I have two funny stories, for public consumption.
(1)
Last Saturday I went with James and Mike to see the movie "Fearless". Mike was late, so James and I bought our tickets. James went first, and when he was done, the fellow at the register called me up. I thought I heard him ask what movie, so I answered, paid for my ticket, and went in. As James and I were waiting for Mike, he pointed out something. The fellow at the register had actually asked, "How are you doing tonight?" And I answered, "Fearless," and stepped right up.
Awesome.
(2)
A couple of us were hanging out after Greg's Olivetti talk last Tuesday. The others were talking about sporadic groups and related stuff, and I wasn't too interested so I opened the door to leave. I was halfway out when I heard Greg say, "It's a well-known result in group theory that ------- is a dick."
I won't fill in the name, but it's pretty funny no matter who you put in there. After I stopped laughing, I said that although the result made sense to me intuitively, I was wondering what the standard proof was. Without missing a beat, Mike replied that the proof is by inspection.
Awesome.
Wednesday, September 27, 2006
Sunday, September 17, 2006
Bugs on a hotplate with metal rulers
A couple of hours ago I drove back to Ithaca from Saugerties. It was amazing outside. Warm but not hot, a little windy, and the leaves are just starting to catch Autumn flame. I went by way of Route 28 and Route 17, and in the middle of this is a weird little road called Route 30.
No, it’s name is not what is weird about it.
The road twists and turns its way around the Pepacton Reservoir, which is this lovely collection of water surrounded by pillows of hills. (I don’t think I actually knew how to spell “reservoir” until just now.) My favorite part is before you’ve gotten a good look at the water, where you’re mostly encompassed by trees, and suddenly everything opens up and you’re crossing this long, thin bridge across the middle of the open water. To either side the view is stunning. Someday I’m just going to stop in the middle of the bridge and take a good, long look. After the bridge, there is one of the zaniest series of turns I have ever seen a road take. I’d almost swear that some of them turn me around more than 360 degrees, which I would have thought to be very unlikely until I drove on this road.
On my trip I listened to two audiobooks, audiobooks being a secret passion of mine. First I listened to Richard Feynmann’s Six Not So Easy Pieces, which is actually a collection of a couple of recordings of his lectures. He must have been incredible to see in a classroom. I’d actually listened to some of this already, so the remaining bit didn’t fill my entire trip. For the rest of the time, I listed to some of Garrison Keillor’s Tales from Lake Woebegone, which are also amazing. He’s amazing. They’re just great.
What I wanted to write about here, though, is regarding some of Feynmann’s material. It was probably around the fifth piece. He had given an introduction to the idea of spacetime, much of which seemed dedicated to convincing people that the idea of a four dimensional vector space is not so crazy. (I didn’t need much convincing, of course.) He was going to start talking about Einstein’s reinterpretation of gravity as causing curved spacetime. To do this, he wanted to describe what he meant by curved space. He was basically trying to give a very informal, but fairly exact, description of Riemannian manifold.
He explained that as far as four-dimensional space is concerned, our imaginations suck. The only method we have open to us is to work by analogy, imagining how we in three dimensional space would look at someone stuck in two dimensional space. (This is very Flatland, which means it’s very smart.) He then took three examples of two dimensional spaces, and asked us to imagine little bugs who live there. He added later that these are bugs who decided to investigate some geometry. He then paused, and commented that these are very unusual bugs.
The first example was our usual Euclidean plane. The second was the surface of a ball, a sphere. (To many people, a sphere and a ball are the same thing. To mathematicians, of course, a sphere is the surface of a ball.) The way to measure distance in these first two examples is just what you think it is. Now we take a weird turn. His third example was bugs on a hotplate, armed with metal rulers. This requires some explanation, but is actually incredibly cool.
Imagine a hotplate the size of a plane. The temperature is different in different spots. At one spot, which we can think of as the origin of the plane, the temperature is pretty low. The further away from the origin you get, the warmer it gets. Now, the bugs on this hotplate have metal rulers to measure distance, and of course as these metal rulers expand or contract based on the temperature. Near the origin, the rulers stay a pretty normal size, but as the bug carries his ruler away from the origin it expands. This means that when he measures distances far away from the origin, the distances won’t seem as large to him as they do to us. The ruler gets bigger! Do you get it?
What I want to say is that when he’s far away from the origin and he sprints for a second, he hasn’t actually gone as far as he thinks he has. But this is ridiculous. The only way the bug can measure the distance is with his expanded ruler. He doesn’t notice anything! But if we’re watching him from above, we will definitely notice the difference. To us, the further away the bug is from the origin, the faster he can move, because distances which look huge to us are not really that big to the bug anymore. As the bug approaches the origin, he will appear to slow down.
The three spaces that Feynmann was describing were models of the Euclidean plane, the spherical plane, and the hyperbolic plane. He goes on to describe how one can measure curvature in these spaces by studying triangles, or by studying circles. If the curvature isn’t zero, weird stuff starts happening. Mathematical X-Files stuff. An important point he mentions is that the bugs can do this themselves. Curvature is intrinsic. You don’t need some higher dimensional cohort to measure it for you. The Euclidean plane has zero curvature, the spherical plane has positive curvature, and the hyperbolic plane has negative curvature. Feynmann briefly mentions that you could have a space where the curvature changes from place to place, like the space we live in. This is a general Riemannian manifold.
Another cool model of the hyperbolic plane is the Poincare disc. (Poincare should have an accent aigu over the “e”, but I’m afraid it might get lost when I move this text around. Plus I can’t figure out where to find it.) Take the open unit disk in the Euclidean plane, and imagine that the rulers shrink as you stray away from the origin towards the boundary. In fact, they shrink so fast that you move more and more slowly as you get to the boundary - so slowly that you never reach the boundary. This model has a nice complex analysis description, which I don’t remember. (It’s probably not that nice anyway.) You can find very nice pictures of this online, or in Bill Thurston’s book on three dimensional geometry and topology.
I was going to write about my ponderings regarding the origin of the term “hyperbolic” to describe these spaces, but I just looked it up in Wikipedia and found the truth, or at least bread crumbs leading to the truth. Now that I think about it, though, why spoil the suspense? Join me in some speculation.
A hyperbole is a figure of speech whose main feature is gross exaggeration. I will share that I just learned that the root of the word comes from Greek for “overshooting” or “excessive”. (I think I could have reasonably been expected to have deduced this.) The main feature of the hyperbolic models I described above is that weird things started happening as you moved towards the fringes, or away from the middle. The weird things were actually of a very specific type, namely exaggeration of length, or its opposite, understatement of length. I think I’ll stop there. I’ve shared enough of my brilliant insight.
No, it’s name is not what is weird about it.
The road twists and turns its way around the Pepacton Reservoir, which is this lovely collection of water surrounded by pillows of hills. (I don’t think I actually knew how to spell “reservoir” until just now.) My favorite part is before you’ve gotten a good look at the water, where you’re mostly encompassed by trees, and suddenly everything opens up and you’re crossing this long, thin bridge across the middle of the open water. To either side the view is stunning. Someday I’m just going to stop in the middle of the bridge and take a good, long look. After the bridge, there is one of the zaniest series of turns I have ever seen a road take. I’d almost swear that some of them turn me around more than 360 degrees, which I would have thought to be very unlikely until I drove on this road.
On my trip I listened to two audiobooks, audiobooks being a secret passion of mine. First I listened to Richard Feynmann’s Six Not So Easy Pieces, which is actually a collection of a couple of recordings of his lectures. He must have been incredible to see in a classroom. I’d actually listened to some of this already, so the remaining bit didn’t fill my entire trip. For the rest of the time, I listed to some of Garrison Keillor’s Tales from Lake Woebegone, which are also amazing. He’s amazing. They’re just great.
What I wanted to write about here, though, is regarding some of Feynmann’s material. It was probably around the fifth piece. He had given an introduction to the idea of spacetime, much of which seemed dedicated to convincing people that the idea of a four dimensional vector space is not so crazy. (I didn’t need much convincing, of course.) He was going to start talking about Einstein’s reinterpretation of gravity as causing curved spacetime. To do this, he wanted to describe what he meant by curved space. He was basically trying to give a very informal, but fairly exact, description of Riemannian manifold.
He explained that as far as four-dimensional space is concerned, our imaginations suck. The only method we have open to us is to work by analogy, imagining how we in three dimensional space would look at someone stuck in two dimensional space. (This is very Flatland, which means it’s very smart.) He then took three examples of two dimensional spaces, and asked us to imagine little bugs who live there. He added later that these are bugs who decided to investigate some geometry. He then paused, and commented that these are very unusual bugs.
The first example was our usual Euclidean plane. The second was the surface of a ball, a sphere. (To many people, a sphere and a ball are the same thing. To mathematicians, of course, a sphere is the surface of a ball.) The way to measure distance in these first two examples is just what you think it is. Now we take a weird turn. His third example was bugs on a hotplate, armed with metal rulers. This requires some explanation, but is actually incredibly cool.
Imagine a hotplate the size of a plane. The temperature is different in different spots. At one spot, which we can think of as the origin of the plane, the temperature is pretty low. The further away from the origin you get, the warmer it gets. Now, the bugs on this hotplate have metal rulers to measure distance, and of course as these metal rulers expand or contract based on the temperature. Near the origin, the rulers stay a pretty normal size, but as the bug carries his ruler away from the origin it expands. This means that when he measures distances far away from the origin, the distances won’t seem as large to him as they do to us. The ruler gets bigger! Do you get it?
What I want to say is that when he’s far away from the origin and he sprints for a second, he hasn’t actually gone as far as he thinks he has. But this is ridiculous. The only way the bug can measure the distance is with his expanded ruler. He doesn’t notice anything! But if we’re watching him from above, we will definitely notice the difference. To us, the further away the bug is from the origin, the faster he can move, because distances which look huge to us are not really that big to the bug anymore. As the bug approaches the origin, he will appear to slow down.
The three spaces that Feynmann was describing were models of the Euclidean plane, the spherical plane, and the hyperbolic plane. He goes on to describe how one can measure curvature in these spaces by studying triangles, or by studying circles. If the curvature isn’t zero, weird stuff starts happening. Mathematical X-Files stuff. An important point he mentions is that the bugs can do this themselves. Curvature is intrinsic. You don’t need some higher dimensional cohort to measure it for you. The Euclidean plane has zero curvature, the spherical plane has positive curvature, and the hyperbolic plane has negative curvature. Feynmann briefly mentions that you could have a space where the curvature changes from place to place, like the space we live in. This is a general Riemannian manifold.
Another cool model of the hyperbolic plane is the Poincare disc. (Poincare should have an accent aigu over the “e”, but I’m afraid it might get lost when I move this text around. Plus I can’t figure out where to find it.) Take the open unit disk in the Euclidean plane, and imagine that the rulers shrink as you stray away from the origin towards the boundary. In fact, they shrink so fast that you move more and more slowly as you get to the boundary - so slowly that you never reach the boundary. This model has a nice complex analysis description, which I don’t remember. (It’s probably not that nice anyway.) You can find very nice pictures of this online, or in Bill Thurston’s book on three dimensional geometry and topology.
I was going to write about my ponderings regarding the origin of the term “hyperbolic” to describe these spaces, but I just looked it up in Wikipedia and found the truth, or at least bread crumbs leading to the truth. Now that I think about it, though, why spoil the suspense? Join me in some speculation.
A hyperbole is a figure of speech whose main feature is gross exaggeration. I will share that I just learned that the root of the word comes from Greek for “overshooting” or “excessive”. (I think I could have reasonably been expected to have deduced this.) The main feature of the hyperbolic models I described above is that weird things started happening as you moved towards the fringes, or away from the middle. The weird things were actually of a very specific type, namely exaggeration of length, or its opposite, understatement of length. I think I’ll stop there. I’ve shared enough of my brilliant insight.
Friday, September 15, 2006
More on riding the bus
Anyone who's been on city buses more than a couple of times has probabaly seen it. Two buses trying to round a sharp corner at the same time. A bus making a right turn, the necessary width of its right turn obscured by a car that has stopped to close to the stop sign. These and other nightmare situations seem to face bus drivers on a nearly daily basis, but they usually manage to steer their vessels through safely.
I find this absolutely amazing.
Now, to me, most of the simple feats involved in driving a bus seem incredibly difficult. It's such a huge contraption, and the roadways are so darn tiny and intricate. But as with most things, I'm sure it's mostly a matter of practicepracticepractice. However, these extraordinary situations I described above seem to be a quantum leap in difficulty above the rest. I've seen some incredible driving, buses passing within a centimeter of each other, or within a centimeter of a car, or a stop sign, or sometimes even a person.
I think that when this happens, when our bus driver performs what, to me, is nearly a feat of miraculous proportions, we ought to give a round of applause. That's right. The entire bus should put hands together in honor of the driver's skill, daring, and composure.
I haven't seen, nor heard, or this happening yet. But on of these days, I'm going to clap. And I hope everyone joins in.
I find this absolutely amazing.
Now, to me, most of the simple feats involved in driving a bus seem incredibly difficult. It's such a huge contraption, and the roadways are so darn tiny and intricate. But as with most things, I'm sure it's mostly a matter of practicepracticepractice. However, these extraordinary situations I described above seem to be a quantum leap in difficulty above the rest. I've seen some incredible driving, buses passing within a centimeter of each other, or within a centimeter of a car, or a stop sign, or sometimes even a person.
I think that when this happens, when our bus driver performs what, to me, is nearly a feat of miraculous proportions, we ought to give a round of applause. That's right. The entire bus should put hands together in honor of the driver's skill, daring, and composure.
I haven't seen, nor heard, or this happening yet. But on of these days, I'm going to clap. And I hope everyone joins in.
Thursday, September 14, 2006
The semester so far
The semester so far is going pretty well. I certainly have lots to say about it, not all of which is appropriate for mixed company to read.
Teaching is finally going pretty well. I'm teaching a course one step above the one I have taught before, but it's good. It keeps me honest. I'm forced to spend considerable time with the material before I teach it, and that's never a bad idea. The first couple of classes were a little bumpy, but after a little talk and work with my dad, the legendary guru of math teaching, I was able to find smoother seas. My planning of each class is now pretty meticulous, with room left for improvisation. The most important thing I've picked up is to present the material very slowly. Start with the simplest examples, the simplest cases, imaginable. Don't throw more than one trick at them at once, until they've seen each of the tricks individually first. It's working very well.
I'm mostly lecturing, with a little group work on Fridays, but I'm comfortable with that. When I'm on, I'm a pretty good lecturer, and I don't think the students get too bored. (One of them does have an intriguing habit of reading a newspaper when I'm speaking, but I'm not sure he's in the right place, anyway.) The students seem nice, although quiet. I teach at 12:20, so everyone's either a little groggy from just eating, or fading fast from not eating yet. Also, I teach in a lecture style room, with stadium seating, so it's a little hard to organize group work, but that's ok. Not all of them seem too ridiculously eager to do the group work, and a lot of them work alone, but that's ok too. They don't have to like it, and they don't even have to do it.
Sometimes their incredible silence is a little irritating, like when I ask them questions as a group. Yesterday, I asked who had seen long division of polynomials before. Silence. I asked if anyone had not seen long division of polynomials before. More silence. I asked if we should all close our eyes, except for me, so that no one would be embarassed to admit he/she hadn't seen it before. Still more silence. I yelled "ARE YOU STILL BREATHING???" There was still silence, but a whole lot of very wide eyes. Some people nodded. Good enough. I told them I was just checking.
I'll probably write later about how my own classes and research are going. Again, there's a lot to say. I also should write about my cat situation at some point.
I'm finally getting settled in with the new kids, and adjusting to all of the people who have left. Let me focus on the new kids. There are FOUR GIRLS! And I think they're all nice! I know the other new folks a little bit, and there's only one of them that I wish dead. Most of them are pretty nice, and only smarter than I am by a tolerable amount. I was even feeling better about the one I wanted to die until just a couple of minutes ago.
Our paths crossed, and he thanked me, saying that even though I didn't know it I had loaned him one of my textbooks for about twenty minutes earlier today. He had left a note, but had finished with it before I got to my desk today, so he removed the note. This is not kosher.
I've said it before, and I'll say it again. I wish I had the mutant power to make anvils appear and fall on people's heads. Anvils or pianos. With pianos, you get that wonderful musical crash when they hit.
Of course, I could have just explained that I'd rather he didn't borrow books without asking, but I feel this is somewhat dangerous territory. The math department is not entirely the happy-go-lucky crew it once was. There are some entirely untrustworthy characters floating around, and I'm not even referring to the new guy who I wish was dead. There is no security at our desks except for the drawers, whose locks are probably not monumental in their sturdiness. This means anything on our desks, such as, say, textbooks, are up for grabs. So far this hasn't been abused terribly much by anyone, but I feel that it's a very delicate balance. Far be it from me to stir the waters.
This entry has taken on a darker tone than I intended.
Teaching is finally going pretty well. I'm teaching a course one step above the one I have taught before, but it's good. It keeps me honest. I'm forced to spend considerable time with the material before I teach it, and that's never a bad idea. The first couple of classes were a little bumpy, but after a little talk and work with my dad, the legendary guru of math teaching, I was able to find smoother seas. My planning of each class is now pretty meticulous, with room left for improvisation. The most important thing I've picked up is to present the material very slowly. Start with the simplest examples, the simplest cases, imaginable. Don't throw more than one trick at them at once, until they've seen each of the tricks individually first. It's working very well.
I'm mostly lecturing, with a little group work on Fridays, but I'm comfortable with that. When I'm on, I'm a pretty good lecturer, and I don't think the students get too bored. (One of them does have an intriguing habit of reading a newspaper when I'm speaking, but I'm not sure he's in the right place, anyway.) The students seem nice, although quiet. I teach at 12:20, so everyone's either a little groggy from just eating, or fading fast from not eating yet. Also, I teach in a lecture style room, with stadium seating, so it's a little hard to organize group work, but that's ok. Not all of them seem too ridiculously eager to do the group work, and a lot of them work alone, but that's ok too. They don't have to like it, and they don't even have to do it.
Sometimes their incredible silence is a little irritating, like when I ask them questions as a group. Yesterday, I asked who had seen long division of polynomials before. Silence. I asked if anyone had not seen long division of polynomials before. More silence. I asked if we should all close our eyes, except for me, so that no one would be embarassed to admit he/she hadn't seen it before. Still more silence. I yelled "ARE YOU STILL BREATHING???" There was still silence, but a whole lot of very wide eyes. Some people nodded. Good enough. I told them I was just checking.
I'll probably write later about how my own classes and research are going. Again, there's a lot to say. I also should write about my cat situation at some point.
I'm finally getting settled in with the new kids, and adjusting to all of the people who have left. Let me focus on the new kids. There are FOUR GIRLS! And I think they're all nice! I know the other new folks a little bit, and there's only one of them that I wish dead. Most of them are pretty nice, and only smarter than I am by a tolerable amount. I was even feeling better about the one I wanted to die until just a couple of minutes ago.
Our paths crossed, and he thanked me, saying that even though I didn't know it I had loaned him one of my textbooks for about twenty minutes earlier today. He had left a note, but had finished with it before I got to my desk today, so he removed the note. This is not kosher.
I've said it before, and I'll say it again. I wish I had the mutant power to make anvils appear and fall on people's heads. Anvils or pianos. With pianos, you get that wonderful musical crash when they hit.
Of course, I could have just explained that I'd rather he didn't borrow books without asking, but I feel this is somewhat dangerous territory. The math department is not entirely the happy-go-lucky crew it once was. There are some entirely untrustworthy characters floating around, and I'm not even referring to the new guy who I wish was dead. There is no security at our desks except for the drawers, whose locks are probably not monumental in their sturdiness. This means anything on our desks, such as, say, textbooks, are up for grabs. So far this hasn't been abused terribly much by anyone, but I feel that it's a very delicate balance. Far be it from me to stir the waters.
This entry has taken on a darker tone than I intended.
Riding the bus
This morning, as every weekday morning for the past several weeks, I rode the bus to school. This afternoon, as most afternoons, I will ride the bus from school to home. I could say that I have a "love/hate relationship" with riding the bus, but that would leave shades of meaning unturned, all in the cause of using a pithy and already heavily burdened phrase. The truth is, usually I like riding the bus, occassionally I hate riding the bus, and sometimes I have no real opinion about it.
I don't like it when the bus is very crowded, or when it's running very late, but this is happening less and less now. This morning I had a wonderful bus experience. There were seats left when I got on, but no pair of seats totally unoccupied. Rather than impose my considerable volume on another passenger, I chose to stand. For the entire bus-ride, I was the only one standing, and I had a blast. It was like surfing (I imagine), or at least like a very fun ride on which you are allowed to stand up. With the bus's current detour route, we make quite a trip around twists and turns, and over a lot of bumps. With no one else standing up, I could hold onto any or as many bars as I wanted, and it was great.
When it came to my stop, another guy and I both tried waving our hands in front of the door sensor at the same time, and we had a little gentle hand-slapping episode. It occurred to me that it's very good that the instructions for the door say "wave your hand", and not "karate-chop your hand". That would have been a very different situation.
The other thing I like about the bus is that, outside of school, it is the place where I find the most general social interaction. I wouldn't go so far as to say that I meet people, but I definitely experience other people, to a small extent. It's not always pleasant, but it's almost always a little interesting. The most interesting thing is how darn quiet everyone is at the bus stop. I think I've talked to people twice so far.
I've only taken the bus sporadically in the past. This is the first time where I'm forced to take it daily. The last time I had to take a bus-like contrivance on a daily basis was in junior high school, and I stopped that pretty soon because of my tendency to be tripped by other riders who were somewhat evil and very much stupid. One of the people who tripped me was working in the Stewart's convenience store near my home a couple of years after we graduated. I only saw him there once, and he tried to have a little friendly conversation and reminisce, as if we were great friends in school. What a bastard.
So far the bus here is much better. No one's tripped me yet, and I've only stumbled on my own once or twice, which is about how often I stumble when not on the bus anyway.
I don't like it when the bus is very crowded, or when it's running very late, but this is happening less and less now. This morning I had a wonderful bus experience. There were seats left when I got on, but no pair of seats totally unoccupied. Rather than impose my considerable volume on another passenger, I chose to stand. For the entire bus-ride, I was the only one standing, and I had a blast. It was like surfing (I imagine), or at least like a very fun ride on which you are allowed to stand up. With the bus's current detour route, we make quite a trip around twists and turns, and over a lot of bumps. With no one else standing up, I could hold onto any or as many bars as I wanted, and it was great.
When it came to my stop, another guy and I both tried waving our hands in front of the door sensor at the same time, and we had a little gentle hand-slapping episode. It occurred to me that it's very good that the instructions for the door say "wave your hand", and not "karate-chop your hand". That would have been a very different situation.
The other thing I like about the bus is that, outside of school, it is the place where I find the most general social interaction. I wouldn't go so far as to say that I meet people, but I definitely experience other people, to a small extent. It's not always pleasant, but it's almost always a little interesting. The most interesting thing is how darn quiet everyone is at the bus stop. I think I've talked to people twice so far.
I've only taken the bus sporadically in the past. This is the first time where I'm forced to take it daily. The last time I had to take a bus-like contrivance on a daily basis was in junior high school, and I stopped that pretty soon because of my tendency to be tripped by other riders who were somewhat evil and very much stupid. One of the people who tripped me was working in the Stewart's convenience store near my home a couple of years after we graduated. I only saw him there once, and he tried to have a little friendly conversation and reminisce, as if we were great friends in school. What a bastard.
So far the bus here is much better. No one's tripped me yet, and I've only stumbled on my own once or twice, which is about how often I stumble when not on the bus anyway.
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