Flatland - Exploring Other Dimensions

Plato's myth of the cave is an allegory that tries to make the point that there may be more to reality than meets the eye, that our experience is simply of a very limited and lowly aspect of reality. Given the knowledge and technology of the time, Plato's use of puppets and shadows in his ancient writing is brilliant in conveying the difference between the real and the apparent, but in the 1800's, Edwin Abbott took this idea much further, and in a way more rigorously and convincingly, as he was able to tell it through a creative exploration of mathematical space:

Compare the experience of a creature from three-dimensional space against one from a two-dimensional world. Easy, and completely reasonable to understand why the two-dimensional figure was skeptical of "more" dimensions, even if it was wrong, right? Well, apply the same reasoning to the difference between four-dimensional space and the three-dimensional space we're all used to. Wouldn't the reasoning you applied in the first case also apply to this one? And just like that time we were introduced to these ideas by the irreplaceable Carl Sagan, when he tried to show us the projection of a tesseract, here's a fun little animation to inspire your imagination and help you explore the beauty of mathematical space that can be understood by the mind even if it cannot be always perceived by the senses :)

Now, are there 'really' more dimensions than the three spatial ones we're used to or is this just a fancy mathematical abstraction? That's a topic for another discussion :p

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Jon Stewart Pwns Bill O'Reilly on Profiling... and Math

We all know that Bill O'Reilly is a petulant, loud-mouth who bullies and yells at people he disagrees with (even when they are invited guests in his show), and it's kind of hilarious to watch him and Rush Limbaugh go at it every now and then, but I'm starting to wonder whether the Jon Stewart vs Bill O'Reilly little debates and repartees are fair... I mean, Papa Bear is like five feet taller, a lot louder, always willing to make up straw man arguments, and overshout everyone... and Jon Stewart, well, he takes the abuse graciously, but also drops an atomic bombshell every now and calls him out on the inconsistency of his bullshit...

Here's just one more example of that ass-kicking:

So, is O'Reilly a masochist or just clueless?

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What Is Zeno's Dichotomy Paradox?

When I first introduce my students to the weirdness of philosophy and how even our most deeply-entrenched beliefs might be subject to serious questioning, I usually like to begin by posing to them Zeno's attempt to refute the idea that motion is possible (here's a fun little animation to get you started), and then continue to defend him against the objections raised by the students.

This has the dual benefit of being both a nice introduction to questioning dogmas and a nice exercise in critical thinking: since the students are convinced beyond a shadow of a doubt that Zeno is wrong, they concoct all kinds of arguments to refute him. The problem, of course, is that it's notoriously difficult to explain why exactly he's wrong, where he's made some kind of mistake.

Now, I get the idea of a limit in calculus, and I get the idea of the sum of a series as opposed to the sum of the individual members of a series, but I have noticed that we seem to have such an aversion to Zeno's conclusions that even professional philosophers and mathematicians tend to engage in a kind of prestidigitation in which they explain a mathematical method for solving the paradox that ends up being more smoke and mirrors than a genuine solution. The problem, and the following animation is an example of this, is that they tend to frame the paradox in a way that's question-begging, one in which they assume motion in order to prove motion.

In the following animation, it seems clear how you can refute Zeno once you grant him motion to the first halfway point, but for Zeno, he wasn't willing to concede actually having crossed that halfway point. For him, the challenge is that BEFORE you reach that halfway point, you must reach the halfway point of that halfway point, and before you can do that, you must reach that other halfway point, lather, rinse, repeat ad infinitum, and Zeno's conclusion is that you NEVER actually get to leave the starting point: you haven't gone anywhere.

As you can see, if you don't grant any motion at all to begin with, the above explanation, nice as it is, doesn't quite do the trick, does it?

When trying to refute someone, you should always ask how that person might respond to your objections, and if you're charitable, right or wrong, that will give you a better sense of the real strength of your own arguments. When it comes to Zeno, I tend to think that if you were to explain to him the asymptotic concept of a limit in calculus, he could just apply his paradox to that and be right back in business with his own paradox, now fortified with steroids...

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Megyn Kelly Makes Karl Rove Her Bitch

I have an ongoing debate with a friend. We're both liberals, and I think one can theoretically separate a person's political ideology from their physical attractiveness, and not like the former while very much liking the latter, or vice versa. My friend thinks I'm full of crap: once you know what's inside a person's heart and soul, especially when they're conservative ideologues, the outside is just as hideous as the inside.

And the object of contention between us has been Megyn Kelly. I think she's totally hot, even though I disagree with most of the things she says. My friend says she makes him want to barf. Here's the awesome part: on election night she did something that should completely vindicate me in my friend's eyes: she made Karl Rove her bitch :)

The Daily Show with Jon Stewart
Get More: Daily Show Full Episodes,Political Humor & Satire Blog,The Daily Show on Facebook

I'm not big on schadenfreude, but that was awesome!

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Hexaflexagons to Blow Your Mind

Whether you are particularly fond of it or not, you have to admit that when math gets involved in origami, amazing things start to happen (as we've seen in this previous TEDTalk presentation).

So imagine what would happen when origami and math meet Vi Hart's quirky style to make mathematical concepts accessible and interesting: pure awesomeness. Thanks to the following lesson on the hexaflexagon, you'll never think of hexagons the same way again...

And as if that weren't awesome enough, here's a second part:

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How to (and Why) Prove a Mathematical Theory

When it comes to the world of our sensory experience, the idea of "proof" is somewhat misplaced, since there is always the possibility of new evidence contradicting something we previously had every good reason to believe, not to mention the fact it could all be a dream or a simulation in The Matrix!

But in the world of abstract ideas, especially numbers, the kind of reasoning used there does lend itself to proofs because you get to specify exactly what you mean by a certain concept, and then all you have to do is follow a few rules of inference to deduce the logical consequences of your idea.

And we owe a debt of gratitude for the whole idea of theory and proof to the ancient Greeks, people like Pythagoras, Plato and Archimedes. But today we get to learn a little bit about the man who is widely considered the father of geometry, a title appropriate to the scope and importance of his work: he formalized the rules of geometry that mathematicians have relied upon for over two thousand years. That man was Euclid.

QED

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Release the Clinton!

First it was the economy, stupid. But apparently not only did trickle-down Republicans not get it, they regressed and went backwards, so it's time for President Bill Clinton to come back and school everyone in what can only be described as a masterful rhetorical and substantive performance.

And because Republicans don't seem to get the economy stuff, Clinton identifies the source of their confusion: it's arithmetic, stupid! :)

Anyway, here's something that has been sorely missing from the Romney/Ryan campaign: honesty, education, wit, charisma, and an avalanche of specifics that no amount of empty soundbites can compete against.

Take them to school!

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Olber's Paradox - Why Is the Sky Dark at Night?

I know that may seem like a silly question at first glance, but there is a long history of seemingly silly questions that end up producing some mind-blowing and really counter-intuitive results that really get us to question our most basic presuppositions about the nature of reality, and in many cases redefine our understanding of the universe. This case is no exception.

So back to the question, sure, the sun is hiding on the other side of the planet, but there are stars in literally every possible direction, so photons are hitting us all the time from every possible direction. But if this is the case, shouldn't the night sky look bright from all these stars?

Well, as the following simple animation brilliantly demonstrates, the sky really is bright from all those stars... we just can't see all that light, not with our eyes anyway, but it is there, and what that tells us about cosmology and the universe is absolutely fascinating!

Bet you didn't see that coming, huh? No pun intended, actually. I was just blown away by the connections. This stuff is awesome :)

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The Birth of Calculus - Newton and Leibniz

Before you run away in fear of having seen the word calculus, let me preface this entry by saying that you don't have to understand how it works, or even be able to do it, to understand and appreciate why it's important and why calculus represents one of the greatest intellectual achievements of human history: the modern world would not exist without it. You may choose to turn a blind eye to it, but that doesn't make it go away, so why not embrace it?

And while you probably know that it was invented, that's right... invented, by two geniuses, Newton and Leibniz, working (mainly) independently of each other, do you know the differences in their approaches? If you've never quite had an appreciation for the beauty and elegance of math, the following short documentary, should be able to change that:

And if you don't know anything about the priority dispute between these two men, here is a short introduction to that regrettable nightmare:

Did you know both Newton and Leibniz were first-borns? From the perspective of evolutionary psychology, it makes a lot of sense why their priority dispute got so heated and cantankerous. Now Wallace and Darwin, on the other hand, were way later-borns, and look how that turned out :)

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Breaking the Code - The Biography of Alan Turing

Computers don't grow on trees, and even though they are ubiquitous today, that wasn't always the case. In fact, they've only been around for less than a hundred years, and although there are certain folks to whom we owe a huge debt of gratitude for laying down the conceptual foundations (the philosopher Leibniz for inventing the binary language system upon which programming depends, for instance, or the enchantress of numbers Ada Lovelace's brilliant insight into the power of computation rather than mere calculation), the individual most directly responsible for modern computation is Alan Turing.

His was a remarkable life, full of genius, insight, inspiration, courage and intellectual creativity. His contributions during World War II (like the fact he broke the Nazi code and was privy to the information sent in secret messages to Hitler and his thugs even before they received it) may be directly responsible for saving hundreds of thousands of lives, and of accelerating the end of the war.

For all of his importance, however, his life was also very tragic. Because of the top-secret nature of his mathematical work during the war, he was never officially recognized. And to add insult to injury, his homosexuality was used to ban him from his own work, to treat him worse than a common criminal, and eventually it led to his suicide. This is his life:

For more on Turing's mathematical and philosophical importance, check out two of my favorite documentaries in this whole blog: Dangerous Knowledge with David Malone, and The Secret Life of Chaos with Jim Al-Khalili.

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What's Up with Pythagoras?

The first time I read Bertrand Russell's History of Western Philosophy, I could not get over the chapter on Pythagoras. As Russell claims, "Pythagoras... was intellectually one of the most important men that ever lived, both when he was wise and when he was unwise. Mathematics, in the sense of demonstrative deductive argument, begins with him." This is probably the aspect of Pythagoras that's well known. The unwise part comes from his weird and wacky mysticism (which he managed to intertwine with his mathematics both to great success and to halt its own progress) and to his dogmatism: you did not want to upset the guy or threaten the beauty and perfection of his rational philosophy.

The guy who did, by pointing out the existence of irrational numbers as a logical consequence of Pythagoras' own rational theorem, angered Pythagoras and his posse, and got himself viciously killed by them.

So how do irrational numbers flow out of the Pythagorean theorem? Well, the awesome Vi Hart explains in her deliciously unique way in today's episode of the doodling in math series.

And if you're curious about some of the rules of the Pythagorean order, check these out:

1. To abstain from beans
2. Not to pick up what has fallen - sorry grandma!
3. Not to touch a white cock - I guess once you go black... sorry, but isn't that racist :)
4. Not to stir the fire with iron - being that iron is a metal, I'll let that slide

And if you want to inspire your friends, family, kids, students, co-workers, you can also show them how the late and beloved Jacob Bronowski explains the mathematical, philosophical and historical significance of this great discovery. Trust me, it's inspirational...

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Measuring the Universe

If you read/watched our recent entry on the majesty, beauty and scientific importance behind the Transit of Venus, you may have come to realize that scientists are a clever bunch: give them two or three things, and they'll give you a whole world full of goodies. How's that for a return on your investment? Suck it, Wall Street!

Right, so the scientific importance behind the transit has to do with the question of how we manage to measure things in space (like the distance between the Earth and Sun), but have you wondered exactly how it is that scientists manage to measure even greater distances, like how far away other galaxies are, or even how old the universe is? Well, wonder no more because here's a quick little introduction to the general idea:

And if you want the more serious treatment about how scientists have figured out the age of the universe, you might want to check out In Our Time, with Melvyn Bragg.

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The Prisoner's Dilemma

One of the interesting aspects of social interactions is that our interests are not always aligned with those of other agents, and when the result of an interaction is a function of your choice and that of someone else whose motives may be different from your own, you face a dilemma... what to do?

Well, you kind of know what you'd like to do, but you also know what he'd like to do, and you know that he knows what you'd like to do, and so you think about what he's going to think you're going to do, so you try to anticipate that by doing something else, except that he probably realizes you're going to try to cheat him, so he's going to try to get you before you get him, lather, rinse, repeat ad infinitum and you're not one step closer to figuring out what to do.

One of the fascinating and paradoxical implications of trying to rationally maximize your own utility function is that, when confronted with certain scenarios, such as the famous prisoner's dilemma, and by virtue of pursuing your own individual interests, you end up screwing yourself more than if you simply decided to cooperate. Oh, but it's never that simple. Anyway, here's a basic introduction to the idea of the prisoner's dilemma:

Now, you may think that because you now understand how this dilemma works, you could beat it if you were to find yourself in such a scenario, but as Dilbert shows, that's not quite right (which is exactly why such scenarios are so interesting and why so many academics in all sorts of disciplines have devoted so much of their time to think about and research such cases):

In political philosophy, one of the most famous instances of a philosopher recognizing the nature of these interactions was Thomas Hobbes, who argued (a few centuries before game theory was actually invented... isn't it awesome how philosophy is so often way ahead of its time?) that the way to solve these problems is to make sure there is some sort of mechanism to enforce cooperation.

Hobbes' particular solution was that all rights should be transferred to a sovereign who would have a monopoly on power (and the violence that could be inflicted on defectors), but as the following absolutely brilliant and gripping game-show example demonstrates, there can be other ways to ensure cooperation:

Was that brilliant or what?!?

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The Transit of Venus

The Transit of Venus has been all over the news recently, mainly because this is the last opportunity to see it for virtually everyone who is alive today. The next one will not occur until 2117.

But do you know its historical and scientific importance? That's right, this isn't just a pretty light show. Over the last four hundred years, this rare occurrence held the key to determine the size of our solar system, and as the following primer shows, the individual pieces required to put the whole together spanned the gamut of scientific and mathematical thinking, and involved some incredible minds:

But this general idea goes far beyond merely determining the size of an astronomical unit: it can be used to detect and measure planets in other solar systems too:

And here it is, in all its spectacular beauty, footage of the 2012 Transit of Venus:

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To Infinity and Beyond

Apparently the title of this documentary also has something to do with Toy Story? Well, I haven't seen that movie, but I can tell you what this documentary is about... Infinity. The most obvious way to start thinking about infinity is through numbers: take any number, and you'll soon realize that there is no such thing as the biggest number because you can always add 1. No, a googol won't do it, nor even a googolplex or Graham's number. Yes, these numbers contain more zeros than there are atoms in the observable universe, but they still fall short, infinitely short, in fact, of infinity.

But the strangeness doesn't end there because infinity may not only be a mathematical conceptual idea: infinity raises all kinds of interesting scientific and philosophical questions and paradoxes (what Immanuel Kant called antinomies of reason) about the physical universe that even modern cosmology doesn't quite know what to do with: is space infinite? If not, what's beyond its edge? Infinite empty space? What about time? If it is, and the amount of matter in the universe is finite, then everything that could logically happen has happened, and will continue to happen, an infinite number of times... And hey, are there an infinite amount of universes? And as you'll see in this fascinating documentary, this is just the beginning of the weirdness:

For a fascinating treatment of the problems of infinity, check out the documentary Dangerous Knowledge.

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Adam Savage - How Simple Ideas Lead to Scientific Discoveries

We live in a world in which we presuppose as a given that great scientific discoveries require the existence of great, expensive technological equipment: lasers, the Large Hadron Collider, microscopes, computers, the Hubble and Kepler telescopes, synchrotrons, super-duper cameras, you name it...

But more than great technology, the real secret to discovery is creativity, because creativity helps you make good use of whatever tools are actually available to you, even if they happen to be, as in the case of Eratosthenes, two sticks on the ground. In the following TEDTalk animated presentation, Myth Buster Adam Savage recounts a few examples (starting with Richard Feynman, moving on to Eratosthenes,  Galileo, and Armand Fizeau's toothed wheel to measure the speed of light,) about how small ideas can give rise to mind-blowing ideas and revolutionary discoveries.

And if Carl Sagan is more your style, you might be interested in his take on Eratosthenes.

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Questions No One Knows the Answers to

Many parents tend to grow increasingly frustrated when their children suddenly become question shotguns, going from one question to another, faster than the parents can dish out answers. Comedian Louis C K can at least turn these episodes into hilarious material for his stand-up. Things, of course, tend to turn particularly frustrating when the shotgun turns into a fully automatic machine gun full of 'why.' I wonder if the frustration is not really about the children but about the parents' sudden recognition of the limits of their own understanding...

In any case, if you happen to find yourself surrounded by a bunch of pre-schoolers outflanking you with their innocent inquiries, maybe you can take advantage of their curiosity and teach them a little bit not only about the things we know (or even about the things we do not yet know), but about how we might be able to start thinking about these questions, and how the very fact of our ignorance can become the platform to future discovery, as the following animation from the TED people shows

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What Is a Leap Year?

As it turns out, there are people who don't quite understand the concept of a leap year. And while I was looking for pictures to post on this entry, I discovered that I wasn't aware that there is some movie by the same name. I wasn't impressed by the trailer, so why don't we take it one step at a time and just resolve for now the more important and interesting question regarding the astronomical concept of a leap year with a quick, fun and educational animation?

And if today is your birthday, happy leap birthday! Hope you feel younger than the rest of us.

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Faster Than the Speed of Light?

Some months ago, news broke indicating that scientific measurements in Italy had determined that some neutrinos could travel faster than the speed of light. The media jumped on the hype bandwagon and declared the "death of Einstein's theory of relativity." Physicists (aka, the experts on these questions) were skeptical because ALL previous experimental and observational data, including neutrinos and light traveling from super nova explosions really really far away and reaching Earth at exactly the same time, confirmed Einstein's predictions, so they concluded there must have been some error in the measurement, the analysis, the equipment, etc. on which the Italians relied. Of course, there were also cultural reasons for doubt: as Sergio Bertolucci, research director at CERN indicated, "I have difficulty to believe it, because nothing in Italy arrives ahead of time." :)

News are starting to pop up everywhere now that there was indeed a glitch in the equipment. Still, whatever the real answer turns out to be, and since everyone seems to be paying attention to the story, this presents a great opportunity to learn some fundamental lessons not only about relativity and other ground-breaking research in theoretical physics and cosmology, but how science itself works and makes the tremendous progress it does, so let's watch this delightful documentary narrated by Professor Marcus du Sautoy:

Sometimes I do wonder whether these physicists are insane in the mem-brane :)

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A Brief History of Mathematical Geniuses

Math, unfortunately, is usually taught, and consequently thought of, as a set of cold, bloodless tools to solve practical and theoretical problems that are somehow removed from real lived experience.

However, the truth of the matter is that math has a fascinating and mesmerizing life of its own, and those who have really confronted its secrets have found it to be a source of love, respect, admiration, inspiration, despair, confusion and wonder. Whatever else it may be, math is not boring... not if you actually pay attention.

In the following set of short documentaries, which you should definitely share with your mathophobe friends, Professor Marcus du Sautoy explores the lives, the insights, the contributions and the influence of some of history's greatest mathematicians.

Newton and Leibniz:
     

Leonard Euler:
     

Joseph Fourier:
     

Evariste Galois:
     

Carl Friedrich Gauss:
     

The Mathematicians who helped Einstein:
     

Georg Cantor:
     

Henri Poincare:
     

Hardy and Ramanujan:
     

Nicolas Bourbaki:
     


What other mathematicians would you have included in this list?

And for a touching and fascinating biography/tribute to Cantor, don't forget to check out the spectacular documentary Dangerous Knowledge.

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