Who needs vases, when you've got orange juice!

Congrats to Isabel on submitting her senior thesis, and on obtaining her thesis submission flower!

The internet went down, taking part of the house with it, but is still working

That's right, at the Mirror Maze, we get internet service via giant icicle. A few days ago, I heard a loud crash, and looked outside to find that the internetcicle had fallen, taking some of the siding of the building with it. If you look closely in the photos, slightly behind the main internetcicle you can see the large blocks of ice that used to be part of it before it fell. I took some pictures the next day, and we debated appropriate captions for the situation. After tossing around various "frozen" puns, we settled on Eric's concise, "the internet is down, but still working."

Fortunately or unfortunately, depending on how bizarre you like your life, the past couple days have melted the ice, but the thing is still hanging there on the cables, so we still have to be careful to mind the internet while taking out the trash.

And, while I'm posting photos, a couple weeks ago Eric, Mitka, and I decided that the best use of City Hall was as a snowball fight venue.

Info, advice, help thinking, or empathy: What do they want?

I've noticed that when people say something about their lives, there can be a huge difference in the kind of response they might be looking for. Eventually, I boiled it down to four categories: Information, advice, help thinking, and empathy.

For example, a few weeks ago, Eric told me that he needed to go to the DMV (technically the RMV because Massachusetts is weird). Here's what my response and the (hypothetical) results could be:

• Information: "You can check what documents you need online."
• Win condition: Eric brings a document he didn't realize he needed.
• Lose condition: Eric was going to bring an extra document just in case, but the website incorrectly told him that he doesn't need it.
• Advice: "I'd go earlier in the day to avoid people coming in during lunch or after school or work."
• Win condition: Eric takes my advice and avoids long lines.
• Alternate win condition: Eric doesn't take my advice, but still makes the best decision for himself. (He doesn't mind lines that much and prefers to sleep in.)
• Help thinking: "What day are you thinking of going?"
• Win condition: By asking leading questions, I help Eric turn a vague idea of needing to go to the DMV into a concrete plan for getting to the DMV and back that fits in his schedule.
• Empathy: "Oof, dealing with bureaucracies is no fun."
• Win condition: Eric feels like I understand how much he doesn't like bureaucracies.
• Bonus points: Having me as an emotional ally makes Eric feel better about his trip.

Unfortunately, people don't tend to communicate which response or responses they want, and sometimes they don't even know themselves until they get frustrated by getting the wrong thing. Information instead of advice: "I don't care, just tell me which one I should pick." Advice instead of information: "I already know what I want, just tell me if this option will do it." Help thinking instead of empathy: "I'm upset; I don't want to think about that right now." Empathy instead of help thinking: "I don't even know if I'm angry about it."

And, a fairly popular one, advice instead of empathy: Person A has a problem. Person B proposes a solution. Person A wants person B to understand their feelings, and so explains why the feelings are still present despite the existence of this solution. Person B understands that the proposed solution was unsatisfactory, and modifies it to deal with the concerns brought up by person A. Person A instead feels that person B shot down their explanation for why their feelings were valid. Person A may attempt to explain why their feelings are valid a second time, at which point person B becomes frustrated that person A keeps shooting down their proposed solution.

Ultimately, as in Cool Hand Luke, what we've got here is failure to communicate. Rather than working together either to understand A's feelings or to improve B's solution, they're talking past each other, often without even realizing why the conversation's "not working."

I've noticed that different people tend to have different types of responses that they default to, and that people seeking a certain kind of response gravitate towards the people who will give it to them. That solution isn't always enough, though. In the above example, person A is already in a conversation with person B, and it's possible that person B would be fantastic at empathizing with person A, but person B is just bad at recognizing that that's what person A wants. It would be great for A to recognize what's happening and say, "I'm not looking for advice right now. I'm looking for empathy," or for B to recognize what's happening and say, "You're not looking for advice, are you? What are you looking for?"

Approximations in physics classes, solved.

If you've taken a physics class, you may have encountered a frustrating situation like this one:

Instructor: Using small angle approximations, approximate $\tan x$ for small $x$.
Student: Okay.
$$\tan x=\frac{\sin x}{\cos x}\approx\frac x1=x.$$
I: Great, now approximate $\tan x-\sin x$.
S: Okay, based on earlier work,
$$\tan x-\sin x\approx x-x=0.$$
I: Nope, that's not a good approximation.
S: What? But you said $\tan x\approx x$ was good!
I: But you get zero, so it's a bad approximation. Instead, you should do something like
\begin{multline*}
\tan x-\sin x=\frac{\sin x(1-\cos x)}{\cos x}=\frac{\sin x(1-\cos^2x)}{\cos x(1+\cos x)}\\
=\frac{\sin^3x}{\cos x(1+\cos x)}
\approx\frac{x^3}{1\cdot(1+1)}=\frac{x^3}2.
\end{multline*}
S: What.

It was never clear to me in physics classes what makes an approximation good enough. It seemed like a hodgepodge of heuristics. At some point, though, I realized that what they are looking for is very simple: The first nonzero term of the Taylor series.

In each problem, there's always some quantity or difference that's "small," and the thing to be estimated (or sometimes its reciprocal) can be written as a Taylor series in terms of that small quantity. The first nonzero term of the Taylor series then provides a good estimate for whatever it is that you're trying to approximate.

The "first nonzero term of the Taylor series" approximations behave nicely. You can multiply quantities together and then approximate the product, or you can approximate them first and then multiply the approximations together, and you'll get the same answer. The same goes for compositions like $\ln(1+\sin x)$. And, the same goes for sums and differences, except when you would get zero. But, of course, the case where you do get zero is often the most interesting one, because then you compute the higher order effects.

In our example, that's exactly what happened. The Taylor series for tangent and sine are
\begin{align*}
\tan x&=x+\frac{x^3}3+\dotsb\\
\sin x&=x-\frac{x^3}6+\dotsb
\end{align*}
The first term of the difference is exactly $\frac{x^3}2$, and the earlier computation becomes an interesting example of how you can compute the third derivative of $\tan x-\sin x$ at zero knowing only the first derivative of $\sin x$ along with the identity $\sin^2x+\cos^2x=1$.

the benefits of non-nonstick

Yasha:  "You can just write that Mitka is using a frying pan as a cutting board, and as a frying pan, simultaneously."

Poems and a plug

Yesterday we had an interdepartmental "mixer" with the literature and philosophy departments. The mixer (when I arrived, very late) turned into exactly what it must have been meant to be, to the point of being corny. The literature majors aired opinions about higher topos theory and the mathematicians discussed postmodernism, with lots of beer involved. I got into a friendly argument with someone who studies (quantitatively, since he's at MIT) poetry. The issue was that I've never learned to read modern poetry. I think modern (and later) poets are hard to read, because they put ideas over linguistic aesthetics, and even the ideas are hard to follow because unlike classical poetry, whose form dictates exactly how it should be read and what hyper-textual connections should be made, most modern poets arrange what they write haphazardly, without a map for the reader to follow. The literature student claimed that the goal of poetry, as of math papers, is to make new connections between heretofore unrelated (or undefined) objects, and that while form is sometimes helpful, the only important thing is to get across the chain of associations, using whatever linguistic tools available, be they form, self-reference or anything else. He recommended me to read Ashbery, and here's a poem I found that I liked: http://www.poetryfoundation.org/poetrymagazine/poem/16013

But I still haven't found any poet who writes in free verse who I like nearly as much as Frost or Shakespeare or Dickinson. If anyone has suggestions, please share.

Also I realized I miss poetry, and decided to try to blog more about it. Here's the result:
http://mvaintrob.wordpress.com/ So far it's in Russian, but if I get around to it there will be other languages involved soon.

A Snow Adventure

We got a foot of snow Thursday and Friday, so Mitka and I decided to go sledding and playing in the snow with friends from the department. Hannah, Mitka, and I had a similar adventure with friends a month ago after we got two feet, and it was pretty awesome. This time, the weather was warmer, so the snow was worse for sledding, but quite excellent for building.

I came with a definite plan: I was going to roll a ball of snow so big that I couldn't push it any further. Alas, my Katamari Damacy adventure was over pretty quick, because it turns out that compactified snow gets heavy very fast. Hans figured that he could improve upon my giant snowball rolling technique, so he rolled a second one. We then had the excellent idea of building an arch between them, and then standing on it.

We claim this mountain with our sled-flags.

at Arches National Park.

 We then figured that if we roll enough giant snowballs, we could have a snow fort. We rolled two more supports. What has four supports and arches? The Eiffel Tower, of course! After feats of engineering unbecoming of pure mathematicians, we built a tower-shaped roof on the snowball-supports, and then punched out windows.

Mitka adds finishing touches to The Eiffel Tower (an artist's impression).

 The result was tall enough to stand up in.

Mitka looks out the window.

All in a day's work.

It was pretty cozy inside.

The view out of an archway.

The view facing up towards the four upper windows.

The Mirrormaze attempts film criticism

恭喜發財!

Last night we all watched a movie in one of Ruthi's movie nights, with wonderful snacks and company. In the past Ruthi showed us such classics as The Godfather and Annie Hall, which are too sacred for us mortals to critique. But this time we saw a movie that none of us had heard of before -- Children of Men, and on our way back through the snowdrifts of Cambridge we compared notes. Each of us saw the same movie, and, surprisingly, each of us saw more or less the same things in the movie. But the really remarkable thing was that each came out in the end with a totally different opinion of what the movie was about, and whether or not it was "good". I hated it (cheesy and moralizing), Yasha saw a cute, enjoyable scifi and Hannah gave it the highest praise I've heard her give a movie -- "it's not fake".

"Children of Men" is a dystopia based on a story by P.D. James. It's the future and everything sucks, in the "Republicans took over and now it's the end of the world" sort of way. Human meddling with the environment has caused universal infertility so the youngest person alive is 18 years old, and the British government (the one surviving government on Earth, we're told), has become a xenophobic, totalitarian farce. Non-citizens ("fugies") are put in cages and shipped off to Nazi-like refugee camps where they're starved to death, and except for nationalist propaganda the one advertisement we see is for "quietus", the perfected suicide drug (an amazing name, by the way). The protagonist gets involved in a rebel organization run by his ex-wife, and unwittingly gets drawn into an intrigue involving the miracle that no one dares to expect any more -- Kee, a pregnant (fugee) woman.

All three of us noticed three things. A: the movie is cheesy. The premise is a trope (an overused and somewhat forced interpolation between modern imperialism and nazis). There is a lot of cliche-ing going on ("Life is determined by chance -- no, it's all for a purpose" is actually a line of dialogue, repeated at least thrice). Key points of the backstory are announced on propoganda slides ("there are no more babies/the rest of the world is in chaos/only England remains"). Hell, there's even a tattered gypsy guiding the protagonists to a boat. B: the movie is highly dramatic. Everyone who can possibly die does so, in the most heartrending way possible. Every pause in the dialogue is predictably interrupted by gunfire or a car chase. Much of the movie has to do with babies: dying, crying and laughing (in that order). And yet, finally, C: the people in the movie are depicted lovingly and realistically. No one (except possibly the main character) is pure hollywood. No one magically knows where to go or what to do; the bad guys aren't all bad; the actresses aren't all hot and, there's not even a single romance!

All of us noticed these things, but we weighted them totally differently. I saw overdramaticized politics and a questionably logical premise, and my mind was set in the first few minutes of the film; I knew it'll be preachy and take itself too seriously. So I turned on my inner critic, and saw that it was preachy and took itself too seriously. Point B above I interpreted as pandering to cheap emotion instead of developing any real psychological content. Point C I saw as a small redeeming quality, but nowhere near enough to redeem the rest of the film. (That, and bad acting). Yasha had lower expectations. He thought of the movie as an action thriller/sci-fi. He liked the action, liked the story (which was, in all honesty, pretty good), and he said that though he noticed it's cheesy, he was moved by the plot. And as for Hannah, she was ok with the events being overdramatized for a different reason. The important part for her was that it depicted people honestly and didn't do the Hollywood thing of making people behave like they're expected in fiction rather than as they do in real life. She made an interesting analogy with grocery stores. Hannah gets food at Harvest and doesn't like going to Whole Foods. She says they arrange all the labels as you'd think they should, and try to make you feel good and righteous about buying them. But at less pretentious stores like Costco, they might have grandiose proportions and arrange stuff to make you get what you want, but they don't try to give you any sort of false idea of what you're buying and who you are: in other words, they don't lie to you. Most movies, she said, are like Whole foods and make characters into something either Hollywoodish or "refined", whereas this one is more like Costco, and makes people into just people.

We all came to the movie with different expectations -- both from movies as a whole, and from the genre we assigned this one -- and based our opinion on whether these expectations were fulfilled or not (or, in Hannah's case, overfulfilled). This brings back a favorite quote of my mom's, which she says is from Aristotle: "a masterpiece is a work that does it best in the conventions of its form". I still maintain that Children of Men is a cheesy action flick which takes itself too seriously (and whatever Hannah says about Costco, when you buy three giant trays of croissants, one is bound to go moldy) -- but I think I have a much richer experience after hearing Yasha and Hannah's points of view. And I can see myself, on a different night, in a different mood -- watching this movie and staying up all night because of its tragedy and beauty (certain scenes are very beautiful -- there are, as Mike pointed out, a couple of takes which last upwards of 10 minutes). Such, I guess, is the mystery of human nature.

Unorthodox Methods for Avoiding Snakes

Happy New Year!

It's the year of the snake according to the Chinese zodiac, which will be the theme for this post. Technically, the year of the snake doesn't begin until February, but let's pretend that we're just going off of a placemat at a cheap Chinese restaurant and don't know such things.

A few weeks ago, I told Hannah and Mitka about a dream I had. In the dream, I was sitting on some grass, and a friend was standing next to me with a pet snake around her shoulders. I reached up to feed the snake, (yeah I know that's not how feeding snakes works in real life) and the friend was like, "You should be careful, this snake is..." At this point the snake lunged at me. I quickly reacted and rolled out of the way, at which point everything disappeared and I found myself in the process of falling out of bed. BUMP as I rolled off the mattress onto the edge of the bedframe, and BUMP as I rolled off the bedframe and onto the floor, shoving my desk, formerly adjacent to my bed, to make room for myself.

Mitka, upon hearing this story: "Well, at least the snake didn't get you."

And so here's my wish for 2013, to be interpreted metaphorically as you prefer: May we avoid all our snakes, even if we have to fall out of bed to do it.

That's right, this whole post was secretly an infamous Long Russian Toast. Cheers!

Concepts of Gender

A couple weeks ago, Hannah mentioned that she thinks that she sees three independent traits of people that often are all called "gender." The first was the gender that a person identifies as. The second and third were the extent to which a person does manly things and the extent to which a person does girly things.

The discussion started with Hannah mentioning that her advisor had at some point in the past changed her first name from Katrina to Katrin. The new name still clearly identifies her advisor as a woman, but sounds less girly.

Hannah said that she sees herself as a woman, but generally does not like things that are considered girly. However, she doesn't see herself as a "tomboy" either, since she doesn't tend to like manly things either. The example she gave was athleticism.

As for me, I identify as a man, and I like to think that I do whatever I feel like doing, regardless of whether it's manly or girly. Realistically, though, I'm sure I'm influenced by what society tells me I need to be doing based on my gender. Nonetheless, I cry at movies, and I'm proud of it. (Also plays, and also one particularly touching dance performance.) I also hike up mountains in the snow uphill both ways with ice axes, and I'm proud of that, too.

Summiting the highest point in Arizona at 12,600 feet.

Cooking potatoes for my housemates at Dabney's retreat.

algebraic representation theory fried rice

Mitka had made too-wet rice and left it in the pot for a few days, and also keeps bringing home olives and not finishing them, so Yasha gave him a deadline to do something with that rice and those olives.  I came home to a steaming heap of fried rice.

Mitka:  "It's algebraic representation theory fried rice!"
me:  "... huh?"
Mitka:  "It's algebraic representation theory fried rice: you take everything and just put it in there, and pretend that what you get is a real thing."
me:  "I think I don't have the mathematical background to participate in this conversation."

Algebraic representation theory fried rice turns out to be tasty.

my visit to the Mirror Maze

First of all, many thanks to Hannah, Dmitry, and Yasha for their hospitality. Thanks to them, I had a very pleasant stay, delicious soups, omelets and fried potatoes, not to mention great conversations. Walking with Yasha around Cambridge was great.

There was a somewhat surrealistic feeling on this visit too. We came to US when I was starting graduate school in mathematics and Yasha was four. So Yasha may remotely remember me taking qualifying exams and now he just took a qualifying exam and, here we are, living, if only for a few days, in the same apartment. Does it count as a mirror image? Or may be it is somewhat related to isotropy of time. (I read Dmitry's earlier post). Well, some additional transformation is involved, I guess...

Talking about the exams and past memories, here is a story I think is worth sharing. I was preparing for an exam (a final for a course). Yasha was five or six years old. I asked him to play by himself, but as every person of his age he would come to me every 10 minutes and ask something. So at some point, I said "If you keep bothering me like this I will fall through on my exam!" (I said this in Russian and "to fall through on a exam" is an idiom meaning "to fail an exam"). To my surprise my warning bought me about half an hour of a quiet time. Then Yasha appeared with a worried look on his face and asked me if I am well prepared now. I said that I am doing rather well, and he is behaving very well, but I might need a bit more time. Yasha started to leave my room with still extremely worried expression on his face.  I became concerned with his look and asked "Yasha, why are you so worried?" "Because, I don't want you to fall through the ceiling on the exam!" - exclaimed Yasha. It turned out that, first of all, he did not know the idiom, but most importantly he had recently visited a university building where a ceiling was repaired. The surface was removed and underside of the upper floor along with some infrastructure were exposed. It may have looked as an aftermath of someone falling through a ceiling! After I reassured Yasha, that no such disaster was about to happen, he became a normal happy child, bothering me every 10-15 minutes.

At our first dinner, Dmitry proposed to raise our glasses "for things we like, people we like, and problems we like". Then we had a discussion whether the "problems we like" would make sense for non-mathematicians, but because all four of us were mathematicians I don't think we were qualified to make a judgement on this. Nonetheless, in my unqualified opinion, this wish can be understood in much broader sense. Everything that we choose to do will unavoidably lead to some problems, but it is great when those are problems we like! As for the people, I definitely spent past weekend with people I like.  Best wishes to you, mirror mortals!

Hannah quotes and a little on OCD

Me: Do you happen to have anything I could use to draw a moustache?
Hannah: No, I don't. But maybe Yasha in his great stash of stashes might have some makeup. (pause) For staches.

Hannah: When my brother was little, he used to want to be a garbage compactor when he grows up. I'm not sure if he wanted to be the truck itself or the operator. [...long discussion about vehicles and drivers ensues, involving Thomas the train engine] 
Yasha: Thomas definitely has his own volition, and an independent operator who's a human being.
Me: That would be cool -- to have your own free will, but just in case you need one you also have an operator.
Hannah: (wistful) Oh man, I have so many operators...


Hannah: (after I point out that she eats M&M's in such a way that there's always as close as possible to the same number of every color):
Yeah. But I really can't eat M&M's. When there's the same number of every color there's no way to decide which color to start with. I can't start with the same color every time because that's uneven. But I can't start with different colors every time -- that's uneven in a different way!
[We convince her to well-order her M&M's -- put the M&M's in a line and eat them from the end one at a time -- this is what I do with menus]
Hannah: I feel like I have to order them in a way that looks random. And it takes some mental effort to remember to take them from the end... [But this way seemed to have something to it -- Hannah repeated  it the next time she ate M&M's].

---

After the conversation about M&M's we compared OCD's. Yasha has trouble throwing things away. I mentally rearrange letters in books to make consonants and vowels alternate in a satisfying pattern. It's fun being crazy when everyone around you is crazy. But maybe more people are crazy than you'd think -- afterwards I talked to my cousin Anna (who needs a blog -- I'm looking at you:). She seemed surprised at our reaction -- "Is there any other way of eating M&M's??"

hurricane preparedness

A few days ago, Yasha told us that there was going to be a hurricane: we'd likely lose electricity, and there might be flooding.
Mitka:  "But why didn't we know about this before now?"
Yasha:  "Because we live under a rock."

me:  "There's nothing to do to prepare for loss of electricity, right?  We should get flashlights or something?  Oh, you probably have flashlights already, because you have everything.  Do you have batteries, too?"
Yasha:  "Yeah.  All my batteries are currently in my lasers."

Today, Mitka did some homework with classmates.
Mitka:  "Harvard's closed tomorrow, so there's no class and Eric and I might get together and do some work."
me:  "If Harvard's closed tomorrow, that probably means it won't be pleasant to walk outside."
Mitka:  "What?  No.  They're probably closed because they'll lose electricity.  And the professors have to drive to campus."
me:  "If people can't drive to campus, you probably won't want to walk.  I mean, in the snow it's fine to walk even when it's not safe to drive---not while the snow is actually falling, but afterward---but..."
Mitka:  "It's not snow.  It's just water."
me:  "Right.  Exactly."

Mitka went to Whole Foods to buy "water and batteries and stuff".
me:  "We decided we'd fill pots with water, remember?  The thing you were actually supposed to buy is cereal."
Mitka:  "But we might not have pots.  I think I'll buy some water anyway."
me:  "We might not have pots?  How would we not have pots?"
Mitka:  "I meant we might not have enough pots."

Mitka:  "Don't forget to move the violin upstairs."
me:  "I just did."
Mitka:  "Oh, okay."
me:  "I mean, during this conversation."

Mitka:  "You know, if there's flooding, your mattress is on the floor."
me:  "Oh.  That's true, I didn't think of that.  I thought, 'Oh, the only valuable thing we have downstairs is the violin.'  But the mattress cost a lot of money, and it's directly on the floor."
Mitka:  "I have a cover thing that might help keep it dry."
me:  "No, I don't think that will help much."
Mitka:  "We can try dragging it upstairs later."

I went to Whole Foods with him, to help carry stuff.  We got one gallon of water, lots and lots of juice, some fruit, cereal, bread, and cheese.  In the checkout line, I loaded juice into my backpack.
me:  "You're buying... a dry salami?  But you don't normally buy salami."
Mitka:  "Yeah.  So we don't starve.  A hurricane is a good excuse."

When we got outside, Mitka remembered that he'd meant to buy toilet paper---we only had one roll left.  The RiteAid was closed.
Mitka:  "Do you want to go back to Whole Foods?  Or we could go to CVS or something."
me:  "I don't really feel like doing anything."
Mitka:  "Okay, but if the stores are out for a week, toilet paper is something we really wouldn't want to run out of."
me:  "That's true."

Mitka:  "We might not have internet tomorrow!  How are we ever going to entertain ourselves?"
me:  "Oh, maybe I should go get my books from my office."

We got home, and MIT had sent a notice.
me:  "MIT is closed tomorrow.  I think I'm gonna go get my books."
Mitka:  "They're not locked, are they?"
me:  "No, no.  They're closed, and I don't expect to go to campus tomorrow, and it'll probably be raining anyway.  I'll just go now and get my books."
Mitka:  "Do you want to borrow my bike?"
me:  "No."
I went up and put my shoes on.
me:  "Okay, bye.  Oh, I'll also get toilet paper."

Decorating!

Some days ago, I mentioned that when I'm older, I might have multiple rooms that are mine. (One can dream, I know.) Hannah said that she didn't find having one's own space to be particularly valuable. I countered, saying that I find it nice to be able to set up my room the way I like it so it is comfy to be there. Hannah said that she doesn't really get anything out of being able to control a space, though she conceded that she does need a space where she can be without anyone else. I suggested that then she wouldn't mind me decorating her room. She declined.

A few days later, Hannah decided that she will decorate her room herself. She did this by printing out a 3 foot by 3 foot photo of her general life advisor on twelve sheets of paper and putting it up on her wall.  I found this to be hilarious, but it made Mitka uncomfortable because having large portraits of revered figures gave him unpleasant associations.

His own room is decorated with artworks, some of them just printed out, and some of them more fancy. Mine has a three-panel painting of Venice, a poster I got with my admission to MIT for undergrad that I kept around for four years, a clock made out of a video card, a calendar, and a large collection of trinkets, pictured below.

Hölder and rural

First I mentioned that my advisor, being German, has a pristine pronunciation of the name Hölder.  Later, Yasha was having trouble saying the word rural.  He kept saying rural over and over because it wasn't working, and I said rural over and over to demonstrate how it was done, and then I started saying Hölder over and over.  Mitka pointed out that rural and Hölder have mostly the same sounds.  Yasha concluded that rural should have only one syllable, and if he just rhymes it with the word curl, it works out fine.

Sunsets at the Beach and Streetlights in Windshields

Some time ago, Hannah asked me if I knew the explanation of the effect one sometimes sees while driving at night, where streetlights look like they have rays of light emanating from them in two opposite directions, which rotate as you drive by. I didn't know, but, after a conversation, we decided that it's very related to the phenomenon that causes the reflection of the sun in the ocean at the beach to look like a path of light heading out to the horizon rather than a small circle of light like one would expect from a mirror.

The thought is that the ocean is choppy rather than flat, and so the surface of the water is often tilted away from the horizontal, up to some maximal angle that depends on how big the waves are. The sun is way brighter that anything else that the ocean might reflect, so, even if only a small portion of the light hitting your eye from a particular point came from the sun, it will drown out any other stuff reflected from there.

If the ocean were flat, there would be some point in the water where light from the sun would bounce off and hit your eye, so you'd see the reflection of the sun there. However, if you move that point in the water either towards the sun or towards you, it doesn't take a very big change in the angle of the water to compensate for the change to ensure that light from the sun still hits you after bouncing off. Since the ocean is choppy, we can expect there to be a little piece of water surface somewhere around there at the right angle to do that for you. It helps that the sun is a reasonably sized disk, so there's some room for error. So, we can expect to see a bright path in the water heading from you towards the sun.

OK, but why don't we see a reflection of the sun everywhere else, too? The problem is that, as you move the point in the water to the side, the angle of the water needed to bounce that light back to you quickly becomes quite large. When that angle becomes larger than the maximal angle of the waves, you stop seeing a reflection of the sun. So, you'd expect the path of the light to be fairly narrow, though it would get wider with a choppier ocean, and also it would get a bit wider near the point of "true" reflection.

So, what's this have to do with windshields? Well, light entering glass gets bent according to Snell's Law. When the two sides of the glass are flat and parallel, like in a windshield, once the light leaves the glass, it gets bent back to the same direction it was originally going. (A lens, on the other hand, exploits having the two sides of the glass be at angles to each other in order to make a net change in the direction of the light.) However, nothing is perfectly smooth and flat, and so some light will hit an imperfection in the surface of the glass where the surface is at a slightly different angle.

Like with the ocean, we can assume that there's a maximum angle of the imperfections in the glass. The details of the double-refraction are different, but the result is the same as with the ocean: after going through the glass, the light will be off by at most a small angle from where it is "supposed to" go. In fact, if we turn the ocean into glass and turn the sun into a streetlight that is below the surface rather than above it, we should see the same "path of light" effect that we see at the beach, though with a thinner path since the glass is less "choppy" than the water.

The only problem left is that the ocean is horizontal, whereas (unless you're very bad at driving) a car's windshield is not, and it's a bit tricky to do 3D rotations in your head to convert one to the other. Fortunately, there's a convenient trick for figuring out which direction to expect the line of light to point. At the beach, the path of light follows the line joining your feet to the point in the water where you'd expect to see the "true" reflection of the sun. You can think of your feet as a magical solar light path attractor: The path of light from the sun will point towards your feet wherever you go.

With a car windshield, we can do the same thing. Instead of your feet, you need to imagine a line from your eye that is perpendicular to the windshield. Since windshields are tilted, this line will go upwards and forwards from your eye and probably hit the plane of the windshield above the actual windshield itself. That point is your magical streetlight line of light attractor: You can expect that the line of light from the streetlight will stay pointed towards your magical line of light attractor as you drive by.

Now, being a theorist, I haven't actually gone out and verified that the direction of the line of light is as I described. I'd be curious to hear if folks see it.

Playing Soccer, with applications to Being Yourself

Some days ago, Hannah commented on the uselessness of the suggestion, "Just be yourself," since it's not like you were considering an alternative.

I figured that, a lot of time, "Just be yourself" falls into the same category as "I'm sure everything will turn out fine" and associated conversation-ending platitudes that secretly mean, "I support you, but I don't really feel like listening to you talk about your problems right now." Naturally, it would feel frustrating to hear that, particularly since there's a bit of hypocrisy in talking like you support someone without actually doing anything to support them. However, I also feel that, on occasion, "Just be yourself" can be meaningful advice.

In a cool paper, folks were asked to do a soccer drill either while paying attention to what they were doing, or while being asked to simultaneously do a distracting task. As one might expect, the novices did best when they focused on the soccer drill. However, surprisingly, the experienced players did better when they had to do a second task at the same time, but only when using their dominant foot.

The mechanism suggested in the paper is that, as people practice a skill, it becomes an unconscious "automated" process. For experts, focused attention on the task interferes with this unconscious process, hurting performance.

Extrapolating wildly from this study, I'd argue that we are experts at tasks like talking or behaving appropriately in social situations (despite plenty of entertaining evidence to the contrary), and so focused attention can likewise be counterproductive.

For example, focusing on impressing someone could lead to bragging, and thereby have the opposite effect. On the other hand, attention is useful when trying to behave in a new manner: For example, a novice at giving lectures might benefit from paying attention to their talking speed. The phrase "Just be yourself" could then be a way of saying, "You're more like an expert than a novice at this particular task, so paying attention to your behavior is more likely to hurt you than to help."

An Open Society

Yesterday, we pondered: What would a society where people's lives are almost entirely public look like?

Mitka brought up the idea of first impressions. In an open society, we might learn things about people we haven't yet met that would color our impressions of them later. Currently, we first get to know people a little on their terms, and then when we do find out their unusual or embarrassing traits, we can fit them into an already-established context.

I countered that, in an open society, few traits would be viewed as unusual or embarrassing. For example, in our current society, finding out that someone had attempted suicide, even if it was a long time ago, would color our perceptions of them, particularly if we find out before actually meeting them. But imagine that all of a sudden you got a list of all of your acquaintances that had attempted suicide. You'd look at the list, which would maybe have a couple dozen names, and you'd probably see some surprises. At first, you'd rethink your impression of the people you see on the list, but after reading through the whole thing, you'd start to rethink your impression of suicide attempts. The next time you meet someone and find out that they've attempted suicide in the past, but aren't depressed anymore, you'd probably react the same way as if instead they were talking about a serious car accident. In turn, your calm reaction would make it easier for people to talk to you about depression and suicide, coming back full circle to having an open society.

In an open society, we'd destigmatize common behaviors, because we'd know just how common they are, and not just in an abstract sense. We'd actually see the people that engage in these behaviors, and realize that they're just like the rest of us, or, more precisely, that they are us. We'd have a much richer understanding of the human experience.

Another example that came up in the conversation was homosexuality. Hollywood-style films sometimes associate being gay with unrelated traits, such as an interest in fashion, whereas in my experience the distribution of traits and interests of gay people is about the same as that of people as a whole. In fact, there's a series of Youtube videos that parodies this disconnect.

In a more closed society, people's impressions of gay people would be more stereotyped. As a result, finding out that someone is gay would change the way you think about them, since you would now associate them with these stereotypes. On the other hand, in a more open society, you'd know that being gay correlates with little else, and hence finding out that someone is gay wouldn't change your impression of them. This, in turn, would make it easier for people to come out as gay, and would help reinforce the open society.

Of course, an open society wouldn't get rid of all stereotypes: If one race for whatever reason had a higher crime rate than another, you'd still judge people based on race, and if one gender for whatever reason performed better than another at a particular task, you'd still judge people based on gender. In fact, it's possible that in an open society it would be more difficult to ignore information in order to make a less discriminatory decision.

Another point that we made is that it's all well and good to destigmatize personal behaviors, but what about behaviors that hurt others? For example, in an open society, if you knew about all of your acquaintances who have shoplifted, it would destigmatize theft. Certainly, if everyone around you is behaving unethically, you don't want to be the sucker that gets left behind. On the other hand, if everyone around you is behaving ethically, you don't want to be the asshole. So, perhaps there is an advantage to hiding our dark secrets and appearing to be better people than we actually are, in order to encourage others to behave more ethically. On the other hand, perhaps being able to talk openly about actions that we're not proud of would make it easier for us to move forward and act better in the future.

It certainly seems that we're currently headed towards a more open society with Facebook and blogs and all, so I guess we'll find out!

yellow house

Mitka:  "Have you ever tried to install... a governing body, inside of your... brain?"
[laughter]
Yasha:  "What do you mean by that?"
Mitka:  "I don't know, maybe this is too farfetched."
Yasha:  "No, I mean, it's possible that I have done this, but you'll have to tell me what you meant by that in order for me to determine."
Mitka, giggling:  "Don't put me in the yellow house!"
[laughter]
me:  "Was that a reference to something?"
Yasha:  "Does 'yellow house' somehow mean insane asylum?"
Mitka:  "It does in Russian.  Or---well, it does in some language."