On American Beauty and Lost Summers

I was watching American Beauty for the third time (if you haven’t seen it, the film is brilliant and this post may contain some spoilers, so go watch it!). The times when I have watched the movie previously are interesting: the first time was a point in high school when I stopped doing homework, questioning its relevance. The second time was in the middle of college applications, when I felt like I was losing touch with the substance I should have in my writing and I had started to litter feel-good anecdotes in my application essays to appease both my parents and possibly, admissions people. The film is a startlingly accurate depiction of American suburbia and its hollow underpinnings, which contrasts greatly with the surreal, bucolic imagery that most of us see on a movie screen. Let’s shuffle back to part of my day yesterday to show how eerily familiar the movie becomes to even those of us not going through a midlife crisis like Lester Burnham.

Here’s part of a conversation with one of my closest friends, a really nice, smart guy (and yes, the first part mentions physics, but don’t fret, the post has nothing to do with physics!).

6:43 PM:

Him: “Ugh, I’m confused here- string fixed at x=0 and x=L, tension=T, density mu” [physics]

Me: “I’m sure there’s a formula in the book for that; I haven’t done standing waves for a long time. There is probably a generalized formula and you can take derivatives to see what happens to phi, delta, etc since the deflection will be zero.” [physics]

6:45 PM:

My friend calls me to ask more in detail about the physics question, but it’s clear he has something else on his mind.

Him: “So I’m going to dinner tomorrow night” and then hesitates and with subtle emphasis in his voice, “…with a girl”.
Me: “Oh awesome, is it the same beautiful girl as yesterday?”

Him: “Nope, a different girl than before! I’ve decided that I’m just going to go on as many dinners as possible with as many girls. I don’t even know with this one; she just reminds me so much of one at Berkeley. It’s thrilling, isn’t it?”

The conversation gradually goes to other things and it’s obvious that he thinks that I am silently judging him, as he shifts the conversation to my life. I tell him how I might go to England over spring break and I ask him if he wants to come, to which he says, “No, I’m a settle down kind of guy.” We both share a chuckle at the irony in this statement compared to what he said only two minutes before. We continue the conversation for a couple minutes but realize it is dinner time for both of us.

7:25 PM:
I think back after dinner and I sort of admire my friend. Maybe he’s not going about his interests the proper way, but he is at least displaying some sort of passion. Here I am, in front of a computer, doing nothing of the sort. Sure, I’ll make kind gestures to someone I think is amazing, or maybe even send a message to talk about a mutual interest and indirectly say that I miss them. While I certainly don’t have the option to ask them to dinner since they might be 3000 miles away, I have my doubts on whether I’d be so bold. Whether my friend has welled up with bravado or newfound confidence I don’t know, but when we both came into Berkeley as freshmen, we were both shy, quiet people bonded by similar interests- and now look how different we’ve become.

7:45 PM:

After twenty minutes of engaging in conversations with friends in which neither of us have anything to say in particular which results in half-hearted jokes and reminiscing, I decided that I should watch American Beauty. This time, I was watching the movie because it seemed more and more like my life was echoing the initial life of Lester Burnham. I was living suburban life, lulled by its sense of comfort and doing nothing important beyond the usual 8-5 grind. You could generalize a midlife crisis as a place where you either have no passion or do not seem to put in enough to follow your passions; even though I’m certainly different from Lester Burnham, was I much better?

8:20 PM:

After watching about thirty minutes of the movie, something dawned on me. The movie is quite the achievement because of the contrast between the quiet, peaceful music and cinematography in the background and the turbulence and vulgarity of the scenes. Even Lester, at some point, when there is elevator music playing at the dinner table, remarks how the music is unfitting of the family’s current state.  In most dramas, the music would be more filled with arpeggios and would reflect the turmoil in the household rather than the beauty behind suburban misery. Perhaps that’s why the film is so well-received by critics and viewers alike, but I digress from my point. The reason the movie is beautiful is because it is a fluid portrait of a society we live in, in motion. Now, if we strip the musical compositions and motion and instead just have individual shots of the film, is it remotely interesting? I would argue that the answer is certainly no, because it is hauntingly similar to the lives that we as college students live, away from college. Sure, some of us are working or doing internships, but what about after work hours?

8:30 PM:

I decide that while watching the movie, I’ll start learning dynamics. Perhaps the movie has made me question how I should live like it had previously, or it has made me concentrate on something that could be important.

9:30 PM:

In the middle of my learning session, my phone alarm goes off. I breathe a sigh of relief; the USA soccer game is on and now I can avoid working and putting in any effort. I can focus on my passion for watching soccer. I vowed to learn more dynamics by waking up early on Saturday morning.
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With all of this being said in a narration of a recent, recurring and dull Friday night, I guess an excellent question to ask yourself would be whether your life is worth watching on a piece of film. Sure, there will be dull moments, but they should lead to more exciting moments or important ones. Looking back, will the hour I spent learning physics while watching American Beauty be memorable? Probably not, but it could be the start of something good, for example, if I learned dynamics well enough, maybe I could be part of a research group that designs a robotic exoskeleton that could help people walk again. I think back to an exchange of dialogue in the movie which is simple yet powerful.

The girl and aspiring model that Lester is infatuated with (who is his daughter’s best friend) meets his daughter’s new boyfriend and questions him.  Lester’s daughter, Jane, says to the girl, “…And you’ll never be a freak because you’re just too perfect.”
Her friend retorts, “Yeah? Well, at least I’m not ugly!” The boyfriend, who appears stoic in this scene, then replies, “Yes, you are. And you’re boring, and you’re totally ordinary, and you know it.”

In reality, I think one of the biggest fears of any individual, whether through introspection or by someone rudely pointing it out, is the fear of being boring. I think we’ve arrived at the awkward junction where I have somehow made this post blend into the usual carpe diem, make the most of the moment, etc post, so I’ll leave it up to interpretation.  Also, I would like to state that I’m not saying I’m bored throughout the summer and in general; it’s just less action-packed than college. I still maintain many passions (in fact, often I feel I have too many), which tend to be the topics of different blog posts; I just have less of a chance to explore them back home.
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Next day, 10:30 AM:

After sleeping through three alarms placed 15 minutes apart and then deciding to just enjoy my morning by reading a book in bed, I officially get up to brush my teeth at 10:30. Has complacency reared its ugly head?

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On Inequalities

Readers, after a post on wealth inequality, I decided to write about something that truly captivates me- mathematical inequalities! All kidding aside, I have always had a passion for mathematics. Knowing this, various friends let me know that if I wrote a blog post about math that I’d never go back. They also told me that I would be completely incoherent. I’ll try to avoid both but there are no guarantees. To those of you who aren’t big fans of math, don’t fret, this post will not be littered with foreign mathematical concepts (but there is a lot of math) and [hopefully] will be interesting!

I guess I’ll start with a bit of background; math was my passion ever since I could read numbers. There were times when it definitely got the best of me: for example, I tried to play around with Newton polynomials in a notebook while biking up hills when I was little (did I mention that I’m terribly clumsy?). For the most part though, it was wonderful because it could accompany me anywhere and I saw it everywhere. My dad had taught me the operations (addition, subtraction, multiplication, division) by grouping coffee beans and now I saw it in food and in particular, breakfast. I actually remember switching from eating waffles to bagels so I could make this (http://georgehart.com/bagel/bagel.html) on the weekends. Really though, I did enjoy my childhood! A friend had once said that I had learned knot theory before learning to properly tie my shoes, which is more a testament to how late I actually learned how to correctly tie my shoes. All anecdotes aside, I’ve always loved the subject because of how most interesting problems have a crux move that changes your perspective completely. Moreover, the elegance of particular solutions based on simple concepts is awe-inspiring and powerful. That being said, I was deterred by math for much of high school because I realized that for math contests, which I thoroughly enjoyed, many people simply memorized formulas from a math book for glory. Still, I maintained an appreciation for math and one of my favorite topics was inequalities, which I thought was particularly elegant due to both its simplicity and power. I figured I would show a glimpse of its elegance (I apologize in advance for the lack of depth to math whizzes and if I use concepts that seem a bit bizarre to those of you that aren’t math lovers). Here’s a word document version of the post in the case that the formatting and haziness of math notation on wordpress makes you cringe: On Personality

My first experience with any notion of inequalities began with learning the arithmetic mean ( P1) and the geometric mean (P2 ).

From there, in a sidenote in an Indian textbook I had, it had said that since for any real number, y,
P3,
then:

P4

The result here seems simple but not that relevant at first glance but it is a delightful first step into inequalities. It is one case of a famous inequality known as the Arithmetic Geometric Mean Inequality (or AM-GM ). I’ve omitted the proof for this post because of its many cases, but I can provide pieces of it in the comments section if you’d like.

This inequality (AM-GM) states that as long as

P39 :
P40

This is definitely a cool result, but I’m sure you’re wondering how it could be applied. Here’s an example of another inequality that can be proven using our newfound inequality.
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Show that:

P41

for positive real numbers a, b and c (Nessbit’s Inequality)

How do we begin?

Well, we know from AM-GM inequality that for any positive real (x,y,z)

P6

since

P7

(cross multiplication and AM-GM)

Let’s write the inequality and similar components as variables:

P8

Initially, maybe this setup seems really arbitrary and just assigning variables rather than being interesting. But in this setup there lies the crux move.

What we can gather from these variable declarations is that:

P9

By adding the last two equations, we get

Since
P10

or
P11

Which is what we set to prove in the first place!
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Ok, so maybe that wasn’t super applicable, but here’s an elegant proof of Cauchy-Schwartz inequality (http://en.wikipedia.org/wiki/Cauchy%E2%80%93Schwarz_inequality) using AM-GM inequality, which was the basis in some sense for the Heisenberg Uncertainty Principle and an inequality you’ve probably seen in a lot of Linear Algebra textbooks (oh, the horror!). If you’re not interested in a proof sort of question with a bit of summation notation, I’d recommend looking at the next example!

We know that based on the definition of AM-GM:

P12 (1)
and we know that if there is some random value, c, that

P13 (2)
since

P14

Another way of saying it would be that
P15

and we also know that

P16

if and only if
P17

Let’s say that:

P18

then
P19

which mirrors Cauchy Schwartz Inequality, which is:
P20

Would you not agree that the result is pretty amazing?
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I remember that when I learned AM-GM, I didn’t realize its power until I saw it used to solve International Math Olympiad problem (which all of you can solve now too!).

Here it is (1964 IMO, Problem #2):

Suppose  are the sides of a triangle. Prove that
P21

It certainly looks complicated at the start, but yet again, just assigning variables and the use of AM-GM makes this mathematical hodgepodge of letters and numbers more elegant.

If we set P22, P23, and P24,

Then the expression becomes:

P25

After the simplification of the messy expansion, we get:

P26

What does this simplified expression remind you of? AM-GM inequality of course! I really appreciate inequalities because you have to creatively use simple expressions to solve much more complex ones; memorization is virtually useless (credit to Art of Problem Solving for posting the solution and not making me write it out on the computer).

I guess I’d like to end on a somber note. The American education system, particularly in elementary school, middle school and high school, seems to be bent towards the regurgitation of formulas, plugging and chugging and repetition to teach mathematics. I definitely don’t have a particular way to resolve this issue, but I’d like to hope that our system will eventually embrace creativity. The question of whether it is possible to teach creativity is certainly a fascinating debate topic that I would love to look more into in the future. I definitely have no background on it, but if you have any insight on it, feel free to let me know. Additionally, if there are any interesting inequality ideas you’d like to point out or something you would like me to explain further than I did in this post, just leave a comment! I apologize to those of you in advance who found the post too long and didn’t see any elegance in my examples; I’ll write about something less absurd soon!
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BONUS!
In case you’re interested, I have posted another problem that is slightly harder than the ones preceding it; not in the number of steps but rather in how daunting it looks. It was on the shortlist for the International Mathematics Olympiad and looks like a problem a lot of people would not even choose to attempt (feel free to skip it if you’re not feeling mathy).

IMO 2010 SL, A3

Let P28 be nonnegative real numbers such that P27 for all P29 (we put P30 ). Find the maximum possible sum of

P31

The problem looks pretty difficult, right?

If we were to take a simple case, where P32  for  P33 ,

then, the expression above, P34 (if you don’t see this, just plug it into the summation and notice how the terms come to ¼ when odd and 0 when even)

Now, let’s consider the other cases for which P33

Then by rearranging the problem statement, we get that

P35and P36 .

By AM-GM Inequality, we see that

P37

(Ah, the Haziness!)

If we sum the inequalities from P33 , we get:

P38

Whee! We have successfully solved a very complicated looking summation problem using basic inequalities!

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