180: Reading The Tea Leaves For Real

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Before we begin, I'd like to thank listener Bret McDermitt for posting another nice review on iTunes.   I'd also like to thank the Math Insider website for featuring Math Mutation in an article on great math podcasts-- check the article out at the link in the show notes!   Now, let's get on to today's topic.

As I looked out the window recently on a windy day, watching leaves and dirt fly around outside, it brought to mind Rudy Rucker's concept of a 'paracomputer', a computer based on observing natural processes that happen to be equivalent to a natural computation.   Let's review the background of this idea briefly so we can explain it.

You may recall from some earlier podcasts the ideas of Stephen Wolfram's controversial tome "A New Kind of Science", which claims that we can view many natural phenomena as similar to a computation such as a cellular automaton.      One popular example of a cellular automaton is Conway's Game of Life.   In this game, each square in a grid can be "alive" or "dead", usually represented by being colored black or white.  A live square stays alive for the next time step if and only if it has exactly two or three live neighbors, otherwise dying of loneliness or overcrowding.   And a dead square is "born" and becomes alive if it has exactly  three live neighbors.    The amazing thing about this simple game is that depending on the initial set of live squares, it can display a huge range of lifelike behaviors, as you can see illustrated on the Wikipedia page linked in the show notes.    This led researchers such as Wolfram to hypothesize that such simple computations might explain the basics of how our universe really works.

If you look at the patterns created by various Life configurations, you can see three basic  types.  Predictable patterns settle into an endless repetition of simple elements, such as a square group of cells simply living forever in the middle.     Chaotic patterns end up looking totally random, like the static on a malfunctioning TV set.   The most interesting ones are what Rudy Rucker calls "gnarly" patterns, which seem to have interesting structure but are unpredictable.   These gnarly patterns are what seem to have promise to represent real life:  for example, a gnarly pattern might look analogous to clouds moving in the sky, the life cycle of a jellyfish, or celestial bodies interacting over the lifetime of a solar system.   These all share the property that their ultimate results are in theory predictable, but so complex to calculate that there's no fundamentally better way to figure out what will happen than to wait and see.

After observing many of these gnarly systems, Wolfram hypothesized the Principle of Computational Equivalence.   This states that at some fundamental level, all these gnarly real-life systems are each capable of acting as a universal computer, and thus equivalent to each other.   So, for example, by coming up with a suitable "mapping" of clouds that represents the initial state of a jellyfish being born, and somehow manipulating the clouds into this configuration, you would be able to, by watching those clouds, predict exactly what is happening at any stage of the jellyfish's life.     You just need to decode the current state of the clouds into the equivalent state of the jellyfish.

And this is Rudy Rucker's idea of a "paracomputer".    The concept is that you figure out some gnarly physical phenomenon that you can control, identify the proper mapping to and from whatever other type of system you want to analyze, and just set it in motion.   Sounds pretty cool, right?   So, all those old fortune-tellers who claim to be able to tell the future from tea leaves might actually have the core of a feasible idea:  all you have to do is figure out enough detail about your life to map its precise state into the tea leaves in a very large kettle, shake the kettle in a defined way so the leaves will represent each future state of your life, read the results in the leaves, and map the final leaf pattern to the details of your life.   Presto!  You now know every detail of your future.

Now if you're picky, you might have spotted a few flaws in this idea.   Wolfram's Principle itself is very controversial, as it seems to collapse what could potentially be many different complexity classes, and claim that a huge range of phenomena are equally complex to compute.    His observations provide some evidence, but nothing close to a real proof.    There's also the question of how you would fully figure out the starting state of your life, to map to the tea leaves:   you might object as the fortune-teller takes out a bone saw, opens up your skull, and carefully extracts the state of each neuron in your brain.    Even after your brain has been carefully put back together without modifying the state, then there's the question of how many tea leaves it would take to map your life:   the fortune-teller might need all the tea in China to properly represent it, especially if you live the exciting life of a math podcast fan.   And finally, there's the question of how long it would take the tea leaves to reach their representation of any point in your future:  would this really be any faster than just waiting for your future to happen?

But, nevertheless, it is kind of fun to think that nature really consists of computers all around us, just waiting to be properly understood.    Maybe soon you'll be able to turn in your PlayStation 3 and play Call Of Duty with leaves blowing in the wind in your backyard.  

And this has been your Math Mutation for today.

References:

• http://www.mathsinsider.com/math-podcasts/ : Math Insider article
• http://www.asimovs.com/_issue_0511/Thougthexperiments.shtml:  Rudy Rucker article describing paracomputers.
• http://en.wikipedia.org/wiki/A_New_Kind_of_Science : Wolfram's "New Kind of Science" at Wikipedia
• http://en.wikipedia.org/wiki/Conway%27s_Game_of_Life : Conway's Game of Life at Wikipedia

179: Bugged By Math

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Before we begin, I'd like to mention that I'm running for my local school board again, in Hillsboro, Oregon. If you're curious about my campaign, or might consider volunteering or donating, be sure to check out my website, http://hillsboroerik.com . But don't worry, I'm keeping this podcast separate from my campaign, so you can keep listening even if you're one of those dorks planning to vote for my opponent. Now on to today's topic.

My 6 year old daughter has been really into insects these days. I try to encourage any science-type interest, though I've had to give her some stern talks about taking this hobby into the house. Thus we watch a lot of Animal Planet shows together. Recently one show mentioned 17-year cicadas, a type of insect whose larvae emerge from the ground into adulthood every 17 years. But as my daughter wondered out loud, how did they come up with the number 17? Why would an insect choose this particular number?

A little web searching revealed a few answers. But to start with, we need to figure out what question we're asking. 17 years is an interesting life cycle for more than one reason. On the one hand, it's pretty long to start with- why would an insect derive an advantage by spending so much time between generations? Then, as an avid math podcast fan, you've probably noticed another interesting aspect of 17: it's a prime number, divisible only by itself and 1. Do the cicadas benefit from both long generation time and prime-ness?

According to one theory, the long generations are useful in cases where the climate is in a state of rapid change. Periodical cicadas evolved about 1.8 million years ago, at a time of climate instability. In these situations, there would sometimes be several years in a row of inhospitable temperatures, either too warm or too cold. You can easily imagine that if bad years come in clusters, waiting a while to return after a year in which you barely survive can give you a bit of an edge, and prevent a Dust Bowl from wiping out your species. Of course, depending on the actual temperature patterns, the long generation time could be good or bad, but one modelling study linked in the show notes showed that in the climate of that time, a 17-year cicada would have a 96% chance of its descendants surviving for 1500 years, while a 7-year variety would only have an 8% chance. That's assuming the main threat is the climate, and is not accounting for hungry predators or pet-seeking 6-year-olds.

The fact that their period is prime is another interesting adaptation. The most popular theory, orginally proposed by famous biologist Stephen Jay Gould, is related to the fact that cicadas avoid predators by a method called "predator satiation". This is a fancy way of saying they just reproduce in such enormous swarms that the predators can eat all they want, and there are still enough survivors to propagate the species. But there's a fundamental problem with this evolutionary strategy: predators could learn to increasre their own population periodically in anticipation of this tasty swarm. If the period was a small, predicatble number, it would not be too hard for some predator by chance to evolve a matching cycle. But with the prime number 17, chances are infinitesmal that a predator would evolve to increase its population by the right period.

Another competing, or perhaps complementary, explanation comes from the cicada version of racial segregation: the subspecies that remain most distinct are the ones least likely to interbreed. For example, a 13-year and a 17-year cicada species would only meet at most every 13*17 = 221 years, meaning opportunities for interbreeding would be rare and the two genetic lines would be more likely to remain separate. There does seem to be a basic flaw in this explanation, if you think about it a bit: a 17-year period seems like an overly-complex way to keep species separate. All it would take is for a pair of species to have, for example, 2-year periods, but emerge 1 year apart-- then they would never meet. However, if there are large numbers of random cicada groups of competing periods, it could be that among all these groups, the large prime-breeders will have the advantage of remaining distinct species. The smaller and non-prime breeders would meet much more often, and most likely blend with each other until they all seem to merge together.

As with many aspects of evolutionary science, it's kind of tricky to figure out empirically whether one or all of these explanations is the right one. But there are lots of links on the web describing computational models that support one or more of these hypotheses. Maybe one day you will figure out an even better answer. Personally, I'm happy as long as they don't end up in my kitchen.

And this has been your math mutation for today.



References:

• http://hillsboroerik.com : My school board campaign website
• http://www.murderousmaths.co.uk/cicadas.htm : An article on periodic cicadas
• http://www.sciencenewsforkids.org/2004/05/prime-time-for-cicadas-2/ : Another article on periodic cicadas
• http://www.cims.nyu.edu/~eve2/cicadas.pdf : A paper on modeling periodic cicada populations

178: Mathematical Immortality

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Before we start, I'd like to thank listeners "Basha De L", "Sir Gustav", and "Leroux-Cifer" for recently posting nice reviews on iTunes. Also a special thanks to listener Mark Webster, who made a donation in honor of Math Mutation to Vittana.org, a charity offering educational microloans. Amazing how much happens when I take a month off from the podcast!

Now on to today's topic. Ever since the dawn of mankind, and probably long before the first caveman figured out that 2 plus 2 equals 4, we have been wondering about what happens to our souls after we die. Does our consciousness continue in some form after we're gone? Or does it just disappear from the universe, never to be experienced again? I'm not going to try to answer this from the philosophical, spiritual or religious viewpoint, since there are thousands of better online flamewars available if we would like to debate those topics. But if we restrict ourselves to the domain of math and physics, can we still construct a good argument for some form of life after death?

One of the most direct arguments on this topic comes from the many worlds interpretation of quantum mechanics, which you may remember me mentioning in some earlier podcasts. This idea was originally proposed in 1957 by American physicist Hugh Everett, and further popularized by Bryce Seligman DeWitt in the 1960s, sadly no relation to me despite his cool name. The many-worlds interpretation derives from the fact that quantum physics describes subatomic events in the form of probability waves, mutliple possible outcomes with different probabilities. A definite outcome does not occur until the system is observed and the waveform collapses to a particular state. A classic example here is "Schroedinger's Cat", a cat locked in a box with a poison capsule set to release only if a particular radioactive atom decays in the next hour. (Luckily PETA was not yet around during the development of quantum physics.) Until the box is opened to collapse the wave function, the cat is neither alive nor dead: the box can only be described by a superposition of states that labels the cat possibly alive and possibly dead.

The many-worlds interpretation decribes this situation in terms of multiple universes. When a quantum event occurs, such as the decay of the atom in the box, our universe actually forks into two universes, one representing each probability. So in one universe the cat is alive, and in the other the cat is dead. This particular example of the cat should make it clear how this relates to our discussion of immortality: while we may open the box, see the dead cat, and mourn it, there is another universe in which the cat jumped out of the box alive and is happily playing with a ball of USB cables. In any case where someone dies in a situation that they could have theoretically survived, there is another universe nearby in which they are still alive.

Clever listeners may have come up with an objection to claiming that this leads to a form of immortality: sure, it says something about life after premature deaths, but what about inevitable causes of death such as old age? Surely in every possible universe, as sad as it is for the podcasting community, I'll be dead 100 years from now. Whie previous generations are out of luck, we do have an answer for this one as well though: the technological singularity.

The singularity idea is that technology has been advancing at an exponential rate for most of the last century. Even when I was in college, I didn't dream that by middle age I would have a device in my pocket that could store a thousand science podcasts and still play Walking Dead games with computing power to spare. Technological growth has been in some ways like a graph of y = 2 to the X power: while it may start slow, it very quickly starts verging towards infinity. If we can sustain this rate of growth, it may be only a matter of decades until we are capable of feats of technology that we would see as almost infnitely powerful by today's standards. When this "singularity" hits, we will be capable of uploading our brain's contents into a powerful computing device, continuing our consciousness indefinitely without the limitations of our frail human bodies. So if in just one parallel universe we can survive long enough to reach the singularity, and upload our brains into a computer with a good extended warranty and reliable backups, we truly are immortal.

If we're talking about the mind as software, we also should not discount another possible form of immortality, distinct from the many-universes-based ones we have been discussing until now. If your consciousness is formed by a finite pattern of electrical firings in your brain, why does this pattern have to occur only in your current brain and current body? There are things all over the universe: stars, planets, quasars, and other stuff- that constantly exhibit many arbitrary patterns of activity among the electrons that make them up. Why shouldn't one of these random patterns be effectively a software program that exactly executes your consciouness, except that it continues after your Earth-based body dies? Sure, this would be a bit of a coincidence--- but if the universe is infinite, perhaps every possible electron pattern will occur somewhere and sometime. (There is the slight monkey wrench here of multiple different-sized infinities, as we disussed in episode 157, but since we don't know the proper classes of infinity for our brains' possibilities or the universe, we're free to speculate.) And this isn't even counting the possibility that your mind is already an intentional simulation that can be rebooted, like we discussed in last week's podcast.

Before we conclude, I'd like to dedicate this week's episode to the memory of my father, Morton Seligman, who recently passed away at the age of 75. If he can just hold out until the singularity in a nearby universe, maybe we'll be seeing him again sometime. Or he might be sitting on an underworld throne next to Hades right now, laughing at us for these silly mathematical discussions while we ignore the real afterlife.

And this has been your math mutation for today.



References:

http://www.vittana.org/: Vittana educational microloans.

http://en.wikipedia.org/wiki/Many-worlds_interpretation: Many Worlds Interpretation of quantum physics at Wikipedia.

http://en.wikipedia.org/wiki/Schr%C3%B6dinger%27s_cat: Schroedinger's Cat at Wikipedia.

http://en.wikipedia.org/wiki/The_singularity: The Singularity at Wikipedia.

177: Does This Podcast Exist?

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Recently a friend pointed me to an interesting article at io9.com, titled "You’re living in a computer simulation, and math proves it." I'm sure you're familiar with the concept of reality being an elaborate computer simulation, an idea most famously illustrated in the Matrix movies, but around in some form or another for many years before that. Even before computers, philosphers were discussing whether life was real or just some kind of dream in the mind of powerful unknown beings. A lot of the recent discussion of the topic seems to have been spurred by a 2003 Philosophical Quarterly article by Nick Bolstrom of Oxford University, "Are You Living In A Computer Simulation?", which was one of the inspirations for the io9 article. According to this particular Internet article, we can prove through math alone that we are almost certainly living in a computer simulation.
Here's how the basic argument goes. Let's suppose it is possible for a sufficiently advanced civlization to create a computer capable of running a simulation of the complexity of our universe. If such a simulation could be created, would these advanced beings create only one, or many of them? Just look at the sales figure for The Sims, a much more primitive simulation game from our time, and I think you'll agree that if you could buy an Xbox disc that runs a full simulation of an Earth-like world for your own amusement, lots of people would do it. So for every real reality, there are millions or even billions of simulated realities. Thus, let's ask the question: given an arbitrary reality, namely ours, is it real or simulated? The odds are millions or billions to one that it's one of the simulated ones. Seems like a pretty convincing argument, doesn't it?

Of course, this is dependent on a few premises that may be a bit questionable. First is the idea that it would be possible to generate a simulation of the complexity of our reality: while extrapolating the rate of advances in computing over the last century makes it sound plausible, we can't really be sure. There may be some fundamental limit to computing power that we will reach sooner or later. Another premise is that creatures advanced enough to create such a simulation would choose to do it-- maybe such advanced civilizations would tend to develop moral philosophies that wouldn't accept the creation of conscious beings for other's amusement. I'd be more inclined to argue from the reverse of this idea, actually: if some powerful beings could arbitrarily mess with the laws of our reality for their own amusement, like some gamer who uses Fallout 3 cheat mods to turn everyone within a mile into two-headed cows just for fun, why aren't we observing a lot more arbitrary violations of physical laws?

A more serious hole in this argument, also discussed by Bolstrom, is that we haven't discussed how many universes exist where civilization never reaches the level capable of creating such a simulation. Maybe it's true that when a civilation reaches this level, it will create a billion simulations-- but at the same time, perhaps only one in a billion billion civilzations reaches this level, which upends the whole argument. It could just be a natural tendency of sentient beings to kill each other off in nuclear wars way before they get to the level of developing a universal-simulation computer program. If this is the case, the odds are a billion to one that we are real rather than part of an elaborate simulation. Sorry, Matrix fans.

Another intriguing trend in this discussion is the search by some physicists for direct evidence that we are actually in a computer simulation. The show notes link to a Mail Online article that discusses such a search. The idea is to find, in the actual physical laws of our universe, elements that would be telltale signs of a computer simulation. For example, scientists could detect effects of cosmic rays travelling in a tiny lattice of regular grid lines, like pixels on a computer screen, rather than being able to truly exist continuously in spacetime. It sounds like a nice idea, but I'm pretty skeptical of such a search: how can we say for sure that such lattice-like behavior, or any unexpected observation from physics, is the result of being a simulation rather than some subtle new law of physics we have not yet discovered? If they had observed relativistic effects experimentally in more cases before Einstein came along, would they claim that time distortion during high acceleration was proof that we were living in a buggy computer simulation, since the observations violated reality, which everyone knows to be described by Newton's laws?

Along similar lines are the arguments that we can infer proof of artificial creation of our universe by the fact that so many physical constants just happen to be fine-tuned to allow sentient life. This one comes up a lot in Creaitionism and "Intelligent Design" arguments as well. But as I see it, this line of reasoning has been thorougly demolished by the anthropic principle. Maybe an equal number of other universes with different constants do exist, but those of us asking the question have to be in the one with the life-friendly constants, otherwise we couldn't be asking it.

Anyway, I'm afraid this podcast will not be able to definitively answer the question of whether we are real or in a simulation. Also, if you're a meta-being listening to this podcast from another universe while monitoring our simulated universe on your computer screen, don't gloat too much at my amusing level of simulated ignorance: you might still be part of a simulation in a meta-meta-universe, and I can't definitively prove it either way for you either. But next time you're messing with our reality for fun, a few more good podcast reviews on iTunes would be nice.

And this has been your math mutation for today.

　

References:

• http://io9.com/5799396/youre-living-in-a-computer-simulation-and-math-proves-it: Article arguing that math proves we are in a simulation.
• http://www.simulation-argument.com/simulation.html: Original Bolstrom article.
• http://www.dailymail.co.uk/sciencetech/article-2216189/Do-live-Matrix-researchers-say-way-prove-do.html : Daily Mail article on physicists trying to prove we are in a simulation.

176: Perfect Maps

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Before we start I'd like to thank listener SkepticHunter for posting another nice review on iTunes. Don't forget to post one too if you also like the podcast!

Recently I installed Google Earth on my iphone, and showed my daughter the cool feature where you can start with a view of the whole planet, and then zoom in to the exact spot where you currently are. Playing with this feature brought to mind an amusing essay I had recently read in Umberto Eco's "How To Travel With A Salmon", called "On The Impossibility of Drawing a Map of the Empire on a Scale of 1 to 1". The essay discusses a project of creating a map that is so detailed, it actually has a 1:1 scale, with each element on the map being the same size as the feature described.

The silly idea of a 1:1 scale map was apparently first proposed by Lewis Caroll in his lesser-known book Sylvie and Bruno, a somewhat enjoyable tale that didn't quite have the narrative flow of Alice in Wonderland, but shared its sense of absurdity. At one point, a character brags about his kingdom's mapmaking skills:

What do you consider the largest map that would be really useful?"
"About six inches to the mile."
"Only six inches!" exclaimed Mein Herr. "We very soon got to six yards to the mile. Then we tried a hundred yards to the mile. And then came the grandest idea of all! We actually made a map of the country, on the scale of a mile to the mile!"
"Have you used it much?" I enquired.
"It has never been spread out, yet," said Mein Herr: "the farmers objected: they said it would cover the whole country, and shut out the sunlight! So we now use the country itself, as its own map, and I assure you it does nearly as well."I think that quote pretty well speaks for itself.

Argentine writer Jorge Luis Borges was apparently a fan of Carroll's, and a few decades later in a short-short story commented on the idea himself. In Borges's vision, the 1:1 scale map is actually constructed, but then abandoned as useless, and left in western deserts to be gradually eroded away by the weather. In another of his essays, Borges points out that if England contains a perfect map of England, then that map must contain a depiction of the map itself, since that map is part of England-- and that depiction must include the depiction of the map, etc, out to infinity.

 

Eco's essay goes into much more detail than Carroll or Borges, discussing the major requirements for a 1:1 map to be useful. He insists that the map be able to exist within the country being mapped; I guess this is because it's not useful as a travel reference if the map must be elsewhere. It must be an actual map, rather than a mechanically created replica: for example, it would be cheating to take a plaster cast of the whole country and call it a map. The map also needs to be useful as a tool for referencing other parts of the country: so it can't be the case that for any location X, the depiction of X on the map lies on the portion of the map directly located at X, since then the map would be no more useful than a transparent sheet over the land. He also discusses various physical constraints such as the materials needed, folding techniques, and the effect on the actual country of having a giant map rolled out over it.

 

I also found an online discussion that pointed to several lesser-known references to such 1:1 scale maps. The rock band They Might Be Giants sings about a ship that is a 1:1 scale map of the state of Arkansas. The British comedy series "Blackadder" has an episode where an incompetent general is constructing a tabletop replica of recently conquered territory on a 1:1 scale. And modern comedian Steven Wright has a joke that goes "I have a map of the United States...actual size. It says, 'Scale: 1 mile = 1 mile.' I spent last summer folding it."

 

This whole discussion also seems to relate to the famous philosophical statement by Alfred Korzybski, "The map is not the territory". He was pointing out that if you are referring to a map, alternate view, or even a mental abstraction of something, you shouldn't confuse that with the thing itself. Naturally, this doesn't only refer to literal maps, but to just about anything you might see, interact with, or think about. This leads to another infinite regression problem, as we cannot actually sense physical things directly: we are always reacting to images on our retina, models in our mind, sets of beliefs we have about reality, or similar abstractions. So even when we think we are directly interacting with actual reality, we are actually dealing with somewhat less fidelity than the ideal 1:1 map.

 

But, in any case, what I consider the most ironic thing about all these riffs on Lewis Carroll's original attempt at absurdity is that the whole concept of 1:1 scale maps is no longer so absurd. Modern software tools like Google Earth really do allow us to depict arbitrary scales, even 1:1, by storing the map virtually and letting us just zoom in on the parts we currently need. Technically I don't think we have quite the satellite resolution to consider Google Earth maps to be 1:1 scale, but we're getting pretty close. And I'm sure philosophers will continue to point out that the Google Map is not the territory-- but for practical purposes, it seems close enough to me. I wonder what other ridiculous absurdities from Alice in Wonderland or Sylvie and Bruno will be rendered mundane by future technologies.

 

And this has been your math mutation for today.

References:

 

• http::/ask.metafilter.com/62666/The-Map-of-a-Single-province-covered-the-space-of-an-entire-City : Online discussion of 1:1 maps
• http://tmbw.net/wiki/Lyrics:Arkansas: They Might Be Giants song lyrics
• https://notes.utk.edu/bio/greenberg.nsf/0/f2d03252295e0d0585256e120009adab?OpenDocument: Borges quote on 1:1 maps
• http://www.hoboes.com/FireBlade/Fiction/Carroll/Sylvie/Concluded/Chapter11/: Sylvie and Bruno chapter with 1:1 maps
• http://en.wikipedia.org/wiki/Map%E2%80%93territory_relation: Map & Territory at Wikipedia

175: Another Great Math Movie

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Before we get to today's topic, I'd like to thank listeners Highland Steve & Maurice Frank, who wrote some more nice reviews. With their help, Math Mutation made the 'What's Hot' list in the Science and Medicine category on iTunes this week. Thanks guys!

I was excited to see a few months ago that the movie Sphereland was finally released on DVD. The original book Sphereland, written by Dutch teacher Dionys Burger in the 1950s, was an unofficial sequel to the classic math novel Flatland. The all-star cast of this film was a nice surprise. Like the Flatland movie of a few years ago, reviewed in Math Mutation podcast 41, it stars Kristen Bell. She plays a talking two-dimensional hexagon-shaped mathematician, a slight departure from her recent TV role as a sex-crazed business consultant on "House of Lies". Danny Pudi, the nerdy college student on TV's "Community", is a litle more typecast, here playing nerdy hexagon astronomer Puncto. Other cast members that seem quite appropriate include "Star Trek Voyager" captain Kate Mulgrew, who plays a 4-dimensional hypersphere, and Danica McKellar, former "Wonder Years" actress and math education advocate, as triangular astronaut Aero. The movie is a half-hour short film aimed primarily at the educational market.

Before we talk about this movie, let's quickly summarize its predecessor and the universe it created. As you may recall, Flatland was the 1884 novel by Edwin A Abbot, describing a two-dimensional world that consisted of a gigantic flat plane inhabited by living circles, squares, triangles, and other geometric figures. The Flatlanders could not conceive of the third dimension, since their entire lives were spent in the two-dimensional world of their plane. But one day the title character, A. Square, is visited by a sphere, who tells him about the third dimension. By this analogy, Abbott hoped to open readers' minds to the concept of a fourth dimension, a new direction in space perpendicular to the three that we know, which we can't conceive due to our lives being confined to our three-dimensional space. We are all like Flatlanders, not realizing that space continues infinitely in a direction we just are not aware of.

Sphereland attempts to expand this story into an allegory about curved Einsteinian spacetime, and introduce readers to the possibility that the fourth dimension not only exists, but that our space is curved in that direction, as physicists now believe is the reality. As is typical for movie adaptations, this film makes the characters a little more interesting than the original book, even adding a romantic subplot. In the movie, it is revealed that rather than living a a generic plane, the Flatlanders live along the rim of a large circular planet, and there are other distant circular planets floating about in their two-dimensional space. Astronomer Puncto starts measuring large triangles, and discovers that some of them have angle sums greater than 180 degrees, and is thought to be a sloppy or crazy scientists by his fellow Flatlanders, since everyone knows that geometry has proven a triangle's angles always total 180 degrees. Convinced that his measurements are accurate, Puncto tracks down Hex, the granddaughter of the Square from the first movie, and together they discover the truth, that their universe is curved in the third dimension. If you recall Math Mutation podcast 35, you will remember that if a triangle is on the surface of a sphere rather than on a flat plane, its angles can total more than 180 degrees. Just as our universe is curved in 4-dimensional space, Flatland is curved in 3-dimensional space, and the Flatland universe sits on the surface of a sphere.

So, aside from the fact that I'm a fan of the original book it adapted, is the movie any good? I thought it was really fun to watch-- as I mentioned, they added a lot of personality to the characters, beyond their basic utility for geometric description that was the priority of the book. On the flip side, there were a lot of fascinating details from the book that could not be covered in the movie, probably due to its half-hour classroom-oriented length rather than being a full-length feature film, so I would strongly recommend that if you enjoy the movie, you read the book too. The movie really only covers one chapter of a much longer work. Burger's original book, for example, contains an element of political satire, describing at length the flat society's evolution from absolute rule by circles to more equitable distributions of power among the geometric figures. There is a very amusing discussion of Flatland's Age of Exploration, when various expeditions discover properties of distant lands, and finally when explorers heading East and West unexpectedly encounter each other, prove that their world is a disc. There is also a discussion of Flatland's animal life, to which the film gives a brief nod when showing Hex's 2-D pet dogs in the opening scene. Actually, I think there's enough material there not just for a feature film, but for an epic TV series. Maybe Showtime will consider producing it after they finish the final season of Dexter.

But perhaps the most important review of the Sphereland movie comes from my 6-year old daughter. She came to me yesterday and asked "Can we watch the movie with the hexagon again?" I was a bit surprised, as I'm pretty sure that a lot of the dialog about curved space and higher dimensions was over her head the first time I watched it with her, but apparently the cute animations & characters really captured her interest. My wife also chimed in, "Oh, so that's why she's been saying weird stuff about the fourth dimension when I asked her about the Horseland cartoons." So on some level, I'm pretty sure the movie really is achieving its educational goal of speaking to children, and not just adult math geeks. Not to say the movie was too kid-focused: I thoroughly enjoyed it as well, and would choose it over Horseland cartoons any day.

So, if you're enough of a math geek that you listen to this podcast, Sphereland is definitely a movie I would recommend. If you're not lucky enough to have it playing at the local megaplex, you can find info on ordering the DVD at spherelandthemovie.com, also linked in the show notes.

And this has been your Math Mutation for today.

　

References:

• http://www.spherelandthemovie.com/index.html: Sphereland movie site
• http://www.amazon.com/Sphereland-Fantasy-Curved-Expanding-Universe/dp/0064635740 : Sphereland book at Amazon
• http://en.wikipedia.org/wiki/Flatland : Flatland at Wikipedia

174: Too Much Math?

Audio Link

Before we start, I'd like to thank listeners JMS and Daniel54600, who posted nice reviews recently on iTunes. Remember, if you like the podcast, seeing good reviews does help motivate me to record the next episode!

Anyway, on to today's topic. Recently I read online about a research study that was published this past summer in the Proceedings of the US National Academy of Sciences, called "Heavy Use Of Equations Impedes Communications By Biologists", by Tim Fawcett and Andrew Higginson. The authors analyzed a large number of recent biology papers, and tried to relate the number of equations in each paper to the citation count, or number of later papers that referenced it. They concluded that the more equations you have, the fewer people will later read and use your paper: each increase of 1 equation per page caused a 28% penalty in citations. So, does this really mean that scientists are afraid of math?

As you would expect, the Fawcett-Higginson paper led to quite a bit of discussion in the blogosphere. Do scientists really hate equations? One caveat that was often pointed out is the fact that theoretical papers with lots of equations tend to be more specialized, which inherently leads to lower citation rates. You can probably think of many other factors in the audiences and targets of papers which might have led to the reported results. On the other hand, there are also numerous articles supporting the implication that biologists don't like math in their papers, one pointing out that some Ph.D.s in the life sciences never had to take a math class beyond calculus. Going through the equations can be difficult-- one blogger linked in the show notes suggests that reading papers should be accompanied by "Eye tracking and mild electroshock therapy. If scientists skim over pages of equations or stare into space for too long while reading a technical paper, they get a gentle jolt of electricity to bring them back to the important equations at hand." A bit of hyperbole, but a good way to highlight the amount of mental discipline it takes to follow a detailed series of equations in a paper.

Reading about this paper brought to mind memories of my graduate studies in computer science, back in the early 90s. One day I had come to my advisor with a proposal for my dissertation topic, involving efficient checkpointing methods for parallel programming systems. After looking over my proposal, he looked up at me and said, "It's a good topic, but can you work in more equations?" I was a little confused, and asked where he thought I had skipped a needed equation. "Nowhere specific, it's just that more equations are expected." Needless to say, that was one of the more frustrating conversations of my aborted academic career.

I think what most discussions of the paper are missing is to ask the question: what is the role of an equation in a paper? Is it a quasi-mystical invocation, as my advisor seemed to think, adding credibility to the thesis regardless of its content? Is it an ego-boosting method for the author, allowing him to claim a deep understanding that is inaccessible to the casual reader? Obviously, neither of these is a very good answer. My answer would be that in a science or engineering paper equations perform a very important role: they allow you to rigorously prove that some result is a consequence of your assumptions, definitions, and experimental work. Mathematics is what you use to start with precise definitions and assumptions, and show their logical consequences. When you can derive a new equation to describe a concept in a rigorous, universally applicable way, that is one of the most powerful results possible in a piece of scientific work.

But do scientists and engineers usually work by spending their days deriving series of equations? Actually, before there is an equation, there is usually some kind of intuition about how the world works. And it is refinining and clarifying these intuitions and experiments in a precise way that lead to the mathematical results. Often when writing a paper, proud of the difficult and rigorous work it took to derive the equations based on your original theories, it is easy to get caught in the trap of wanting to put those equations up front as the primary focus of your communication. It's not the fault of the authors that they make this mistake, as it has been programmed in us starting as early as high school geometry. Were you in a class where you were presented a series of proofs that seemingly sprang in ordered, fully understood form from Euclid's mind? I would bet that each of his results certainly came from many hours of doodling and experimenting with different pictures and measurements. How many triangles do you think the ancient Greeks drew before they were ready to prove the Pythagorean Theorem?

When you're writing a paper in any scientific or engineering discipline, you need to keep in mind that your primary job is to communicate with your audience-- to enable the reader to understand the ideas in your paper. If you have performed some insightful math to verify the surprising and more general consequences of your initial intuitions, it is important to tell your readers about that, BUT also important to help them understand your original intutions that led them to the equations. The first reading of a paper by an individual reader will nearly always be a casual attempt to get the basic idea; only after they have been convinced intuitively of the value of the new contribution will they take the time and mental energy to follow all the details. I've sat through way too many lectures at conferences that presented a dense series of derivations and equations one by one, rather than describing at a high level what they are talking about. The results were probably really good, but without any intuition to grab on to, it's nearly impossible to stare at a long series of equations on the screen or in a paper without zoning out.

So I wouldn't take the Fawcett-Higginson result as a statement against equations-- but as a call for theoreticians to keep their audience in mind when trying to describe their results. If describing the intuition first and then putting the detailed equations in an appendix makes the paper more understandable, they should not be afraid to do it. It doesn't detract from the math to provide the audience with an intution about what inspired it, and often can make it much more likely for the work to ultimately be understood and built upon. And isn't that the real point of a scientific or engineering paper?

And this has been your math mutation for today.

References:

• http://www.pnas.org/content/early/2012/06/22/1205259109  : Abstract of the original paper
• http://blogs.scientificamerican.com/observations/2012/06/28/scientists-theyre-just-like-us/  : Online discussion
• http://altmetric.com/blog/interactions-foggy-with-a-risk-of-mathematics/  : Another online discussion
• http://theoreticalecology.wordpress.com/2012/06/26/more-equations-less-citations/  : Yet another online discussion