I made it through. The exam was easier than I expected, though not easy, and I have a feeling I did as well as I needed to do. At a price, though. I have no brain left. I actually just heard the sound of a mosquito buzzing and saw a flicker of movement near my eye and, without thinking-- slapped myself in the face. I wish I were kidding. The worst part is that I don't even know if I caught the damn bug, since I can't see my own face...
And after the exam, I came home, and sort of stared at the walls for a while. I had some food (breakfast at around 6:30, being in the exam building starting at 8, only leaving at 12, I was hungry!) and zoned out for a while, and then I went over to Miklos' house!
That's right. He hosted a dinner for the Spectral Theory class today, because he wanted to get to know us better. It was a perfect event for a post GRE afternoon. No one else in the class took the exam (though several other students at this program did), so no one there wanted to talk about the test (amen), and it was the weekend, so we were relaxed enough not to talk about math all the time - but also dorky enough to make lots of jokes about it. Miklos made us "liba maj" (fois gras) in the traditional Hungarian way - gently cooked and served thinly-sliced on bread with raw onions, or various pickled vegetables -or even with his wife's homemade raspberry jam (sounds weird, but it was delicious). Then, there was some wine, some math jokes, some talking, some storytelling - eventually gnocci and then dessert got added to the mix, and all in all a lovely evening.
And now, I'm going to sleep. I still have quite a bit of work to do - after all, my classes didn't stop while the GRE happened. But my head is so much clearer. The underlying anxiousness is slowly abating as I realize the test is over.
I'm now writing this while lying on the couch, and my head is actually on my arm, not even propped up anymore. Now one eye is closing. I do believe it's time for bed.
Saturday, October 19, 2013
Friday, October 18, 2013
Things That Are Much Worse Than the GRE That I Have Already Weathered And Survived: A Pep Talk
It's tomorrow morning at 8:30 a.m. At 11:30, I will walk out of there, with the pressure of the last four months off of my shoulders, one way or another. It's just a damn exam. I'm so excited to be done.
And, as I go to sleep here tonight, I want to give my brain something fun to think about, as mentioned in the title. So, below are some harrowing experiences I have had that were both a) more painful than the GRE Math Subject Test will be tomorrow and b) lasted longer than the aforementioned test will. I don't know if it's for me or for you, but it's here.
1. Surviving through the morning after the day that I hiked Half Dome with my family in Yosemite - when I thought none of my bodyparts belonged to me anymore.
2. Getting my wisdom teeth out.
3. Sitting through graduation practice at high school and college in stuffy rooms with not enough air.
4. Bowing incorrectly to a monk in Thailand and sitting through the rest of a silent, 2-hour service in the temple when people kept intermittently turning around to stare at me.
5. That one time I was on a plane from Frankfurt to San Francisco, and 12 hours into the flight we circled and circled and circled around San Francisco, then had to land in Oklahoma City, they barely had enough fuel for such a big plane, then we had to fly back and circle and circle and circle around Denver before landing and I had to fly to SFO from there.
6. Having strep throat a few weeks ago.
7.The day of crew practice when I almost fell out of the boat due to incorrect form and was thoroughly embarrassed/shamed and also very sore from all of the rowing.
8. Studying for this damn test for the past several months.
I was on the exercise bike today with my GRE book and I was listening to the Pirates of the Caribbean soundtrack - and as I turned to the page about the integration of complex functions, the music swelled with the first appearance of the main theme in that film, and I all of a sudden was out of myself and observing the situation, realized the dramatic music and me, sitting on a STATIONARY BIKE with a book about MATH, so worried about one damn test. And I started laughing so hard I had to stop pedaling. Life cracks me up sometimes.
So, my plan. Go to sleep. Wake up. Have some breakfast. May or may not involve bacon. And treat myself as awesome. That's what I did all through my undergrad - it worked for me there. No reason for it not to now.
And, as I go to sleep here tonight, I want to give my brain something fun to think about, as mentioned in the title. So, below are some harrowing experiences I have had that were both a) more painful than the GRE Math Subject Test will be tomorrow and b) lasted longer than the aforementioned test will. I don't know if it's for me or for you, but it's here.
1. Surviving through the morning after the day that I hiked Half Dome with my family in Yosemite - when I thought none of my bodyparts belonged to me anymore.
2. Getting my wisdom teeth out.
3. Sitting through graduation practice at high school and college in stuffy rooms with not enough air.
4. Bowing incorrectly to a monk in Thailand and sitting through the rest of a silent, 2-hour service in the temple when people kept intermittently turning around to stare at me.
5. That one time I was on a plane from Frankfurt to San Francisco, and 12 hours into the flight we circled and circled and circled around San Francisco, then had to land in Oklahoma City, they barely had enough fuel for such a big plane, then we had to fly back and circle and circle and circle around Denver before landing and I had to fly to SFO from there.
6. Having strep throat a few weeks ago.
7.The day of crew practice when I almost fell out of the boat due to incorrect form and was thoroughly embarrassed/shamed and also very sore from all of the rowing.
8. Studying for this damn test for the past several months.
I was on the exercise bike today with my GRE book and I was listening to the Pirates of the Caribbean soundtrack - and as I turned to the page about the integration of complex functions, the music swelled with the first appearance of the main theme in that film, and I all of a sudden was out of myself and observing the situation, realized the dramatic music and me, sitting on a STATIONARY BIKE with a book about MATH, so worried about one damn test. And I started laughing so hard I had to stop pedaling. Life cracks me up sometimes.
So, my plan. Go to sleep. Wake up. Have some breakfast. May or may not involve bacon. And treat myself as awesome. That's what I did all through my undergrad - it worked for me there. No reason for it not to now.
Thursday, October 17, 2013
"Hm. Let's do something simple - consider the infinite 3-regular graph ."
Mathematicians are weird.
The thing is, you have to wait around to figure out what the things are that make sense to you without thinking, which are the ones you have to take a few seconds to figure out, and which are the ones that feel beyond you when you first encounter them, but really you need an hour with a blank piece of paper with an afterthought of follow-up questions. For example, something "simple" today had me simply reeling as I couldn't get it into my head (having to do with matrix multiplication, which is something I learned in high school - though not in such an abstract way) and then, later in the day (at the colloquium talk I just left) something that came to my mind without thinking: picture a lattice, like the kind on top of a pie. Basically, looks like a checker board or chess board - vertical and horizontal lines with squares in between. Now, there can be other kinds of lattices - ones that don't have little squares, but rather, little triangles, or parallelograms, or -- hexagons. In the talk today (about percolation - the way liquids move through particles, like how far water gets if it has to seep through a bunch of densely-packed sand), it turns out that in percolation theory, many results can be proved when using hexagonal lattices but not with the traditional square ones. I asked why. "Because," the speaker responded, "The faces of the hexagons are the vertices of triangles." And I had the picture in my mind immediately. (Grab a piece of paper - draw a dot on the center of each hexagon you draw, and connect them - you'll see what I mean. This result is not all that deep or anything - I was just happy to visualize it so quickly.)
Okay, okay, so the more I think about it, the more pathetic a victory that tiny moment was. However, I've been thinking today. Quite frequently, in my head, I think about how I don't feel like I'll ever be worthy of the title 'mathematician' because there are so many people who know so much more about math than I do - so very, very much more. But all of a sudden as I was feeling sorry for myself like this this morning, I thought of a tiny fish saying that it can't swim because it doesn't go as fast or as deep as the mighty narwhal, for example. What a stupid thing to say. Of course it can swim! Damn well, too!
Hold on, though. Maybe, in this metaphor, I might actually be a sea lion- because, I can swim - and in some areas of the ocean, I may even be a force to be reckoned with - but I like to spend a lot of time on land (a.k.a. playing music, reading, thinking about philosophy, psychology, economics, etc) and not in the water (i.e. the murky and seemingly infinite depths of math). In fact, I can't be in either place indefinitely.... Oh dear. Metaphor carried me away. I'll swim back to the point.
So, the point is, as you can clearly see, the GRE is in less than 48 hours and I'm on my last mental legs. But at least, I understand how hexagons can be like triangles. Guess that's a small victory.
The thing is, you have to wait around to figure out what the things are that make sense to you without thinking, which are the ones you have to take a few seconds to figure out, and which are the ones that feel beyond you when you first encounter them, but really you need an hour with a blank piece of paper with an afterthought of follow-up questions. For example, something "simple" today had me simply reeling as I couldn't get it into my head (having to do with matrix multiplication, which is something I learned in high school - though not in such an abstract way) and then, later in the day (at the colloquium talk I just left) something that came to my mind without thinking: picture a lattice, like the kind on top of a pie. Basically, looks like a checker board or chess board - vertical and horizontal lines with squares in between. Now, there can be other kinds of lattices - ones that don't have little squares, but rather, little triangles, or parallelograms, or -- hexagons. In the talk today (about percolation - the way liquids move through particles, like how far water gets if it has to seep through a bunch of densely-packed sand), it turns out that in percolation theory, many results can be proved when using hexagonal lattices but not with the traditional square ones. I asked why. "Because," the speaker responded, "The faces of the hexagons are the vertices of triangles." And I had the picture in my mind immediately. (Grab a piece of paper - draw a dot on the center of each hexagon you draw, and connect them - you'll see what I mean. This result is not all that deep or anything - I was just happy to visualize it so quickly.)
Okay, okay, so the more I think about it, the more pathetic a victory that tiny moment was. However, I've been thinking today. Quite frequently, in my head, I think about how I don't feel like I'll ever be worthy of the title 'mathematician' because there are so many people who know so much more about math than I do - so very, very much more. But all of a sudden as I was feeling sorry for myself like this this morning, I thought of a tiny fish saying that it can't swim because it doesn't go as fast or as deep as the mighty narwhal, for example. What a stupid thing to say. Of course it can swim! Damn well, too!
Hold on, though. Maybe, in this metaphor, I might actually be a sea lion- because, I can swim - and in some areas of the ocean, I may even be a force to be reckoned with - but I like to spend a lot of time on land (a.k.a. playing music, reading, thinking about philosophy, psychology, economics, etc) and not in the water (i.e. the murky and seemingly infinite depths of math). In fact, I can't be in either place indefinitely.... Oh dear. Metaphor carried me away. I'll swim back to the point.
So, the point is, as you can clearly see, the GRE is in less than 48 hours and I'm on my last mental legs. But at least, I understand how hexagons can be like triangles. Guess that's a small victory.
Sunday, October 13, 2013
Those unexpected words: "Undress the manikin!"
A few weekends ago, I went with some of my friends to an 'Escape Room' here in Budapest. This is a very local thing, a phenomenon that makes total sense with a culture so steeped in (and skilled at) mathematics, logic, and problem solving*.
It goes like this: you get a group of 2-5 friends and find a little hole-in-the-wall Escape Room (sometimes called Trap Room) place. The person at the desk welcomes you in quietly, asks which room you want to visit or asks for your reservation number. You answer, and the group files into a dusky room, usually in a basement, usually with strange things painted on the walls - dusty, rusty, strange things hanging around the room. Barrels. Rope. Broken mirrors. Piles of clothing. Old photographs. And the man from the counter sees everyone is inside, says "You have one hour," and shuts the door. You hear it lock. You have one hour to figure out how to open the door again.
And you find your way out! It might be small mathematical calculations from strange diagrams painted on the walls, getting a cork out of a wine bottle with only a rag, finding a marble in a dusty corner that then becomes a vital part of a contraption to free a tiny key from a glass box -- clues are hidden around the room and hopefully after an hour, you have opened every possible part of the room (in the case of the one that I did, there were several rooms contained in one, and you had to thoroughly explore all three to find the key code to re open the original door). And our group - fantastically mixed, by the way. It's good to have people who are good at different things, and we had exactly that. Math people and non math people, tall people, small people, creative people, determined people. It was perfect. Everyone did something to help the group. And boy, did it teach you to think of everything as important and worthy of investigation.
At one point, the person who could watch our attempts to get out through the camera in the ceilings of room noticed our time was running out and that we were attempting something fruitless and said those words over the loudspeaker:"Undress the manikin!" We had rifled through the manikin's clothes, found various clues in them, but none of us had dared to undress him completely! I will never look at manikins the same way again. A few of the people in this group are fellow students and when we're stuck on a problem, we've taken to saying that to each other as an expression of being at a loss for what to do next.
I hope to go back and do that again sometime soon, maybe with some different people. I can build my team with care. :) I think it's a brilliant idea, this type of thing. If it were perfect for me, it would also involve a rope climb, or a complex pulley system where we had to lift our team members up - something physical. I feel it's a shame that we don't really use our bodies anymore. Not for real things. Even the gym feels fake and mechanical. Helping someone move, helping in a garden, raking the lawn - I love those things. I just wish they could be incorporated into my daily life more - in a natural way. (I think I'll be one of those wirey sixty-year-olds who bikes to work even in pouring rain...)
All in all, though, a brilliant thing to be able to do. I hope to participate in several more and get myself out of many other locked rooms before this semester is over.
--
*I realized tonight after listening to some fellow students that "problem solving" is a misleading phrase when spoken by mathematicians (or wanna-be mathematicians) to mixed audiences. Here are a few "problems" the way I (we?) use the word: 1. Consider a square constructed by 8x8 smaller tiles, each square in shape (i.e. like a chess board). If one tile is removed from the upper left hand corner of the board and one from the lower right hand board, is it possible to cover the rest of the board in dominos (with no overlaps or pieces hanging off the edge), given that a domino covers two adjacent tiles? 2. How many strictly increasing integer sequences begin with 1 and end with 1000? As you can see - not real life problems. We're as good at those as the rest of you.
Wednesday, October 9, 2013
And it's October!
I'm not quite sure how all of a sudden, it's the 5th week of the semester. I was checking on my flight itinerary for Christmas, and at the top of the webpage, it says "75 Days Until Departure". Wow.
In other news, Miklos is at a conference this week. This means various things to me, but really one important thing - we didn't have a NEW assignment due this week in class. Instead, it was suggested that we spend the weekend typing up revisions of our old assignments. As I may have mentioned before, in Spectral Theory, we write up proofs for an assignment, turn them in via email (fancy math programming - well, not all that fancy, but seeing as I'm still quite new to it, I feel fancy whenever I do it) a few days before class and then during class, get them handed back and see what we did wrong, and those who had the proofs right (or at least mostly) go up and present them for the rest of us. So, the tech copies we have so far are the uncorrected versions, and this kind of correction takes a while. For example, to get the humble vector brackets to appear around something modest, like: <x/2, 5y>, in LaTex (the programming language), one has to type:
In other news, Miklos is at a conference this week. This means various things to me, but really one important thing - we didn't have a NEW assignment due this week in class. Instead, it was suggested that we spend the weekend typing up revisions of our old assignments. As I may have mentioned before, in Spectral Theory, we write up proofs for an assignment, turn them in via email (fancy math programming - well, not all that fancy, but seeing as I'm still quite new to it, I feel fancy whenever I do it) a few days before class and then during class, get them handed back and see what we did wrong, and those who had the proofs right (or at least mostly) go up and present them for the rest of us. So, the tech copies we have so far are the uncorrected versions, and this kind of correction takes a while. For example, to get the humble vector brackets to appear around something modest, like: <x/2, 5y>, in LaTex (the programming language), one has to type:
\langle \frac{x}{2}, 5y \rangle
As you might imagine (with computers being as brilliant as they are) every time I accidentally typed 'angel' instead of 'angle', I spent twenty minutes finding the error that was keeping my document from compiling. Basically, once you are fairly fluent at it, this language makes creating typed math documents much, much easier (for articles, journals, books, - and homework, sometimes). And look how nice it ends up!
 | |
| (who knows if I will have readers this picky, but about two thirds down, YES 'a' can be negative - it should read a=/=0. Haven't fixed that bit yet. :) ) |
I'm still new at it, so I'm not the greatest - my spacing is still sometimes wonky, and I know there are things you can do to make everything that much more neat, or fancy, or whatever- but still! Five weeks ago, I couldn't do this at all, so progress has been made!
As I was going back and editing my assignments, I really did see how much I had learned. The pace of my courses here is such that I never feel like I'm coasting. Back at Mills, I certainly didn't feel like I was coasting the whole time, but I always felt like I was standing on solid ground by the end of a lecture, even if the beginning of a lecture had me feeling a bit wobbly with new concepts. (Well, a few times even at the end I didn't feel like all was well, but the rest of my foundation was so solid that it didn't matter) Here - I don't feel that stability. And until you take the moment to go back and think over what you've done, it feels like you haven't made any progress, if you are used to measuring progress in your own confidence. But I have learned things.
But right now, quite honestly, the main thing on my mind is the GRE. The big bad exam. It's a week from Saturday - on the 19th. And right now, the main thing I'm thinking is that I want to take it now. As I sit and work through ages of tedious integrals, breaking my brain to remember trigonometric identities and recall how to find the equation of a surface in three dimensions - I really realize how much more interesting the math I am doing now is. But studying is important, I realize, which is why I do it. And even if I feel a bit like I'm going mad in the process, it'll be done soon.
I can barely imagine the free time I will have when it's over!
Speaking of free time, I should get back to work now, but I'll leave you with a bit more math humor. Check it out here. Good ol' xkcd. Always good for some fun.
Tuesday, October 1, 2013
Some days are just like this, I suppose.
Every cup I reach for falls over, ever fork slips out of my hand, every idea fades as soon as it's been formed - emails are sent with typos and even my whiteboard won't stay on the wall. And since this is my physical and mental environment, you can imagine how I am reacting to everyone around me today: specifically, I feel like slapping everybody in the face, hunching my shoulders, and going to take a nap for the rest of the day. But it's okay. As soon as I realized that today was just going to be like that no matter what I did, it got a bit easier, and I could giggle a bit at just how ornery I was and how uncooperative the rest of my immediate universe was.
And so now I'm home, I've managed to carve out some headspace for math with some beverage that I have not yet spilled and I think I'll be able to concentrate a bit. I wanted to tell you first, though, about something that happens sometimes in Algebraic Topology. So, Algebraic Topology (ALT for short) is every Tuesday and Thursday from 12-2, that mystical time of day when everyone is either lost in thought because they are daydreaming about lunch, or falling asleep because they just had lunch, or bothering everyone by eating lunch right there and then in class. In general, the rule in Hungary is to start the class actually at a quarter after whichever hour was on the schedule, and then at the halfway mark to take a fifteen minute break. Quite civilized. So, we get started around 12:15. The gears start turning in a woefully slow manner, we brush off the dust of the weekend and start to think about higher-dimensional donuts and loops and other mind-boggling things. By the time the break rolls around, the professor has usually revealed something so similar to what can only be described of as a thought-twister (akin to the familiar tongue-twisters) and has promised us a proof after the break that will resolve our worries and pain in comprehension of the topic at hand.
We shuffle out to the hallway, the open window, the bathroom, or the coffee machine in a daze. We all show up back in the classroom before the break is over - little lost sheep, don't know where else to go. And usually, some kind of deep discussion ensues for the last five minutes of the break before we actually do more math. Today, it started this way - a comment from a friend of mine who sits in the back row and waits for the math to impress him, but in a somehow charming way. "Professor," he says, lounging in the corner. "What if I just... don't believe you?"
I turn around at the question. And I see that instead of the typical sassiness on his face, he actually looks a bit like he just lost a battle for the comprehension of the last topic. (Let me just say that that last concept involved assuming that one can visualize a cube in n dimensions and then hold that in mind while you do weird morphy math to it and glue it somehow to an n+1 - dimensional cube...) "Believe me about what?" The professor asks. This professor is about 34, named Boldizsar (effing badass mathematician name, if you ask me), and seems like he breathes this infinite-dimensional business and could read our ALT textbook the way I do Calvin and Hobbes.
My friend in the back row continues. "Just -- everything. What if I think you're making this all up??" His utter confusion is so plain on his face that this question echoes and skitters around the room and we all laugh, including Boldizsar. For the next ten minutes, we follow along this slightly meta route and talk about mathematics and the larger world, how some it relates to physics, how it's all just a model anyway, how some mathematicians can't even talk to any other mathematicians because their particular area of study is so specific that it doesn't relate to anything else.
And even though I'm about to go and labor over the technical and rigorous definitions we don't do in class but that I need for understanding ("That's the worst and most boring part of mathematics." - quote Boldizsar, when we asked him why we don't do these technical things in class), and even though it hurts and I don't really want to do it on some level - at the end of class, I had another moment when I remembered why I do this. A concept was presented that was so mind-bogglingly abstract for me -- well, let me first briefly clarify. (Am I just avoiding researching the definitions by writing this? I'll let you decide. ) Some/most of what we do in ALT is mind-bogglingly abstract by any normal standards. But when you don't really understand something that is abstract, it doesn't really seem all that insane. That not knowing, that confusion, fits rather easily in my head. For example, I don't know what a black hole is, and when I think about black holes, I think about how I don't understand them, and that's about it. But if someone were to talk to me about what a black hole is really like in terms that I understand, the more I understand what goes into that concept, the harder it is to really think about because the idea is becoming more well-formed.
So today, among all the n-dimensional cubes and infinitesimally knotted pieces of 1-dimensional rope, one concept came up where I knew all of the bits that came into the definition of this new thing, but I never knew they could be put together in that way before. I felt like my jaw actually hit the desk when the idea crashed into my brain. (I write with my face so close to my notebook paper that this is actually quite possible - confirm with old classmates of mine). I understood each of the pieces so well and put them together and it felt like having the idea fully formed was stretching the inside of my brain. That's what math does to my head. It's kinda fun.
And so now I'm home, I've managed to carve out some headspace for math with some beverage that I have not yet spilled and I think I'll be able to concentrate a bit. I wanted to tell you first, though, about something that happens sometimes in Algebraic Topology. So, Algebraic Topology (ALT for short) is every Tuesday and Thursday from 12-2, that mystical time of day when everyone is either lost in thought because they are daydreaming about lunch, or falling asleep because they just had lunch, or bothering everyone by eating lunch right there and then in class. In general, the rule in Hungary is to start the class actually at a quarter after whichever hour was on the schedule, and then at the halfway mark to take a fifteen minute break. Quite civilized. So, we get started around 12:15. The gears start turning in a woefully slow manner, we brush off the dust of the weekend and start to think about higher-dimensional donuts and loops and other mind-boggling things. By the time the break rolls around, the professor has usually revealed something so similar to what can only be described of as a thought-twister (akin to the familiar tongue-twisters) and has promised us a proof after the break that will resolve our worries and pain in comprehension of the topic at hand.
We shuffle out to the hallway, the open window, the bathroom, or the coffee machine in a daze. We all show up back in the classroom before the break is over - little lost sheep, don't know where else to go. And usually, some kind of deep discussion ensues for the last five minutes of the break before we actually do more math. Today, it started this way - a comment from a friend of mine who sits in the back row and waits for the math to impress him, but in a somehow charming way. "Professor," he says, lounging in the corner. "What if I just... don't believe you?"
I turn around at the question. And I see that instead of the typical sassiness on his face, he actually looks a bit like he just lost a battle for the comprehension of the last topic. (Let me just say that that last concept involved assuming that one can visualize a cube in n dimensions and then hold that in mind while you do weird morphy math to it and glue it somehow to an n+1 - dimensional cube...) "Believe me about what?" The professor asks. This professor is about 34, named Boldizsar (effing badass mathematician name, if you ask me), and seems like he breathes this infinite-dimensional business and could read our ALT textbook the way I do Calvin and Hobbes.
My friend in the back row continues. "Just -- everything. What if I think you're making this all up??" His utter confusion is so plain on his face that this question echoes and skitters around the room and we all laugh, including Boldizsar. For the next ten minutes, we follow along this slightly meta route and talk about mathematics and the larger world, how some it relates to physics, how it's all just a model anyway, how some mathematicians can't even talk to any other mathematicians because their particular area of study is so specific that it doesn't relate to anything else.
And even though I'm about to go and labor over the technical and rigorous definitions we don't do in class but that I need for understanding ("That's the worst and most boring part of mathematics." - quote Boldizsar, when we asked him why we don't do these technical things in class), and even though it hurts and I don't really want to do it on some level - at the end of class, I had another moment when I remembered why I do this. A concept was presented that was so mind-bogglingly abstract for me -- well, let me first briefly clarify. (Am I just avoiding researching the definitions by writing this? I'll let you decide. ) Some/most of what we do in ALT is mind-bogglingly abstract by any normal standards. But when you don't really understand something that is abstract, it doesn't really seem all that insane. That not knowing, that confusion, fits rather easily in my head. For example, I don't know what a black hole is, and when I think about black holes, I think about how I don't understand them, and that's about it. But if someone were to talk to me about what a black hole is really like in terms that I understand, the more I understand what goes into that concept, the harder it is to really think about because the idea is becoming more well-formed.
So today, among all the n-dimensional cubes and infinitesimally knotted pieces of 1-dimensional rope, one concept came up where I knew all of the bits that came into the definition of this new thing, but I never knew they could be put together in that way before. I felt like my jaw actually hit the desk when the idea crashed into my brain. (I write with my face so close to my notebook paper that this is actually quite possible - confirm with old classmates of mine). I understood each of the pieces so well and put them together and it felt like having the idea fully formed was stretching the inside of my brain. That's what math does to my head. It's kinda fun.
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