your mechanic, your physician or even a math teacher.

Unlike literature, history, politics and music, math has little relevance to everyday life. That courses such as “Quantitative Reasoning” improve critical thinking is an unsubstantiated myth. All the mathematics one needs in real life can be learned in early years without much fuss. Most adults have no contact with math at work, nor do they curl up with an algebra book for relaxation.

Those who do love math and science have been doing very well. Our graduate schools are the best in the world. This “nation at risk” has produced about 140 Nobel laureates since 1983 (about as many as before 1983).

Let’s address this passage point by point.  First, while it’s true that many people don’t use math in their everyday life, this is not a criticism that is unique to mathematics.  I’d contend that most people in the list he mentions don’t use U.S. history in their everyday life (except for perhaps the lawyer), nor would most of them use English literature or biology.  Does it therefore follow that none of these things should be taught in schools, either?  Ramanathan seems to be suggesting that the purpose of education is to impart only the skills that will be needed for the vast majority of the student population when it reaches adulthood.  This is fairly ridiculous, though, both because the range of human interest is so vast that what would comprise such a necessary intersection would seem to be not nearly deep enough (as he himself says, the math component could be learned in the “early years”), and also because it’s not entirely clear what skills children learn as students will turn out to be the most important to them in their future careers.  Perhaps Ramanathan is an advocate for having students focus on an area of interest earlier in their academic life, which might explain this position, but it’s never made clear if he believes this to be a sound alternative.

I'll concede that this guy probably doesn't need to know much math. I doubt whether the same can be said, though, for the people who create his games.

What’s more, students sometimes fail to realize how important mathematics is to their future career plans until they’ve already written themselves off as hopeless students of the subject.  By shutting academic doors prematurely, students are also shutting the doors to potential career opportunities.  The fact that so many students hate math I think speaks more to the way math is taught than the fact that it is taught at all.

And while we’re on the subject of career opportunities, in a time when job reports are at the forefront of the news, we should be encouraging students to go into technical fields, not telling them that since some of them might not use math, there should be less math in schools.  This seems to me to be a fairly nihilistic viewpoint, and in the interest of neutrality, I’d say the same thing about a professor in a different discipline advocating a similar platform.  As a graduate student in mathematics, I can’t remember the last time I directly applied knowledge I gained in a history class, an English class, or a chemistry class.  I do, however, see the value in my having taken such classes, even though my career path probably won’t benefit from that knowledge in any way.

This brings me to the next point: I don’t see why it’s at all obvious that mathematics has any less relevance to everyday life than literature, history, politics, or music.  The relevance of any of these disciplines to one’s everyday life depends highly on the life one is living, and while it may be true that on average mathematics appears less in popular discourse than these other subjects, it doesn’t follow that it is therefore less worth of study by a general population.  By way of analogy, just because news coverage may spend more time talking about Lindsey Lohan than the American presence in Afghanistan, does it follow that Lindsey Lohan is inherently worthier of investigation than the American presence in Afghanistan?  (Note that I don’t mean to equate Lindsey Lohan with literature, history, or music…politics, maybe.) The only unsubstantiated myth worse than the one that “courses such as ‘Quantitative Reasoning’ improve critical thinking” is the one that “[u]nlike literature, history, politics and music, math has little relevance to everyday life.”

Lohan's portrayal of a mathlete in Mean Girls, unfortunately, shatters my analogy.

Finally, let me address the final point in the quote.  Ramanathan remarks that American graduate schools in mathematics are the best in the world, but fails to mention what they lack: American graduate students.  The best in the world these schools may be, but that’s because the students are the best students in the world, not because they are Americans who have come up through the American education system.

Also, the statement that students who love math and science excel in it isn’t supported with any evidence, and it’s not at all clear that it’s true.  In fact, I’m sure there are a number of students in this country who enjoyed math but didn’t stick with it because they had an insufficient support system in their education.  It’s simply not true that a love of math is a universally good enough support system for a student who wants to study the discipline.  What good does it do to say “Among students who love a certain discipline, they will learn it well enough because they love it, and for everyone else, it’s not important anyway”?  If that is one’s philosophy, why have education at all?

The Nobel Laureate claim is also not completely relevant, since there is no Nobel prize in mathematics, and there are Nobel Laureates for disciplines that have nothing to do with mathematics.  If you want to measure a country’s math aptitude by big prizes (which itself seems like a rather dubious metric), a more natural thing to consider would be the Fields medal.  The difficulty here is that the sample size is smaller, but in the interest of comparison, here are some things worth noting: Since 1983, there have been 3 recipients of the Fields medal from America.  By contrast, from 1962-1982, the number of American Fields medalists was three times this number.  Moreover, in the last 20 years, only 2 Americans have won the fields medal, as compared to six French mathematicians and six Russian mathematicians.

The world is moving towards a state of more complexity, not less, and this will require a stronger mathematical background on the part of the world’s population.  Rather than burying our head in the sand, as Professor Ramanathan seems to be advocating, we should be seriously considering how mathematics can best be taught to a 21st century student body.  What good does it do to pander to a general population that already hates mathematics (due in no small part to the way they were taught, I’m sure)?

In the future, will the Washington Post print more insightful musings on the current state of math education in this country?  I certainly hope so.

More successful than any of the three previous films I listed (from a box office standpoint, mind you, not necessarily a mathematical one), Ron Howard’s 2001 biopic of mathematician John Nash won four Academy Awards, including the coveted Best Picture and Best Directing trophies.  I saw the film when it came out, but didn’t remember much, except for being underwhelmed, and so it was with low expectations that I decided to rewatch the film this weekend.  Should you need a refresher, please take a look at the following trailer.

If the trailer strikes you as unnecessarily saccharine, then you’re in luck, because it captures the tone of the film fairly accurately.  But tone isn’t primarily what I’m interested in here; as you might expect, I am more interested in how this film portrays mathematics and mathematicians.

Let’s address common misconceptions individually.

- People who are good at math are socially awkward.

If the only mathematician in this film was John Nash, then the film certainly would be perpetuating this misconception.  Even though the film chronicles Nash’s relationship with and marriage to an actual female (!), I don’t think anyone would try to argue that Crowe’s portrayal of Nash is anything but socially awkward.  Consider, for example, this example of his attempt at a pick up line from a scene early in the film: “I don’t exactly know what I am required to say in order for you to have intercourse with me.  But could we assume that I said all that?  I mean, essentially we are talking about fluid exchange, right?  So could we go just straight to the sex?”  At the very least, perhaps Nash could have benefited from a dating coach.

On the other hand, Nash’s idiosyncrasies are presented as the exception, rather than the rule.  We meet a number of other mathematicians in this film, mostly Nash’s colleagues from graduate school, and they all seem normal not only in relation to Nash, but in absolute terms.  None of these characters is explored in much depth, but their presence does offer a counterpoint to Nash’s strange behavior, and for that I am grateful.  In the end, therefore, the film both seems to perpetuate this stereotype and go against it.  After all, it shows people who are socially well-adjusted and good at math; John Nash just happens to be the latter without being the former.

- Mathematicians are single-minded of purpose, and have no interest in anything besides mathematics.

Again, although Crowe’s portrayal of John Nash somewhat exemplifies this misconception, the supporting characters provide a contrasting view.  Nash certainly fits the mold of a mathematician completely involved in his work.  He is so determined to find his one unique idea, he doesn’t even both going to class, claiming that classes “destroy the potential for authentic creativity.”  He is not a slacker, though; his time spent away outside of the classroom is always spent inside a library or his dormitory.  The only reason he even scored a date with his future wife is because she asked him, not because he took the time to pursue her (although I’m not sure whether or not this was the case in their actual history).

One could go so far as to argue that it is this obsession with work that ultimately drives Nash insane.  After all, when explaining as an old man how he learned to live with his illness, he says “I still see things that are not here. I just choose not to acknowledge them. Like a diet of the mind, I just choose not to indulge certain appetites; like my appetite for patterns; perhaps my appetite to imagine and to dream.”  In other words, he associates his illness with the very same creative spark that made him a good mathematician.

Other mathematicians are able to find the work-life balance a little more easily.  Consider, for example, the wise word’s of Nash’s friend Sol, played by the dude who was Chandler’s annoying roommate.  When he goes to visit Nash after he’s released from the hospital, Sol tells him that “there are other things besides work.”  To this, Nash can only respond, “What are they?”  It’s unfortunate that Nash is this way, but at least the film shows that in general, mathematicians need not be.

This guy knows his math.

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

What can I say?  Again, Nash meets this criteria quite nicely, but at least his contemporaries do not.  Then again, there’s something to be said for the fact that Nash is often portrayed as a better mathematician – does this mean that to be a truly great mathematician, one must be insane?  On this, the film’s answer is a decisive “Maybe.”

Take Nash’s relationship with fellow colleague Martin Hansen, played by Josh Lucas.  The film initially uses Hansen as a foil to show how superior Nash is as a mathematician.  When they first meet at their student orientation, Nash says, “I read your pre-prints. Both of them. One on Nazi scientists and the other one on, uh… non-linear equations, and I’m extremely confident that there’s not one seminal or innovative idea in either one of them.”  Boom, roasted!

Later on, however, Hansen is shown as the head of the Princeton mathematics department, a position I’m pretty sure one can’t obtain without being a great mathematician.  So does Nash really have the more brilliant mind, or is he just more full of himself?  Hansen’s character isn’t explored in much detail, so the answer to this question, unfortunately, is not forthcoming.

- It is important to do math on windows, mirrors, or other unconventional writing surfaces.

Check. Come on, guys, seriously?  Isn’t it difficult to read things that are written on a window?

Clearly, somebody needs to take a trip to Office Depot.

- Mathematics is so inherently difficult and complicated that only gifted people have a hope of doing well in it.

Again, by virtue of the supporting cast, the film manages to shed itself of this misconception.  While Nash does relatively well for himself mathematically, so do his peers, many of whom are presented as being less original than him.  Having a career in mathematics is shown to be just as much a product of hard work as it is natural ability.  Then again, the film only shows mathematicians who were graduate students at Princeton University, so there may be an implied sense that every character must have been gifted to some extent in order to get that far.  I’m willing to cut the film some slack on this, though, since there are more egregious offenders out there (sorry Good Will Hunting, but I’m looking at you).

- Mathematics involves nothing more than tedious calculations and memorization of dry facts.

Nash’s passion for mathematics is one of the strongest parts about this film.  He makes it very clear that the beauty of mathematics lies in ideas, and that people with new ideas are the ones who make the greatest mathematicians, not the ones who can multiply numbers the fastest.  As an older man, he can also be seen telling a group of students that mathematics “is an art form, no matter what these people tell you.”  The film is ambivalent on many of these issues, but on this one, I’m thankful to say that it bucks convention.

So, where does this leave us?  On one hand is John Nash, who nicely fits nearly every misconception about people who study math, and on the other hand sits a larger supporting cast that doesn’t play to these stereotypes, but isn’t fully developed.  In the end, the entire affair left me feeling, well, not much of anything.  But to be fair, my expectations going in were much worse.  Credit is due to the film for showing Nash articulate the beauty of mathematics, but I would’ve preferred to see more focus on Nash’s contemporaries.  Nice try, Ron Howard, but as far as math movies go, I think Stand and Deliver still has one up on you.

This graph tells you, for example, that during opening weekend for The Polar Express in 2004, the screens showing the film in 3D made nearly 7 times as much money as the screens showing the film in 2D.  As you can see, the drop from this film is quite precipitous, and among recent films, it looks as though 3D and 2D versions of a 3D film are making around the same amount of money.  Given the extra effort involved in making a film 3D (either by shooting in 3D or by adding the effects in post production), some might take this as a sign that the 3D bubble is ready to burst.

Of course, given the higher ticket prices for the 3D experience, attendance numbers may paint an even bleaker picture.  Suppose that on average, prices for a 3D film are 20% higher than prices for its 2D equivalent (this is a fairly conservative estimate – I wouldn’t be surprised if the actual surcharge was closer to 30%).  In that case, if we scale the 3D revenues by a factor of 5/6 (so that a 20% increase yields the original price), we get a better idea of attendance ratios for 3D vs 2D screenings.  Looking at the original data and the scaled data side by side gives us the following picture:

Click to enlarge!

Notice that my graph has not omitted the success of My Bloody Valentine 3D.  This brings me to a related point – it seems a little inconsistent to exclude this movie from the graph on the basis of it being an outlier, but not to exclude The Polar Express on the same basis.  It’s natural to want to do this, if you are arguing that 3D is on the verge of another extinction, but it doesn’t seem fair to exclude one and not the other.

Excluding both films as being outliers (which seems like an entirely natural thing to do – after all, when The Polar Express came out, there were only 59 screens that showed it in 3D that opening weekend, and this small number of screens highly skews the revenue per screen ratio), we get the following adjusted graph:

Click to make larger!

This graph is less compelling.  While there is a general downward trend, there are many movies that have bucked that trend.  Also, given the recent uptick in 3D box office performance, it’s hard to say conclusively how audiences are responding to 3D from this graph alone.

What’s more, it’s not entirely clear to me how much can be concluded from only looking at a film’s opening weekend.  For example, on the graph above we see that Avatar had an opening weekend 3D/2D revenue ratio of 1.70 (adjusted to 1.42), but according to this article from The Hollywood Reporter, by January nearly 80% of Avatar’s domestic gross had come from 3D screenings, which should put the 3D/2D ratio closer to 4 ($4 earned in 3D for every $1 earned in 2D).  Of course, Avatar may be somewhat of an exception, given its staggering box office returns, but even so, I wonder whether word of mouth could help a film that really shines in 3D and push the opening weekend ratio up during subsequent weeks.  I tried to find total 3D vs. 2D revenue numbers for these films, but was unsuccessful – if anyone knows where I can find this information, I’d be much obliged.

This brings me to another question: do better reviewed films perform better in 3D?  By comparing these opening weekend ratios to the films’ freshness ratings on Rotten Tomatoes, we can try to get an answer to this question.

Here are two scatter plots showing the freshness rating vs ratio of opening weekend 3D revenue to opening weekend 2D revenue.  The first plot includes the two outliers, while the second ignores them.

Click to make bigger!

Click to embiggen!

While ignoring outliers helps, in neither case is there a strong correlation between a movie’s perceived quality and its relative performance on 3D screens during opening weekend.*  Again, I’d be curious to see how these numbers change if the ratio of 3D revenue to 2D revenue for the entirety of a film’s run was considered, rather than just opening weekend.  For now, though, I’ll content myself with this.

I should point out that the data I used for the (admittedly brief) analysis above was found here, since I couldn’t find the source data used in the Slate article.  Because of this, there may be slight differences in our numbers (although judging from our graphs, there doesn’t seem to be a huge difference).

In conclusion, regarding the proclaimed death of 3D, I think it’s too early to make a conclusion.  I will tell you, though, that paying $17.50 to see Piranha 3D seems a little ridiculous, even for the most ardent Jerry O’Connell fan.

* For the statistically minded, the correlation coefficient in the case where outliers are included is roughly .026.  When the outliers are ignored, it increases to around .158.

]]> http://www.mathgoespop.com/2010/09/3dead.html/feed 2 http://www.mathgoespop.com/2010/09/scott-pilgrim-vs-gravity.html http://www.mathgoespop.com/2010/09/scott-pilgrim-vs-gravity.html#comments Wed, 01 Sep 2010 15:00:31 +0000 Matt http://www.mathgoespop.com/?p=663 More than three weeks after its opening, Scott Pilgrim Vs. The World appears to be limping towards the end of its theatrical run.  For whatever reason (some blame marketing, others blame Michael Cera exhaustion, for others the fault lies with a crowded weekend of opening releases) this action comedy with a video game aesthetic . . . → Read More: Scott Pilgrim Vs. Gravity]]> More than three weeks after its opening, Scott Pilgrim Vs. The World appears to be limping towards the end of its theatrical run.  For whatever reason (some blame marketing, others blame Michael Cera exhaustion, for others the fault lies with a crowded weekend of opening releases) this action comedy with a video game aesthetic and a heart of platinum has failed to find an audience.  Critical response has been very positive, and everyone I know who’s seen the film has enjoyed it, so it’s unfortunate that this level of support hasn’t translated into higher revenues.

While I’m sure many people have their own explanations for why this film didn’t resonate with a larger crowd, I would like to posit my own: that our culture’s math anxiety runs so deep, we instinctively run when there’s even a hint of mathematics afoot.  For if you look closely enough, you can find some mathematics in this film.

As you probably know, the plot of the film involves Scott Pilgrim fighting his way through the seven evil exes of Ramona Flowers, the girl of his dreams.  And don’t let his wiry frame fool you, for Scott Pilgrim has played enough Street Fighter to know how to throw a serious punch.  This, combined with his sharp wits, make Scott a deadly opponent.

That’s not to say that these fights are a breeze, however.  Consider Scott’s fight against Ramona’s third evil ex, Todd Ingram.  Because of his vegan diet, he has certain psychic and telekinetic powers that give him a distinct advantage.  Luckily for Scott, though, he is not the sharpest of the evil exes.

Near the end of their battle, in an act of unparalleled physical strength, Todd Ingram executes a most righteous uppercut on Scott Pilgrim that sends him flying through the air.  He lands some time later in a pile of garbage.  When I watched this scene, one thought came to mind, a thought that I could not dispel for the rest of the movie: how high up did Scott Pilgrim go?  Such a question may seem unanswerable, but with some careful measurement and a simple application of the classical equations of motion, an answer is entirely within our grasp.

Consider the setup: Todd and Scott begin in a face off.

Todd then hits Scott and he flies upward with some initial velocity v₀. After a certain time, Scott returns to earth.

In the film, Scott remains airborne for approximately 23.6 seconds.  If we ignore air resistance, since it takes him about as long to go up as it does to come down, this means he reaches the top of his trajectory around 11.8 seconds into his flight.  We can then find the initial velocity, because we know that when Scott Pilgrim reaches the top of his trajectory, his velocity will momentarily be zero.

More specifically, the velocity at any time t is given by

v(t) = v₀ + gt,

where g is the acceleration due to gravity, typically approximated by -9.8 m/s² (since Scott Pilgrim is Canadian, I think it’s only fair that we perform this analysis in the standard units of his homeland).  In particular, v(11.8) = 0, so we have

v₀ = 9.8 x 11.8 = 115.6 m/s,

so Todd sends Scott flying at a speed of approximately 115.6 meters per second, or a whopping 258.6 miles per hour.

Given this initial speed, how high up will Scott travel?  For this, we can use the equation governing the vertical position (which is the integral of our velocity equation).  If we set Scott’s initial height to be 0, then his height at time t is given by

y(t) = v₀ t+ gt²/2.

Since we now know the initial velocity, we can plug in values at t = 11.8 (when Scott reaches the highest point).  This gives us y(11.8) = 115.6 x 11.8 – 9.8 x (11.8)²/2, which is approximately 681.8 meters (if you prefer, around 2,237 feet).

Of course, this number isn’t very meaningful without context.  As you can see from the picture below (courtesy of Wikipedia), Scott flew way above the Empire State building, and until recently would’ve flown above the highest skyscrapers in the world.  In other words, that punch sent him up fairly high.

Click for a larger view

Admittedly, this analysis is oversimplified by the fact that we ignored air resistance.  If we try to take this into account, what sort of answer will we obtain?

One can obtain analogous answers in the case where air resistance is included, but the mathematics involved quickly becomes more complicated.  Let me simply summarize the results, and for those brave souls who want more explanation, I will provide it at the end.

First of all, even though Scott’s trip takes 23.6 seconds, with air resistance included we can no longer assume that he reaches the zenith of his trip in half the time.  This is because air resistance on the way down will slow his acceleration, so it will actually take longer for him to come down than it did for him to go up.  I have calculated that in this case, instead of reaching the peak at 11.8 seconds, he actually reaches it at closer to 8.1 seconds.

As for the peak itself, once you know the time the peak is reached it’s not hard to compute.  In this case, I found that the peak distance was around 649.1 meters.  Lower than in the previous case, but not as much lower as I would’ve thought.

Perhaps the most interesting thing to note is the change in initial velocity.  To keep Scott in the air for 23.6 seconds with no air resistance, we saw that Todd must propel him with an initial velocity of 115.6 m/s.  With air resistance, however, to keep Scott in the air for that long requires a much greater initial velocity – around 455.1 meters per second, which is over 1,000 miles an hour.  This is also well above the speed of sound, so Scott should have produced a sonic boom as he traveled upwards – the fact that he doesn’t in the film shines a light on Canada’s best kept secret: it lies inside a perfect vacuum.

Those interested in the details of the air resistance problem, read on.  But be careful, there is mathematics ahead!

*

To solve the problem in this case, I began with Newton’s second law:

,

where is Scott’s acceleration as a function of time, and v is his velocity.

The drag force provided by air resistance, ignored in our earlier calculations, is directly proportional to the square of the velocity.  Moreover, when Scott is traveling upwards, the drag force points in the same direction as the force of gravity.  This means that the net force on Scott’s body must be , for some constant c.

When Scott reaches the zenith and begins to descend, the drag force points in the OPPOSITE direction of the gravitational force, so on this side of the trip we see that the net force is .  Note that this net force will shrink in magnitude until the object reaches its terminal velocity, which we will denote by .  Since the sum of the forces is 0 when an object reaches terminal velocity, we see that .

With this setup, we can now prove the following claim.

Claim: Suppose an object is thrown vertically upwards with some initial velocity and lands at a later time .  Then the time at which the object reached the highest point on its trajectory, denoted , satisfies the following equation:

Why is this true?  Well, on the upwards trajectory, isolating the acceleration we see that

.

By separation of variables, this gives

Since , this implies that for some constant k.   Solving for v we find that

Moreover, since , we see that .

Integrating once more and using the fact that the initial height is 0, we find that for ,

A similar argument works for the downwards trajectory – the only difference is that now the force of gravity and the drag force are in opposite directions.  This has the effect of turning the velocity into a function of tanh, rather than tan.  The end result is that for , we have

Since , by integrating once more we can also conclude that for ,

Note that at we have two ways of writing the position, depending on which formula for we’d like to use.  Setting the two equations equal at , canceling out the common factors of , and exponentiating each side, we are led to the equality , which proves the claim.

This allows us to find numerically given g, , and  .  As usual, we approximate g by 9.8 meters per second per second.  For the terminal velocity, I consulted this website to find the terminal velocity of a skydiver.  I averaged the first four estimates that were in a similar range, and came up with an estimate of 54.7 m/s as a terminal velocity for Scott.  Once again, we use 23.6 as the value for .  Graphing for these values of the parameters, you will see that there is a root near t = 8.1.  This gives us the time at which Scott achieves his maximum height, and this in turn can be used to find that maximum height, , and the initial velocity, .

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.