Math in the Movies: Up

As many of you are no doubt aware, Pixar’s latest film opens this weekend. I have yet to see the film, so I’m sure I am spoiling nothing by telling you that part of the film involves an old man flying through the sky by means of balloons that are attached to his house.

Do not try this at with your home.

Given that I have yet to see the film, you may wonder how I could possibly hope to connect it to mathematics. Thankfully, I don’t have to – the work has been done for me by Alexis Madrigal over at Wired.com, who wrote an article discussing the feasibility of using balloons to take to the skies in one’s own home.

His assumptions are that the house weighs roughly 100,000 pounds, and that the balloons are spherically shaped with a diameter of three feet, which may seem large at first, but seems more reasonable if you look at other shots of the floating house.

How many balloons does that look like to you?

Based on these assumptions, and the weights of air and helium, Mr. Madrigal estimated that it would take roughly 105,854 balloons to lift the house.

The fun really begins in the comments section, however. The model presented in the article was quite simple and easy to understand, but perhaps overlooked some of the subtleties in the problem. Here are my favorites:

You forgot to factor in the weight of the string for the balloons. The balloons at the outer edge of the cluster would need longer strings = less overall buoyant lift/balloon once you factor the extra string weight. So you’d need more, or larger balloons.

Then there’s this one:

Actually, there’s a math error. The density of helium will be greater when in a balloon as the pressure in the balloon is greater than atmospheric pressure. This means that the lifting force per balloon is less and that you’ll need even more.

Of course, there’s also the question of how the house broke free of its foundations. The trailer makes it look as though the balloons themselves were able to uproot the house – if this is the case, one would need significantly more balloons in order to rip the house up from the ground.

Perhaps this is why the in-house estimate from Pixar for the number of balloons that would be required to make the house fly is nearly 250 times larger than the estimate given by Mr. Madgrigal. Believe it or not, the feasibility of lifting a house with balloons was brought up at Cannes, where the film premiered, to which co-director Pete Docter replied:

We have scientists at Pixar, and one of the first questions we asked them was how many balloons it would actually take to float a house. They calculated it would take 26.5m balloons. So we reckoned we would be safe from people trying it themselves.

Whether you think the number of balloons required is in the tens of millions, or just a measly 100,000 or so, both numbers are significantly more than the amount of balloons pictured on screen – which, according to a recent article by Daniel Terdiman at CNET, is closer to 10,000. This article is also good for those of you who may not be impressed with the simple calculations involved in trying to determine how many balloons would be required to lift a certain amount of weight.

Instead of analyzing the plausibility of using balloons to lift a house, the CNET article discusses the real world difficulty that comes with trying to animate upwards of 10,000 balloons so that they move together in a realistic way. The dynamics of the balloons are all highly connected – the motion of one balloon will have an effect on all the others – and modeling a system with that level of complexity is certainly no easy task. This is where mathematics flexes its muscles a bit more.

Everyone is happier with a houseful of balloons.

According to the article, the machinery at Pixar was initially only able to handle animation for about 500 balloons – to ramp this number up twenty-fold certainly must have taken some hard work and some mathematical wizardy. Producer Jonas River sums it up best:

The audience looks at (the balloon cluster) and says, “Oh, that’s pretty.” But they have no idea how much work went into it. We worked on that for over a year. (Then) the kid takes off his hat and runs his fingers through his hair. My mother will never know that took 15 people six weeks.

So, whether you’re in it to get ideas for how to build your own flying balloon contraption, or because you’re curious to see how Pixar is again pushing the boundaries of what is possible in animation, there will be something to satisfy your mathematical curiosity. The movie currently holds an impressive 98% freshness rating on Rotten Tomatoes. How much of this high rating can be explained by the film’s mathematical sophistication? Probably very little – but at least a man can hope.

May 30, 2009 | No Comments »

Baby Animals Just Want to Do Math

Over the past few months there have been several studies aimed at understanding the mathematical sophistication of some of our friends in the animal kingdom. This is a topic I have discussed before, but these new findings are interesting and worth mentioning.
The most recent experiment involves the cutest animal discussed so far: baby chicks. Don’t let their looks fool you, my friend, for under that puff of yellow down sits a mind capable of mathematical wizardry. Surprisingly, researchers found that chicks were not only able to perform simple mental calculations, but could do so from a very young age.

How do you tell if a baby chick can do math? Well, apparently the little ones try to stay close to familiar objects (for example, their mother). Moreover, given the choice between a small group of familiar objects and a larger group of familiar objects, researchers noted that chicks tended to gravitate towards the larger group.

But what if some calculation is required to determine which is the larger group? Researchers put the chicks in a glass cage and then hid yellow balls behind one of two screens. Sometimes they would then transfer some balls from one screen to another, in a process that the chick could see. However, the chick couldn’t see how many balls were behind each screen, so the only way to keep track would be to keep track of how many balls moved from one side to another, and how many were initially on each side – in essence, to perform some basic mental arithmetic.

Surprisingly, the chicks were up to the challenge, and consistently went towards the larger group, even though the two groups were hidden from view. Here’s a link to a video that shows the basics of the experiment design.

Apparently, chicks love yellow plastic balls.

Of course, the word “baby” has several meanings. Baby chicks are described in this way because they are young, but the adjective baby could just as well describe tiny things (think of baby corn, baby back ribs, or baby math blog readership). With this interpretation, baby chicks aren’t the only baby animals that want to do math – some baby fish are joining the party as well.

This is a combination Math/French joke.

Please meet the graceful mosquitofish, a species poised to revolutionize mathematics as we know it. Or, if not that, at least it can do some simple counting, according to researchers from the University of Padova in Italy.

What makes us think these fish can count? Well, the fish were put in a tank and given the choice of several doors to swim through. One of those doors had a larger group of mosquitofish (no doubt they were all studying for the Putnam exam together). First the researchers trained the fish to associate the correct door with a certain number of geometric shapes. The fish were then put in an empty tank and were allowed to move freely through any of the doors.

The results? More often than would be expected by chance, the fish chose the door with the number of shapes that they had been trained to enter. Moreover, to try and pin down the effect of the number of shapes, rather than any other parameter, researchers “placed sets of shapes that varied in size, brightness, and distance…only the number of shapes stayed the same.”

Does this mean that these tiny fish have some rudimentary method of counting small sets? Do they have a number sense? What does it even mean to claim that a fish can count? With further research, maybe the answers to some of these questions will become clear.

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Our last foray into mathematics within the animal kingdom comes to us from what is undoubtedly the coolest looking animal mentioned so far: the rhesus macaque.

Researchers at Duke University were able to have “widespread success” in getting rhesus macaques to calculate differences of whole numbers.

The main idea is similar to what was done with the chicks, although slightly more was expected from the macaques: they were first shown a collection of dots on a computer screen. The dots were then covered by a square, and some of the dots flew off screen – the monkey could see how many dots were removed, but not how many dots were remaining. The article linked above has a video showing this animation.

Afterward, the monkeys were given a choice between two collections of dots – one with the correct number of dots remaining, and one with the incorrect number of dots remaining, and were asked to pick a collection. Researchers found that the macaques performed just as well at identifying the correct difference as the human college students that were used as a control. (Then again, the macaques were rewarded for their correct answers with Kool-Aid – no such incentive is mentioned for the human controls.)

Could the secret to mathematical ability be locked inside the belly of this anthropomorphic glass pitcher? The question remains open.

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With all of these stories, there is an important question to ask: why should we care? Who cares if chicks can count, or if macaques can subtract dots? More generally, why should we be bothered with questions regarding the mathematical ability of other species?

One important answer is that clues about the abilities of other species may help give us clues as to how our own ability to do math has evolved. More specifically, we can attempt to address the question: what is the role that evolution has played in the development of mathematical ability?

A few of the articles mention potential evolutionary benefits to mathematical ability. For example, in the case of the mosquitofish,

…the ability [to count] in fish is probably a “last resort” strategy that has evolutionary underpinnings, [lead study author Christian] Agrillo said.

That’s because non-numerical cues probably come more easily to fish as they make rapid-fire decisions.

Being able to count may require more brainpower than simply judging numbers based on size. But counting might sometimes be necessary as the fish seek safety in numbers to shield themselves from predators, Agrillo said.

This “safety in numbers” phenomenon may also help explain the chicks ability to keep track of small sets of numbers. If there is an evolutionary advantage to moving towards a larger group, then it’s reasonable to guess that chicks may have developed a basic ability to keep track of relative sizes, even under difficult conditions such as the ones present in the study.

What about the macaques? In this case, there may also be an evolutionary advantage to having a knack for mathematics. The authors note that “For instance, research has shown that apes can determine at a glance roughly how much food is present in an area and decide whether to stay and eat or to move on.” This ability to estimate would require at least a certain level of mathematical sophistication, one which could arguably depend upon the ability to perform simple subtraction calculations.

So, there are evolutionary arguments for the development of mathematics – but to what extent it can be said that these animals are “doing math” is a good question. And as for how to bridge the gap between their level of mathematical sophistication and abstract thought and ours, I’ve no doubt there is plenty of research waiting to be done.

I would start by looking into the Kool-Aid.

May 22, 2009 | No Comments »

Math Gets Around, and So Does Disease (Both Real and Virtual)

Most of us are familiar with the story of Chicken Little, the young chicken turn Disney sellout who one day has a major panic attack because she (or he, depending on the version you’re told) believes that the sky is falling.

No doubt this fable has conditioned many of us to be wary of chickens that try to warn us of impending crises. But given the recent media frenzy surrounding swine flu, perhaps we should turn our attention away from the chicken, concerns over avian flu notwithstanding, and focus a bit more on the humble pig.

There is some debate on this issue: while everyone seems to be in agreement that the swine flu outbreak is, so far, milder than many had anticipated, health officials have cautioned that we may not yet be out of the proverbial woods (or pigpen, as the case may be). At the same time, however, one can just as easily find articles that argue that maybe this whole thing has been overhyped.

Cute family film, or a foreshadowing of the coming apocalypse?

This divergence of opinion shows that we still have a great deal to learn about how to deal with potential pandemics. An improvement in our understanding will, among other things, necessitate a better understanding of how disease spreads within human populations, as well as a better understanding of how humans respond to health crises. Since we can’t really release diseases into the population to observe what will happen, the only thing we can do to guide our response is look at historical data, or try to model what would happen in the event of an outbreak. In either case, mathematics can provide us with valuable insight.

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There are a few decent survey articles online that discuss how math can be used to model the spread of disease. For some discussion that involves only basic Algebra, the Wikipedia article on Mathematical modeling of infectious disease isn’t a bad place to start. Another good survey article by Professor Matt Keeling (now of the University of Warwick) can be found here.

The simplest (and crudest) way to model a disease would be to assume some sort of exponential growth: for example, person A gets a disease, and passes it on to 3 friends, who each pass it on to 3 friends, who each pass it on to 3 friends, and so on. This is the sort of model that people love to point to when trying to put fear into the hearts of the masses. For instance, if we postulate that swine flu spreads in approximately this fashion, with each infected person infecting another 3 people every day, it would take only three weeks for the entire world to become infected.

Of course, we know that such a conclusion is unrealistic – this means that this is not a very good model. To get a decent model for the spread of disease, we need to take into account certain parameters of the population and of the disease in question. For example, what are the birth and death rates in the population? How contagious is the disease, and once someone is sick, for how long can they infect other people?

One such model that keeps track of these and other parameters is dubbed the SIR model, so named because it partitions the population into three groups of people: Susceptible, Infectious, and Recovered. The Susceptible group has not caught the disease in question, although they are able to catch it if they come into contact with it. If a person catches the disease, that person moves into the infectious category – that person is now able to infect anyone he or she comes into contact with. After a certain period of time, the person will recover, and move to the recovered class. Once recovered, we assume the person cannot become infected by the same disease again.

Using differential equations, one can model the long term behavior of diseases under the SIR assumptions. The result is that the proportion of infected people follows a damped oscillatory pattern with time – in other words, when the disease is introduced, outbreaks are common, and result in an infections to a larger proportion of the population. As time progresses however, outbreaks become less common, and the disease stabilizes to an endemic state where a certain fixed proportion of the population is infected at all times.

SIR model over time. Image taken from the article cited above.

Once things have stabilized, the proportion of susceptible individuals is given by 1/R0, where R0 is the the average number of other individuals each infected individual will infect in a population that has no immunity to the disease (called the Basic reproduction number). In other words, the more infectious a disease, the smaller the susceptible population will be in time. This meshes well with our intuitive understanding of how diseases spread.

It’s worth noting that the values of R0 have been estimated for various diseases. For example, we believe that the R0 for HIV/AIDS lies somewhere between 2 and 5, while measles is between 12 and 18. This speaks to the fact that it’s much harder to infect someone with HIV than it is to infect them with measles.

This is by no means meant to be an exhaustive discussion of the ways in which mathematicians model disease. And, as with any model, the SIR model has its limitations. There are generalizations of the SIR model – interested parties can read about some of them here. However, in general, finding an appropriate model is one of the difficulties in trying to understand the spread of disease.

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Constructing a model isn’t the only way we can use math to better understand the spread of disease. We can also analyze historical data from diseases that have already occurred to try and find patterns.

Some may find a treasure trove of data from the 1918 flu epidemic. Others may search for answers by looking at data from Ebola outbreaks. Moreover, some have found patterns by studying data from diseases that don’t even exist!

Well, that’s not entirely accurate. The disease does exist, but only in the far away land of Azeroth, where orcs and humans battle for baubles and trinkets, form guilds, and go on quests. It’s true – World of Warcraft has given us useful data regarding how people respond during an epidemic.

Whatever disease she has, I’m pretty sure you don’t want it.

Here’s the story, courtesy of this article on the phenomenon:

In 2005, the game’s designers at Blizzard Entertainment decided that some players’ characters had become too powerful, so they created a virus — called “Corrupted Blood” — that would make the game more challenging for the most powerful players.

Turns out that even in the virtual world, things don’t always turn out as planned. The virus quickly infected any nearby character, regardless of its relative strength.

The programmers imposed a mass quarantine…yet many players ignored the quarantine, spreading the virus. Eventually, more than four million of the game’s six million players worldwide were infected, and millions “died.”

The silver lining in this cloud of virtual death was found by researchers studying the spread of disease, including Rutgers professor Nina Fefferman. The article continues by saying that “Fefferman and a colleague studied the plague as it spread…[focusing] on how a pandemic would spread and affect society and the economy. Since then she’s been called upon by health agencies all over the world to consult on her findings, including the U.S. Department of Homeland Security and the U.S. army.”

Certainly the traditional avenues for analyzing pandemic data are open as well, but it is surprising to think that interesting conclusions can be drawn from the reaction to virtual diseases. This also allows WoW players to counter the argument that online games are a waste of time by asserting that they are, in fact, helping to prevent mass extinction at the hands of a new strain of disease.

And for that, World of Warcraft players, you have my heartfelt thanks.

May 12, 2009 | No Comments »

When will this stop?

Ok, now it’s just getting annoying. Odd day? Give me a break.

My thoughts on this irritating trend can be found here, here, and here.

May 7, 2009 | 4 Comments »