The Seams of Death

The Seams of Death is this problem I've been working with on and off for the past few months, getting more or less nowhere. "Seams" is what the problem is about, and "Death" is how I feel when I'm getting nowhere. But it's a really cool problem, so I can't hate the problem itself. If there are any interested mathematicians reading this, any tips or a solution would be fantastic! So, the problem:

Seam carving is a cool semi-intelligent algorithm for scaling images. Let's say you want to reduce an image's width but not the height. If you just scale it regularly, the features will be compressed; if you crop it, you will just lose some from the edges. Seam carving instead removes a seam, a string of pixels from the top row to the bottom row, each connected either vertically or diagonally to the next. In particular, it removes the seam of lowest energy (for some predefined energy function like contrast or color or something more complex).

This problem is not actually about carving itself. Rather, I'm interested in finding the total number of such vertical seams in an image (say of width w and height h). An exact closed form solution, not a bound. It's not hard to construct a decent bound. If the edges didn't exist, then for each pixel we'd have 3 ways to go to the next row-down to the right, down to the left, and straight down. Starting from any of the top pixels, then, we'd get (w*3^(h-1)).

It's also not hard to construct a recurrence/algorithm to recursively/dynamically calculate the count. Most simply, you could just start at the bottom and fill upwards. For instance, for a 3x5 image:

5 8 9 8 5

2 3 3 3 2

1 1 1 1 1

The sum of the top row is 35, so that is the number of seams in the image.

But I want an actual equation, or at least a sum that I can evaluate in sub-linear time. It turns out that counting the paths that fall of the edges is not as simple as I thought it would be. In computer science, edge cases are always important, but for this particular question, the edges are the entire problem. :P

The Infinite Harp

I strum on a harp

With diamond eyes

Majestically gazing

Into open skies

Where soaring birds

Will jolt in surprise

As I ride up high

And this dragon flies.

 I strum on a harp

With silver strings

Touched with sparkling flecks,

Of topaz and emerald,

Of ruby and sapphire,

Of opal and amethyst,

Musical pictures,

Some in discord,

Some in harmony

Tuned so finely

That if I gently pick

A dragon's glowing scale,

It will breathe fire,

A flower will blossom,

A dwarf will hammer,

A spirit will weep

For friends long gone,

An elf will dance

Till the break of dawn.

I strum on a harp

With a silver tapestry

Of queens and kings,

Of myth and legend,

Infinitely near and far.

I strum on a harp

With a golden frame

Crafted into infinity,

A serpentine dragon,

Twisting and turning

Its head to its tail,

Specks of jade glistening

In each shining scale.

I strum on a harp 

Filled with invisible strings

So high, so clear,

That it sounds almost

Like a grasswhistle

Shrill across a summer day,

So low, so deep,

That its base rumbles

Like an underwater volcano. 

So near, so close

That when I pick just one

The circle of sound

Ripples over its neighbors,

So fine, so many 

That you can integrate them

And get silver tapestries,

Of magic and spells,

And flowers and elves,

Infinitely close and far.

I strum on a harp

That was made just for me,

Shaped out of gold 

Into infinity

 Two golden loops 

For two playing hands,

Each with an infinitude of strings

Each with its own pitch and key

 Spelling a glistening melody.

I strum on a harp

Molded slowly, carefully,

Into infinity,

The number of strings in each loop,

The number of notes I can play,

The number of things I can do,

The number of thoughts I can say,

As I strum away,

One loop for each hand,

With numbers on the edges,

One number for each string,

So infinitesimally small,

One dot for each digit,

That they almost look

Like fine embroidery,

From nothing in the center

To infinity on either side,

 A fractal of jade on gold

Framing silver tapestries

Of spirits and souls,

And dwarves and dragons,

Infinitely close and far.

"Do we need to know this?"

The very question belies something wrong with the curriculum of a student. The specific fault may lie on a variety of parties or a combination thereof, but clearly something is amiss. If the student doesn't want to learn, then why are they in the class? If most of the material of the class is relevant but some is not, why hasn't it been made clear which parts are important? Or if none of the material is of any interest, why is the student being forced to take the class?

These questions are especially relevant in college, where presumably students are taking their lives into their own hands and charting their paths through the world. It should be a given that they are only taking the courses that they want to take, be it out of interest or acknowledged necessity, both predicated on their own belief in the benefits the new knowledge or experience will hold for them. :)

Fountains of Time

I walked to the corners of the Earth

And found four fountains there,

Each with water falling down

Like locks of Neptune's hair.

 I caught each current in a cup

And brought it home with me,

But little did I realize

What I would later see.

The draught of beauty I had found

Had colored ashy grey,

The lightning shots of energy

Had slowly dripped away.

The cup of everlasting youth

Had grown extremely old,

But burning curiosity

Had turned to wisdom gold. 

:)

Origami: The Beautiful Mathematics of Folding

Origami is just so relaxing, especially when I'm folding a pattern I know by heart, like this crane. :) And of course, the results are (hopefully) wonderfully beautiful. But to me, origami seems as much a science as an art. Put a square piece of paper through one algorithm, and you'll get a crane. Put it through another, and you'll get a ninja star. Use a triplet of folds here to build a foot, leave a hidden strip and you can make claws. Modular origami is fundamentally built upon mathematics, and may even be more of a mathematical endeavor than a physical one. Tons of brilliant people even create new designs through analysis akin to engineering, to the point of being able to take a stick-figure design and generate a possible folding pattern. Wow.

Computer Games

For those who haven't noticed, the range and skill of computer gaming is growing by the day. Most popular deterministic low-move games like checkers or connect four have already been solved, which directly means that computers can play perfectly.

As you step outside those bounds into games with a large number of possible moves, like chess and go, computers have to rely on more or less the same techniques as humans--pattern recognition, looking several moves ahead, and certain predetermined sequences (like opening moves, for example). Add some luck into the mix with games like poker, and perfect play is even less feasible, even if there are less moves to make.

How good can computers get? For move-based, deterministic games, they can theoretically have perfect play, given sufficient time and resources to solve them. With luck involved, they can be perfect within the constraints of the situation, at least, using heuristics to combat unpredictability in the same way we do.

What about games that essentially have heuristics built into them--Jeopardy, crosswords, and puzzles? These games may require not only a large memory but also several guesses and connections, for both human and computer. What is the best way to create fast, reliable algorithms to solve such puzzle games? Matt Ginsberg, the creator of Dr. Fill, argues that in most circumstances, it's best to work inside the realms that computers do best at (and when attempting to beat humans, those realms where they do better). For crosswords, and many other related problems, that realm is memory and search.

"I have a database of pretty much every crossword clue that's been used since around 1990. It's got the body of wikipedia so it looks in there just to see if something comes up, but not a whole lot, it's got a very careful list of words that are probably good crossword words," Ginsberg said. "It just tries to say well I've seen something like this before."

Humans are built evolutionarily to be pattern finders, so we have a default advantage in that area. A species without much in the way of fearsome weaponry or agility had better use its brain to recognize the meaning of a rustling bush or a particularly colored berry. However, our memory is fairly limited, mostly in short term but in long term as well; it makes sense, therefore, for computers to focus on memory and search in relevant areas.

But will this be the dominant paradigm for AI and computer decision making? As we learn more and more about neuroscience to find out how and why we are so good at pattern matching and heuristics, might not we be able to grant those same neural algorithms to computers? After all, once a game is solved, it can be programmed perfectly...once our brains are solved, why can't they?

OMG! WNBMM (Part 3)

One of the biggest reasons we make algorithms is so that we can do things faster. One of the biggest reasons we study algorithms is so that we can make better algorithms to do things even faster than that. In particular, we like algorithms to do better on large scales, so we worry about big-O/big-theta complexity and so on. For loops, that often means solving recurrences.

Consider these two recurrences:
A(n) = A(n/3) + A(2n/3) + n
B(n) = A(n/5) + A(3n/5) + n

They look pretty similar, right? But if you look at the solved complexities, A(n) is in O(n logn), while B(n) is in O(n)!

Why does this happen? It turns out that if you look at the tree of operations (where A(n) breaks into A(n/3) and A(2n/3) after doing n work at that level), and so on, it seems a lot like Pascal's Triangle, which in turn is related to binomial expansions.

In fact, as you work at the sum for the recurrence, it miraculously turns into a sum on binomial expansions, more or less. But while (1/3 + 2/3)^k will be 1 no matter how big k gets, (1/5 + 3/5)^k will drop to 0 (the limit for k being the height of the tree, which is in log n, as usual).

A(n)'s geometric sum is just adding 1 log n times, so of course it keeps the log n. In contrast, since B(n)'s geometric sum converges even as it goes to infinity, the log n disappears! :) SO COOL!!!

Fancy Ratty Desserts!!

I, Ninja

Qualifications:

• Wears dark colors 
• Is quick of mind and calm of heart
• Can make a 4-tipped or 8-tipped paper shuriken
• Can throw a paper shuriken with a (distance/target radius) ratio of about 5
• Can throw a metal shuriken with a (distance/target radius) ratio of about 20 
• Can hide his thoughts and emotions completely 
• Can appear and disappear from a group/crowd at will
• Can intercept a person/group en route within a distance of a mile
• Can run and climb with moderate agility and speed

INTJ

According to Myers, Briggs, and Kiersey, that's who I am! For those of you who don't know, Myers-Briggs is a personality test that measures people based on 4 scales:

Extraversion-Introversion (Self-explanatory)

Sensing-Intuition (S look to concrete senses, data; N look for theory, abstract patterns)

Thinking-Feeling (Self-explanatory)

Judgement-Perception (J like matters settled, P like to keep things open)

Apparently, I am INTJ, which Kiersey has labeled as "Mastermind." Check it out for more fun categories, like "Champion" and "Provider." I've taken this test 3 times now, once in 10th grade, once in 12th grade, and once just now. I've consistently been assigned INTJ, though apparently I'm less extreme than I once was. :) I actually wonder whether that is because my personality has actually changed or because I have a more nuanced self-image now through serious introspection. More food for my thoughts. :D

Seven Stars Bakery Round Durum Bread

I officially have a new favorite bread! Before, I used to love the sourdough we'd get at home--a little crusty, a little dense, quite sour. But this durum bread is simply fantastic! Very crusty, decent shelf life, very airy and chewy, I've used it for everything from sandwiches to instant croutons. On a long train ride + plane flight coast to coast, it is my go-to snack because of its chewiness and simple flavor. NOM!

Lost in Translation

People in my life have often told me that what matters is not my love of learning or what I know, but instead my ability to convey my thoughts to other people. I get reminded of this every now and then, when I fail to communicate enough with my friends and family, but today I received quite an unexpected shock of proof at Brown.

I am pretty good at thinking, I think; I'm pretty sure that's why I'm at Brown in the first place. So my algorithms class is not only a lot of fun but also somewhat understandable. In fact, I'm pretty sure I understood this past problem set completely, in terms of the concepts, correctness, and runtime. Apparently that didn't matter, or at least was insufficient: I was in the bottom 20%.

Why? Something along the lines of style, presentation, coherence, precision, concision. That is, my lack thereof. This was somewhat of a first time occurrence for me. To have understood something so completely and yet utterly failed in my endeavor...so bizarre. Well, I suppose I am at college to learn, and this is an important lesson. :)

OMG! WNBMM (Part 2)

Fibonacci! 1,1,2,3,5,8,...
You can do a lot of cool things with the Fibonacci sequence. So many, in fact, that people have written whole books on them. Obviously then, I don't have time for them all here. However, here are just a couple things that I found out about it today!

Let the nth Fibonacci number be a_n.

1) If you add a_n and a_n+3, the sum will be divisible by 2. Similarly, the sum of a_n and a_n+4 is divisible by 3, and the sum of a_n and a_n+8 is divisible by 7. Let's call this property P(k). So P(2) works, P(3) works, P(7) works. But this doesn't work for some numbers. As examples, P(4) and P(11) don't work at all, and instead of a_n and a_n+6, instead it's a_n and a_n+10 for 5. Interesting.

2) Any 2 consecutive Fibonacci numbers a_n and a_n+1, n > 2, are relatively prime! This is somewhat surprising but even more surprisingly easy to prove if you write them in terms of the Euclidean Algorithm. :)

Seven Letters, First Turn

Word games are so much fun! Especially Scrabble. Especially when you get a 7-letter word on the first turn! This had never happened to me before, that I can remember, and suddenly in the last year it's happened around 5 times! :) Maybe I'm getting better at finding them? (Probably not. :P)

Hi Koo Fun

These algorithms 

Are really rather complex 

Yet intuitive.

With red pen in hand 

From 6 straight to 11 

I graded homework. 

I'm saved again by 

The pigeonhole principle 

Making things exist. 

Today was sunny, 

But yesterday was cloudy. 

Both were very cold. 

The cow jumped over 

The moon and shimmering stars, 

Missed, and flew away.

OMG! When Numbers Blow My Mind (Part 1)

There is a power of 3 that ends in 01. 

There is a power of 3 that ends in 001. 

There is a power of 3 that ends in 0001.

There is a power of 3 that ends in however many 0's you like and then a 1.

:O

For the intrepid reader: Why?
Take the first 101 powers of 3 (3¹, 3², ... 3¹⁰¹). There are only 100 possibilities for the last 2 digits (00 to 99), so some 2, let's say the nth and mth powers, have the same last 2 digits! But that means that 3^m - 3ⁿ, or 3ⁿ (3^m-n - 1), will have 00 in the last 2 digits, so it will be divisible by 100. Well, 3ⁿ certainly doesn't have any factors in common with 100, so that means that 3^m-n - 1 is divisible by 100. In other words, 3^m-n will have a remainder of 1 when we divide by 100. But that means that it ends in 01! YAY!!! :)

The cool part is that you can do exactly the same thing for 1,000, 10,000, etc., so you can have as many 0's as you like!

You can prove some crazy things with powers and modular arithmetic.

Biological Codes...On the Back of my Book

Molecular Biology FTW!

Meditative Thawing: Coming in from the Cold

It was sub-freezing all day today! Not a particularly new phenomenon, at this point, but still. SO COLD!!!! >.<
While practically my entire body is covered and my hands are in my pockets, my face and ears are somewhat exposed (and I refuse to wear one of those full face beanies).

So every time I come inside from the cold, I stand still, close my eyes, and focus on my face. At first it's rigid, just as it was outside, with a numb rubbery feeling when I touch it with my fingertips. First my mouth and jaw thaw out, followed by my eyes and then nose. My ears are farthest away, so the cold tingling sensation as they heat up lasts for almost a minute.

And thus I am rejuvenated! Revitalized! Reheated! So I continue off to wherever I was going, until I have to face the frigid winds again.

Ratty Meal 1: Portobello and Yogurt

Stage 1: Portobello Delicious

1. Combine less than a teaspoon each of steak sauce, garlic-infused olive oil, and balsamic vinegar, along with any other desired seasoning (I used crushed basil, salt, and pepper).
2. Either brush portobello slices with sauce or mix sauce in a bowl and roll slices in it until thoroughly coated.
3. Microwave on high for ~4-5 minutes. This is usually enough to both cook the mushrooms and evaporate most of the water.
4. Make some kinda sandwich/burger/wrap. Bonus yums for melting cheese over them. :D
5. Enjoy!

Stage 2: Cran-Oat-Yogurt Goodness

1. Depending on desired sweetness, add strawberry yogurt first to bowl, followed by plain yogurt. Shake bowl gently to even out and smoothen the surface.
2. Sprinkle copious amounts of oats and dried cranberries on top, and drizzle with honey.
3. Enjoy! :)

Aim, Shoot, Capture

For an activity seemingly innocuous, photography sure has a lot of hunting vocabulary associated with it. The OED has references for "shoot" with respect to taking a photo dating back to 1890, a time when people were starting to build more cameras with mechanical shutters and automatic machinery. Perhaps the simultaneous advances in weaponry caused some kind of association between them?

Or is it perhaps something much more metaphorical? To shoot and capture a picture is to snatch an image to take home and enjoy, perhaps as a hunter enjoys a trophy head, or maybe to isolate and freeze a moment of time, thus killing it and yet immortalizing it. Especially with today's digital technology, moments that would pass away into the mists of time and even slowly vanish from physical canvas can now be stored, more or less permanently, in the digital ether! :D In fact, maybe that is the point. Moments that would otherwise escape us like fleeting gazelle, never to be seen again, are caught and preserved, so that the story of that moment can be displayed and told and remembered.

Mind Control: Wait. Start with Yours.

People talk about mind control as if controlling other people's minds would be something incredible and powerful. Of course, it would be. However, it's not quite as unbelievable as people often imagine, because in quite a real sense, we are able to control others' thoughts, at least to a low extent, by directing their thoughts. Advertisers, magicians, and politicians, among others, all make their livings with this ability. It's pretty amazing! :)

This brings me to the second point, that control of our own minds is not as concrete as people often imagine. The two relevant virtues that I can think of now are willpower and critical thinking. I often wish I had stronger determination, self-awareness, self-control.

I have procrastinated with full knowledge that if I were to start my work now, I could actually finish, but if I wait I will not be able to. I have fallen asleep while working in bed, steadily sinking minute by minute into the covers. I have been unable to get out of bed, hitting the snooze button after waking up for a few more minutes of slumber, to the point of missing class on more than one occasion. I have given myself completely reasonable new year's resolutions and similarly completely failed to maintain them. Perhaps this problem is just mine, and I ought to see a psychologist and cure myself; I rather think, though, that this is a more widespread phenomenon.

Second, critical thinking is a skill that is valuable even to those who are not critics. In an information-flooded era, it is more necessary and yet difficult than ever to be able to distinguish the sound facts and logic on which to base opinions. There are those who ignore information entirely, and base their opinions and beliefs on their past, general assumption, and previous bias. Then there are those who are information-children, innocently taking all the "facts" around them as true, when in fact many are examples of false reasoning, propaganda, and plain trickery.

Critical thinking is the ability to identify good and bad facts, logic, arguments, and therefore opinions, and even those who have it (most of us) simply fail to use and hone it as often as we might. Again, critical thinking in an information era is difficult and exhausting. The benefits, however, are improved thought, quality of life, and common sense, for free! :) Indeed, we all have common sense, hence the name. Critical thought, half of the ability to control your own mind, allows you to keep it, amidst everything trying to deny it. :D

Grading: A Potluck of Emotions

Yes, yes, grading is supposed to be objective. I'm not trying to say that I let my emotions influence the grades I give, but the process of grading is really a roller coaster on the wheels of my nerves.

1. First comes excitement, at the prospect of a new experience and seeing how the students dealt with the problems we gave them.
2. Then comes joy, as one person seems to completely understand what's going on.
3. Then comes disappointment, as someone else doesn't.
4. Then comes sadness, as I'm forced to take off points because someone just forgot to do a problem, apparently.
5. Then comes ecstasy, as someone not only seems like he/she knew exactly what I was looking for but also has the neatest handwriting I have ever seen.
6. Then comes bewilderment, as I find someone who obviously didn't understand the problem at all but really really tried hard, put in a lot of work, and came up with a completely unexpected and utterly wrong answer.
7. Repeat 2-6 in random order for about 100 more papers, intermixed with
1. Anxiety as I realize that it's been 2 hours and we've finished less than half the papers, and
2. Liberation as I really get on a roll and zip through 5 in a row at breakneck speed.
8. At last, there is a great sense of anticipation as the "finished" pile grows large and the papers to be graded slowly vanish from the table, followed by a final sense of satisfaction and accomplishment.

Whew! Let's do it again next week! Yayyy! :D

Rhythm: Beats in the Brain

OED: "Regularity in the repetition in time or space of an action, process, feature, condition, event, etc.; periodic or cyclical change or movement."

Rhythm is pervasive. From music to heartbeat, so many facets of life can be described in terms of rhythm that it seems to be fundamental to human life itself. I've often found songs (especially pop songs) whose lyrics are either incomprehensible or even offensive but whose rhythm is so compelling that I can't help drumming my fingers on any available surface. In fact, I practically always am drumming away to the beat whenever I'm lounging alone or with friends, listening to some tunes. 

And I'm not even a drummer, or a musician by any real standard! So I can imagine how much more important rhythm must be for people whose lives actually revolve around it. I have one cousin who is so in tune with the beat that she got our entire table of 7 pounding away with vim and vigor...at the wedding reception for one of my other cousins. Yet it didn't matter! Another table started copying us, but they couldn't stand up to our mad drumming skills. Even my niece's stuffed animal got in on the action. :)

Of course, life itself has its own rhythms, of days, of sleep, of work. My tempo is a lot faster this semester with all the classes and TAing and research, but hopefully I'll manage to take some time breathing very slow. :)

25:25 Hindsight, 25:25 Foresight: Bifocals Unnecessary

They say hindsight is 20:20. This has always seemed a bit of an overestimation. Memory is notoriously fault-ridden itself, so if we can't even remember the past with perfect resolution, how do we expect to be able to think about it with perfect clarity? Just as we become less and less able to predict the farther futures, we are less and less able to remember the distant past. We fumble to recollect and reconstruct our thoughts, and of course others' motivations are an even greater mystery.

Still, for all that, we do pretty well. We may repeat the mistakes of new and ancient history, but we definitely learn from them too--otherwise we wouldn't exist! :) Any hominid that barely escapes a lion jumping out of a bush that isn't cautious about the next rustling bush he sees isn't going to last very long. We certainly believe that we are smart enough to elect different, hopefully better, leaders if the current ones aren't helping much (although certain elections provide distressing counterexamples). In view of all the learning we do, I'd estimate something closer to 25:25 hindsight. Good enough to survive, learn, and thus progress, but bad enough for history to repeat itself in the worst ways imaginable.

Now what about foresight? Presumably, if people are making such a big deal out of hindsight, it should be pretty bad in contrast. In fact, I'd score it at about the same, 25:25. The future is foggy, certainly, but bad viewing conditions don't imply that our eyesight is worse, do they? Even 20:20 individuals have a hard time seeing in pouring rain. Indeed, when the world seems to be falling around us, foresight often goes out the window as well.

But think about it. We can predict and control our futures to a remarkable degree! In the immediate future, we have enough precision to be able to answer specifically when people ask where we'll be and what we'll be doing in a few hours, and in generalities for days, weeks, even months. We have enough abstract foresight to plan for decades, going through school and college in the belief that it will improve our future. So all in all, we're probably about 25:25 in both foresight and hindsight, though admittedly the future is foggier than the mists of the past. But really, I think that's good enough. :)

Cheese!

Cheese is delicious! Cheese is nutritious (kinda?)! Cheese is smooth, cheese is crunchy, cheese is sweet, cheese is tangy, cheese is everything that you could want out of a foodstuff that you want to be able to eat with something else. At Whole Foods, there is a whole cheese section devoted entirely to this incredible dairy product. When you think "Cheese and Crackers," it sounds delicious, but when you think hard about it, at least 70% of the awesome there is coming from the cheese, if not higher. Cheese is as pervasive as alcohol around the dinner table, whether a cozy family dinner or a fancy Christmas party. Just saying "cheese" makes you smile a little, which is why photographers always ask you too.

Once, when I was walking around South Station, passing the time as I waited for a train to Providence, I saw that I thought captured the glory of cheese.

Indeed, cheese dips are used for everything from pretzels to nachos. And then there's fondue! OMG, fondue. I think the fact that chocolate and cheese are the two most common fondued foods is really a very telling sign of their importance. However, I shortly saw another sign about cheese within the next 5 minutes that really hit the mark:

The Light Fantastic: Lamps, Lanterns, and Chandeliers

I recently became slightly obsessed with chandeliers, during my trip to India. Not even just chandeliers, actually. Even those weird creative lantern things that you see hanging from certain buildings like this cool pine cone in the Rockefeller library at Brown.

And then just this semester, I realized that some lamps are actually really cool too. I helped one of my friends assemble this tree-like one about a week ago. I think the only thing that could have made it better would be if the stalks were lit up too. :)

 

The OED defines a chandelier as "an ornamental branched support or frame to hold a number of lights (originally candles), usually hung from the roof or ceiling." I supposed that what I'm looking for is something that satisfies either part of that condition, that is, either "an ornamental branched frame to hold lights" or "a frame to hold lights hung from the ceiling." Or both, of course. :)

Light, of course, is totally amazing itself, but that's not what I'm talking about not, though I certainly will another day. Rather, I'm talking about the problem of bringing light into the indoor environments we have created, and how the creativity some designers have brought to their solutions is really cool! Chandeliers are stereotypically beautiful, but even structures with no glass at all can not only effectively distribute light but also look fantastic doing it. I'm aware that other kinds of furniture can be equally amazing, but the combination of light and the sheer variety of artistic forms that chandeliers, lanterns and lamps come in is breathtaking. Here are a few more pictures I took in India--I hope you enjoy them as much as I did!

Bookshelf: The Organizing Adventure!

As transient as snow in California, a well-sorted bookshelf is rare and does not last very long. Yet the process of removing all the books into a messy pile, planning and organizing, and then putting them back in some semblance of order is really fun and fulfilling, for some reason. This is true of any large-scale organization project, of course, but a bookshelf sits in balance between complete disarray and acceptable manageability. And there are so many serious decisions to be made!

Should I sort first by the owner of the book (i.e. me, my sister, my mom, or my dad), or should I just go by genre? Within fiction, should I separate out all the series? Do brain teasers and number theory books fall in the same category? For each section, should I organize by color or size or author or title? Does any of this even really matter? What are the chances of this bookshelf being organized in two weeks? What's the point of this, anyway?

The point, Vishesh, is that it's fun, and you like order, and when you turn the disarray back into beautiful shelves of books, you will feel so accomplished that you won't feel obligated to do anything else for the rest of the day. Also, books are really awesome. :D But that's for another day.

Frozen Pasta, Foamy Bubbles, and Awkward Turtles

What do these three things have in common, besides being really random? They are all subjects of serious, dedicated scientific research, and that makes me so, so, so happy. :D Every now and then, I chance upon an article that makes me stop and think: "There is someone in the world who studies that for a living!?"

The tolerance of science to any and all possible research is apparent in practically all fields. In fact, these three are even ordered in terms of the relevant scientific discipline. What's that you say? Food, Toys, and Biology don't seem ordered at all? Food and Toys aren't even really scientific fields? Sorry, I what I meant was Chemistry, Physics, and Mathematics.

Frozen pasta is a really complicated process, when you think about it. Trying to achieve, after freezing, thawing, and eating, the same shape, texture, and flavor as regular pasta is quite a feat. Freezing causes a frost which, upon melting, can soak the pasta and dilute the sauce. Moreover, the coating of lipids (oil) on the pasta apparently modulates chemical changes that can result in altered pasta rigidity and sauce retention! Wow. Serious business! (In fact, pasta sauce research goes back a while. In the 1980s, Howard Moskowitz revolutionized the food industry, beginning with his research with Prego.)

Foamy bubbles may seem chaotic, but in fact they follow very specific scientific rules governed by physics. Joseph Plateau formulated laws that describe the typical formation of foams. Bubbles that do not satisfy these laws are unstable, and they either quickly change to fit these laws or simply pop.
1) Soap films are smooth surfaces, and the average curvature is constant on any point on a single soap film.
2) Soap films meet in 3's along an edge at angles of 120 degrees (inverse cos of -1/2).
3) Edges meet in 4's at a vertex at angles of ~109.5 degrees (inverse cos of -1/3) .
Foams are so organized! :)

Turtles usually fall within the biological domain, but this study of how awkward turtles (turtles on their backs) are able to right themselves--using mathematics! In summary, turtle shells fall somewhere on a scale between flat and tall. Flat-shelled turtles are usually flat because they live most of their lives in the water and need to be hydrodynamic. Flat shells have 2 stable equilibrium points (the back and the bottom), and so if such a turtle is on its back, it is difficult to right itself by just wobbling, because the back is stable. Interestingly, aquatic turtles usually have an abnormally long neck, and this allows them to reach the ground with their head to push off. Medium turtles usually have stable equilibria on the sides of their shells, but these are less stable than the back of a flat turtle, so they can usually get around this issue by wobbling their head and feet.

The tall turtles (the roundest) are special, because they have only 1 stable equilibrium point, which is on the bottom-that is, on their feet. Any other point on their shell is unstable, so they need very little wobbling to right themselves. This may sound cool as is, but the really cool thing is that mathematicians are very interested in 3D objects with just 1 stable equilibrium point, to the point that this class of objects has a name: monostatic. The coolest thing about the fact that these turtle shells are practically monostatic is that monostatic objects are almost non-existent in nature--one study by Domokos included a systematic study of thousands of pebbles, and none of them were monostatic. So this property really is very special. Yay turtles! :) And come on, honestly. How awesome is it that there are not only scientists working hours and hours seeing how stable pebbles are one by one but also scientists that spend their hard working hours on awkward flipped turtles!?!?!? :D

FroYo: Never Too Cold

There are a few things that have been substantially harder to procure in college than back at home, especially various kinds of food. Perhaps this is mostly because I hesitate to spend much money outside of my meal plan, but fruits like berries, pomegranates, grapes, clementines, peaches, and mangoes have been very elusive.

The other main foodstuff (besides delicious homemade Indian food, of course) that is much rarer without a car is frozen yogurt. Back in California, I would go to Pinkberry and Yogurtland at least twice a month, if not more. The weather is completely irrelevant, although high summer leads to increased consumption; I don't care if it's freezing here, as long as it's not freezing where I'm eating. So many flavors, so many toppings, so much deliciousness! I actually tend to stick to "original" in terms of flavor, only because I love the combination of that simple tartness with tons of juicy and colorful fruits. :D

Fortunately, I found one place nearby (just one, really?) called Juniper last year, and this past summer a Froyoworld opened, but I have yet to try it. On the whole though, there is just not enough bang for the buck compared to the traditional haunts. The tart freshness is missing, the fruits less flavorful and plentiful (just as they are in general, really). As you can see in the picture, I still manage to get a decent array of yellow, red, and indigo. But really, I can't wait to come home. :)