K 3 3 G

Daytime observations

Posted on March 30th, 2007 by ephemere.
Categories: science/math, anecdotes.

(title inspired by Taylor & Wheeler’s Parable of the Surveyors, Spacetime Physics; crossposted from Miamor)

(title inspired by Taylor & Wheeler’s Parable of the Surveyors, Spacetime Physics)

Imagine a room of students waiting for an exam to start. They talk in hushed tones about concepts and equations none of them really understand, reciting formulae like mantras. The room is cold with excess air conditioning and barely stifled dread.

The proctor — a graduate student — walks in carrying the exam papers and questionnaire sheets. He doesn’t look at the students; instead he goes to the blackboard and begins writing.

It’s standard procedure to write the starting time, the finish time, and the duration on the board, but this is ridiculous.

My first reaction was: “There’s something wrong here. That’s not a well-defined sequence and its limits don’t make sense. And the integral is really ugly. I don’t even think you can do that.”

And then I paused and thought, “No, really, there’s something wrong here; he shouldn’t be doing that.”

It wasn’t just the math that was off; it was his writing it in the first place. I found it terribly arrogant; sure, intimidate the undergraduates (most of whom have only limited knowledge of derivatives, much less sequences and limits) with unnecessarily arcane symbols. Be inconsiderate and obnoxious and rub it into their faces that they are worth so much less than you are.

Showing off is self-destructive because, of course, intellectual humility is always, always fundamental. You should never lose sight of that, never flaunt what you know — if you do so, you’re just displaying your ignorance (of the fact that there’s so much more you don’t know, for one thing). Once you get to the point where you feel you can start showing off your mastery of your field, you hit a wall and learn less and less every day. Or worse — lose everything you have.

The proctor collected our answer sheets ten minutes early. I didn’t bother correcting him; he didn’t look amenable to corrections anyway.

–

So this semester is over and I have one summer and two semesters left before I graduate. I’m starting to find some sort of grounding in economics: interesting problems, situations I’d like to explore. It’s like being given a whole new playground in which to wreak havoc…! There’s much to be said about being different; as an economics major who can read (and understand, and work on) physics papers, I straddle two worlds.

And there are so many things to learn!

–

In other news:

This made my day. It’s a formula that… graphs itself!

Tags: anecdotes, economics, opinions, people, science/math, students

0 comments.

If We Had Blackboards Like These…

Posted on December 6th, 2006 by sofimi.
Categories: everything else, science/math, tech, ink & paper.

ASSIST: A Shrewd Sketch Interpretation and Simulation Tool
(also known as Assist Sketch Understanding System and Operation around the net)

I’ve never really embedded a YouTube video before because it kinda ruins the look of most blogs, but this is too geeky an opportunity to pass up. The video below features a man, possibly an MIT professor, demonstrating a computer program hooked up to a whiteboard. The program interprets what is drawn on the board as objects within a physics simulation scene. When he presses the “Run” button, the computer animates the drawing according to the laws of physics.

I hear that the Microsoft Physics Illustrator for TabletPC is quite the same thing. I wouldn’t know since I don’t have a tablet PC!

We’ll have to be content with the Line Rider game, which I’ve been hearing about lately. I’ve tried it once, just now. Draw some lines and have a little critter on a sled ride on ‘em. Check out a blog dedicated to the whole thing.

Footnote: I noticed you guys are unleashing your “inner geek,” while I have a stronger “outer geek” tendency, if you catch my drift!

Tags: computer, demo, drawing, everything else, geek, ink & paper, paper, physics, programming, science/math, simulation, tech, video, visualization

6 comments.

The nonconformist’s plaything

Posted on December 1st, 2006 by ephemere.
Categories: science/math, ideas.

Hello, sorry for the silence! I’ve been semi-away from the ‘net these past few days, since I had to do a thesis-related presentation last Wednesday and spent the days before that with my nose buried in papers. Slacker though I am, I wouldn’t have needed to cram quite as much as I have if not for the fact that I decided to change thesis topics just last week.

Yes. I’m not the most reasonable of people.

It was worth it, though; from a thesis problem that (to me) wouldn’t go very far, I’m now studying a very interesting combination of statistical mechanics and game theory. My thesis problem is concerned with something known as the minority game. Just to clarify, game theory does not have anything to do with MMORPGs or RTSs or things that go boom. It deals with the patterns and strategies involved in certain situations where your reward depends on not only your choice but also everyone else’s decisions.

The minority game is the mathematical formulation of the El Farol bar problem, which deals with the dilemma of anti-social people who want to drink alone… No, just kidding. This is how it goes. Let’s pretend that there’s a certain bar that you, along with a lot of other people, go to exclusively (meaning you don’t go to any other bars). Every night, you have the choice between going to the bar (action +1) or staying at home (-1).

However, you, being a strange type of person who goes to a bar to drink and not to pick up hot bois or chikzorz, don’t want to go to the bar if it’s too crowded. If given the choice between going to a crowded bar and staying at home, you’d rather stay at home. In other words, you want to be in the minority: if few people go to the bar you want to be one of them (since the bar won’t be crowded), and if few people stay at home you’d rather stay at home as well as to avoid the pack of sheep crowding the bar. This is why the game is called the minority game — the minority wins.

Also, let’s assume that everyone else thinks like you (what? no really, let’s pretend… stay with me here). Nobody wants to go to a crowded bar, but nobody has any idea how many people are going to the bar on a certain night anyway. So how can you decide what decision-making strategy to follow? Also, what does the bar attendance for each night look like? Are there winners and losers, or does everyone more or less make as many bad decisions as good ones?

And that’s where statistical mechanics methods come in. My thesis problem involves the solution of a set of rather intimidating equations that I don’t quite understand as of the moment, but anyway…

Minority games, though they’re a class of very simple models, have a lot of applications. You can use them to analyze traffic (should I take the shortcut route or the normal route?), lines (like at the grocery or bank or — gasp — the horrifying enrollment queues at my university), and financial markets (buy or sell?). Also, you can modify minority games to add a lot of other choices, as long as you stick to the rule that the minority always wins. They’re very useful in disciplines like economics and social science.

…Though, of course, if I were given a choice on something to do during a Friday night I’d pick something like stargazing or going to an amusement park or playing my current game obsession. My professor actually kept interrupting me while I was presenting this–

“Why don’t you like crowded bars, anyway?” he said. “Who goes to bars to drink? You go to bars to pick up people, right? Hmm, or… you can just bring your date home. That’d probably be just as fun, if you live alone.”

Really now. You should’ve heard what he had to say about the prisoner’s dilemma.

Tags: complex adaptive systems, el farol, game theory, games, ideas, minority game, models, science/math

4 comments.

Mathematical wallpapers

Posted on November 21st, 2006 by ephemere.
Categories: science/math.

Though some of the wallpapers K33G uses for its Vistered rendition are just images I deem pretty, others have less random origins. The tiling wallpapers are actually modified versions of patterns called Penrose tilings, which were developed by the mathematician Roger Penrose.

(If the name Roger Penrose sounds familiar, it might be because you’ve read one of his books. He’s written several books, one of which he co-authored with Stephen Hawking of Hawking radiation and A Brief History of Time fame.)

What are Penrose tilings? They are a class of objects that tile the plane aperiodically, using only two tiles. You might ask what the fuss is all about, but think of the kinds of geometric tiles (floor tiles or wallpaper tiles) we’re all familiar with. Common sense tells us that, since we’re dealing with a limited number of shapes that fit together, the pattern has to repeat itself eventually. On the left is an example of a periodic tile. It’s pretty obvious that if you repeat it you’ll see the same pattern over and over again.

A Penrose tiling, on the other hand, can go on to cover an infinitely large area without repeating itself. You don’t even need complicated shapes for that to happen. The first kind of Penrose tiling, announced in 1973, was made using six shapes, but then a year later Penrose had reduced the shapes needed to two rhombi: a thick one, called a “kite,” and a thin one, called a “dart.” These rhombi follow specific rules regarding angles — they’re all multiples of pi/5 — and are tiled according to certain rules.

First, no two rhombi must be tiled to form a parallelogram — if that happens you’ll get a periodic tiling. Secondly, the rhombi must be marked and tiled in such a way that only edges (or corners) with the same mark can meet. The most common marks used are colors to label the corners, and arcs of different colors to label the edges. After all that hard labor, plus a little coloring, you get something like this:

It’s interesting to see how these abstract problems of mathematics — the problem of covering an infinite plane with a finite number of shapes, for instance — have such beautiful solutions. Penrose tilings also turned out to be useful in the study of quasicrystals, transitioning from esoteric theory to practical applications. Something like that ended up being useful…! Math is funny that way.

Hmm, I should look for a good M.C. Escher image to turn into a wallpaper next…

P.S. Of course the wallpapers we’re using are imperfect examples of Penrose tilings — if they were really Penrose tilings we’d need very large images. The trick is that the image was cut so that the edges would appear as if they fit together, when in fact they aren’t supposed to. It’s very hard to see in action on the page, but if you look at the original background images you’ll see that the edges form shapes that aren’t the original rhombi. ;p Images taken from Mathematical Imagery, The Encyclopedia of Astrobiology, Astronomy and Spaceflight, Science U, and Tilings.

P.P.S. Oh, and the black snowflake wallpaper is a Koch snowflake. Heeee.

Tags: ephemere, geometry, penrose tiling, science/math, tiling, wallpaper group

9 comments.