Origami Black Hole
3 days ago
Thursday, May 12, 2005
30 pages
I just finished writing and printing out my 30-page paper. Five pages of that were contents, appendix, bibliography, etc. But the other 25 were writing. And now I'm done, and I am very proud of myself. I wonder if my professor will read it. She will be receiving 50 papers, all at least 10 pages in length. But she is a professor of education, so she probably will. And it's 2:30 AM. That's late.
Sunday, May 08, 2005
Williams-Mystic
So, the suspense is over: I was accepted to Williams-Mystic! Golf clap. I think the admission rates for this program are exceedingly high. Williams-Mystic is "the maritime studies program at Mystic Seaport" which is a Williams program wherein a group of students, several of whom are from Williams, go to Mystic, CT and take four classes about maritime things and learn to sail boats and whatnot. It is a semester program. I will be going on this program. This means that I might drop the psych major. Sorry about that, psych major. I'd choose a thesis over a psych major. In case you want to know more, here is the link: Williams-Mystic (but their site is not so excellent, sorry to say).
Saturday, May 07, 2005
The Möbius strip
Today, I decided to do a little Möbius strip project that I have been thinking about for a while. While I was doing this project, I realized that I didn't understand exactly how the Möbius strip works, so I decided to do a little experiment.
("What are you doing?" asked my S.O. "I'm doing a math experiment," I said. "There are no math experiments," said my S.O., "math is not an experimental science." "Then I'm performing a miracle," I said.)
I took a strip of paper and taped it together to make a Möbius strip, with one half-twist. Then along the left side of the strip, I made a purple line, and next to it on the right side of the strip, I made a yellow line. Note that this means that along the whole strip, there were adjacent purple and yellow lines, and left and right mean nothing; they're just to orient you.
I was wondering what would happen to the lines when I cut the strip the long way down the middle, because I figured that it couldn't possibly just end up with the yellow line switching to purple or something; it had to be that the yellow line would be on one side and the purple line would be on the other. And that is what happened! Wow.
So I figured the next thing should be to see what happens when you add more stripes. So to the right of the purple line I drew a blue line, and to the left of the yellow line I drew a red line, both all the way along the whole strip. So now on one side of my strip there were parallel blue and purple lines, and on the other strip there were parallel yellow and red lines.
("Logical fallacy," you say: "A Möbius strip has only one side." Correct, but once you cut it in half the long way, it ends up having two sides, and it's not a Möbius strip anymore. Pity, I know, but it's the truth. Instead of one half-turn, a cut-in-half Möbius strip has two full turns (four half-turns), I think.)
The idea of doing this was that you could not end up with red, yellow, purple, and blue each on their own side, because there are only two sides. So something drastic had to occur. This is probably common knowledge, but here is what happens when you cut an already-cut Möbius strip in half again: You get two interlocking twisted strips! They each have four half-turns. And they're not just singly linked, like the Olympic Rings, but are kind of doubly linked.
Naturally I tried cutting them in half again, and I got four rings which are all interconnected in a kind of knotted-up way. That wasn't very interesting, but I knew you'd be wondering. I should try it with a bigger piece of paper next time. For a more interesting experiment, try this, suggested by Mathworld (link below): "In addition, two strips on top of each other, each with a half-twist, give a single strip with four twists when disentangled." It's true! It happened to me! Whoa.
For anyone who is very interested in the Möbius strip (and you had better be interested, Ms. 3.95) check out the animated Möbius strip on Mathworld. You have to have some sort of Java to see it, I think, but if you do, you can click and drag to spin it around, and if you kind of drag and let it go as though you are throwing it, it will spin in the direction in which you threw it, so long as you keep your cursor within the boundary of the graphic. Awesome. I think that I was talking about this awesome graphic when someone gave me the idea for the project that inspired me to investigate the particulars of the Möbius strip -- a project that I will soon be sending to you, Ms. 3.95, which is why you had better be interested, because today, I decided to do a little Möbius strip project that I have been... yes, you see, it all comes together at the ends, just like a Möbius strip.
("What are you doing?" asked my S.O. "I'm doing a math experiment," I said. "There are no math experiments," said my S.O., "math is not an experimental science." "Then I'm performing a miracle," I said.)
I took a strip of paper and taped it together to make a Möbius strip, with one half-twist. Then along the left side of the strip, I made a purple line, and next to it on the right side of the strip, I made a yellow line. Note that this means that along the whole strip, there were adjacent purple and yellow lines, and left and right mean nothing; they're just to orient you.
I was wondering what would happen to the lines when I cut the strip the long way down the middle, because I figured that it couldn't possibly just end up with the yellow line switching to purple or something; it had to be that the yellow line would be on one side and the purple line would be on the other. And that is what happened! Wow.
So I figured the next thing should be to see what happens when you add more stripes. So to the right of the purple line I drew a blue line, and to the left of the yellow line I drew a red line, both all the way along the whole strip. So now on one side of my strip there were parallel blue and purple lines, and on the other strip there were parallel yellow and red lines.
("Logical fallacy," you say: "A Möbius strip has only one side." Correct, but once you cut it in half the long way, it ends up having two sides, and it's not a Möbius strip anymore. Pity, I know, but it's the truth. Instead of one half-turn, a cut-in-half Möbius strip has two full turns (four half-turns), I think.)
The idea of doing this was that you could not end up with red, yellow, purple, and blue each on their own side, because there are only two sides. So something drastic had to occur. This is probably common knowledge, but here is what happens when you cut an already-cut Möbius strip in half again: You get two interlocking twisted strips! They each have four half-turns. And they're not just singly linked, like the Olympic Rings, but are kind of doubly linked.
Naturally I tried cutting them in half again, and I got four rings which are all interconnected in a kind of knotted-up way. That wasn't very interesting, but I knew you'd be wondering. I should try it with a bigger piece of paper next time. For a more interesting experiment, try this, suggested by Mathworld (link below): "In addition, two strips on top of each other, each with a half-twist, give a single strip with four twists when disentangled." It's true! It happened to me! Whoa.
For anyone who is very interested in the Möbius strip (and you had better be interested, Ms. 3.95) check out the animated Möbius strip on Mathworld. You have to have some sort of Java to see it, I think, but if you do, you can click and drag to spin it around, and if you kind of drag and let it go as though you are throwing it, it will spin in the direction in which you threw it, so long as you keep your cursor within the boundary of the graphic. Awesome. I think that I was talking about this awesome graphic when someone gave me the idea for the project that inspired me to investigate the particulars of the Möbius strip -- a project that I will soon be sending to you, Ms. 3.95, which is why you had better be interested, because today, I decided to do a little Möbius strip project that I have been... yes, you see, it all comes together at the ends, just like a Möbius strip.
Thursday, May 05, 2005
05.05.05
That's what today is -- 05/05/05, or as I like to call it, 05.05.05. If you were awake at 05:05 this morning, that would be really special -- 05:05:05 05.05.05. But if you missed it this year, there's always 06:06:06 06.06.06 next year, and every year until 2012 (12:12:12 12.12.12). Professor Morgan pointed this out in -- you guessed it -- Math 305. Too bad the class meets at 9:55, although that has its fair share of 5's.
Today Vojislav and I were working in the math library on topology, and Professor Morgan was talking with us as he was about to leave. Then it apparently occurred to him that Vojislav is from last year's Geometry Group, I am going to be in this year's Geometry Group, and he was going to an open house hosted by a member of the 1990 Geometry Group (the one that published the paper proving that the double bubble is perimeter-minimizing in R2 in Euclidean space). So he invited us to come along. So half an hour later, Vojislav and I were on our way to Pownal, VT where we got to see this guy's new house. It was a nice house, with a big field going down to a marsh. And Professor Morgan took us to get ice cream afterwards. I had graham cracker flavored ice cream (on his recommendation). It was yummy.
We went out to look at their field, which slopes down towards the marsh, and I found myself trying to figure out what had happened in the past to make it look like this, just as we did at the Mountain School. I determined that the part by the house had been a field for a while, because it was flat and mown. There were about six trees dispersed through this field, and since they all looked different, I decided that they were different species of trees that had been planted, rather than growing naturally from seeds. On the edge of the field at the bottom there was a peculiar swath leading down to the marsh with no trees on it, which really perplexed me. To the left was a big tree on the corner, which means that it was on the edge of the field, and behind the big tree to the left were trees about 10 inches in radius, which were all about the same size. By contrast, on the right of the cleared swath was an area full of little trees, pretty tall (20-30 feet) but all under 4 inches in diameter. So I figured that area had been pasture much longer than the part on the left, and had at some point been left to grow into trees, at which time all the little trees grew in.
For anyone who kept reading, try this, which is our Geometry Group homework:
If you finish that, try this:
Next hint: It's not a circle. But that was a good guess.
Today Vojislav and I were working in the math library on topology, and Professor Morgan was talking with us as he was about to leave. Then it apparently occurred to him that Vojislav is from last year's Geometry Group, I am going to be in this year's Geometry Group, and he was going to an open house hosted by a member of the 1990 Geometry Group (the one that published the paper proving that the double bubble is perimeter-minimizing in R2 in Euclidean space). So he invited us to come along. So half an hour later, Vojislav and I were on our way to Pownal, VT where we got to see this guy's new house. It was a nice house, with a big field going down to a marsh. And Professor Morgan took us to get ice cream afterwards. I had graham cracker flavored ice cream (on his recommendation). It was yummy.
We went out to look at their field, which slopes down towards the marsh, and I found myself trying to figure out what had happened in the past to make it look like this, just as we did at the Mountain School. I determined that the part by the house had been a field for a while, because it was flat and mown. There were about six trees dispersed through this field, and since they all looked different, I decided that they were different species of trees that had been planted, rather than growing naturally from seeds. On the edge of the field at the bottom there was a peculiar swath leading down to the marsh with no trees on it, which really perplexed me. To the left was a big tree on the corner, which means that it was on the edge of the field, and behind the big tree to the left were trees about 10 inches in radius, which were all about the same size. By contrast, on the right of the cleared swath was an area full of little trees, pretty tall (20-30 feet) but all under 4 inches in diameter. So I figured that area had been pasture much longer than the part on the left, and had at some point been left to grow into trees, at which time all the little trees grew in.
For anyone who kept reading, try this, which is our Geometry Group homework:
Given that a circle is perimeter minimizing for given area in the Euclidean plane (i.e., R2), prove that a semicircle is perimeter minimizing in the Euclidean halfplane (i.e., just the part above the x-axis), and that a quarter circle is perimeter minimizing in the first quadrant.I have proven it. It is a very short proof.
If you finish that, try this:
Gaussian space is endowed with variable density, so that points near the origin are denser than points far away from the origin. The distribution follows a normal distribution (bell-shaped curve). So area is worth more near the origin, but perimeter also costs more near the origin. What is the perimeter-minimizing shape to enclose a given percentage (say, 10%, or 30%, etc.) of the Gaussian plane?Hint: it's a shape you've heard of.
Next hint: It's not a circle. But that was a good guess.
Monday, May 02, 2005
HRUMCXII
On Saturday we hosted the HRUMC XII at Williams. For those of you who don't know, that is the Hudson River Undergraduate Mathematics Conference, year 12. Ken Ribet came to speak. Do you know who he is? Probably not. You will soon enough.
A long time ago, a guy called Fermat said that there are no nonzero integer solutions to the equation an + bn = cn for n > 2. He said he had a clever proof that was just too big to fit in the margin he was writing in, and so mathematicians tried for many years to prove what was known as "Fermat's Last Theorem." Then in the middle of the 20th century, over in Japan two mathematicians called Taniyama and Shimura were doing some seemingly unrelated mathematics, leading to the conjecture that modular forms and elliptic curves were essentially the same thing. Then another mathematician gave a plausibility argument that if Fermat's last theorem were false, it would create a bizarre function that would not fit into the structure of the so-called Taniyama-Shimura conjecture, rendering Fermat and Taniyama-Shimura logically equivalent -- but he couldn't prove it. So Ken Ribet came along and proved it. And then Andrew Wiles proved the Taniyama-Shimura conjecture, which proved Fermat's Last Theorem, which is what all the fuss was about with Fermat's Last Theorem finally being proven.
Here are some pictures of us Williams people with Ken Ribet:
Professor Pacelli, Professor Ribet, Neil, me, and Brian.
Ken Ribet and me.
I talked to the people who were giving a talk on "A Problem Oriented Approach to Geometry." Here was their abstract:
I also gave my talk on Latin squares. People actually came to watch it. It was astounding. People actually came into the room right before my talk started, to watch it. It was amazing. Who would've though that people actually would come? And the guy chairing the session memorized the number in my talk. That was nice of him, but I felt kind of bad because it was a really long and somewhat useless number. It was actually a pretty engaging and informative talk, if I do say so myself, though not as enlightening as talks that were of a higher level. And 500 people came to the conference. That was a lot of people. And it rained. And Kate Kraft was there. But she didn't come to my talk.
Oh, and the number in the title is the total number of 10x10 Latin squares. I assume you were wondering.
I got into Williams-Mystic! Yay! Take that, Williams-Exeter Programme in Oxford! Take your humanities majors and your stellar GPAs and leave all the engaged mathematical learners back in this country; I'm going to Mystic! (Williams-Mystic site)
A long time ago, a guy called Fermat said that there are no nonzero integer solutions to the equation an + bn = cn for n > 2. He said he had a clever proof that was just too big to fit in the margin he was writing in, and so mathematicians tried for many years to prove what was known as "Fermat's Last Theorem." Then in the middle of the 20th century, over in Japan two mathematicians called Taniyama and Shimura were doing some seemingly unrelated mathematics, leading to the conjecture that modular forms and elliptic curves were essentially the same thing. Then another mathematician gave a plausibility argument that if Fermat's last theorem were false, it would create a bizarre function that would not fit into the structure of the so-called Taniyama-Shimura conjecture, rendering Fermat and Taniyama-Shimura logically equivalent -- but he couldn't prove it. So Ken Ribet came along and proved it. And then Andrew Wiles proved the Taniyama-Shimura conjecture, which proved Fermat's Last Theorem, which is what all the fuss was about with Fermat's Last Theorem finally being proven.
Here are some pictures of us Williams people with Ken Ribet:
Professor Pacelli, Professor Ribet, Neil, me, and Brian.
Ken Ribet and me.
I talked to the people who were giving a talk on "A Problem Oriented Approach to Geometry." Here was their abstract:
We think it may be possible to organize geometry courses around a set of problems. This approach seems unusual. We have not collected enough problems for a complete course but we have started such a collection. We will share examples of problems we think are interesting, challenging and instructive.So before the conference I e-mailed them and told them that I had learned not only geometry, but also everything else after algebra and all the way up through BC calculus via a problem-oriented approach. I gave them the URL for the Exeter teaching materials and they wrote back and said something like "that's very interesting, thanks, I'll check it out sometime." They did not sound very interested. But when I went to talk to them on Saturday, they were very excited about the curriculum, said it was the best thing ever (an exaggerated paraphrase) and said that they were going to print out the whole thing (no joke, all 400 pdf pages). I told them it was most definitely the best math curriculum ever because people actually learn math, and they don't forget it, even over the summer, even now that I haven't done that math for two years I still remember the problems from ninth grade and the strategies to solve them. It is the best math curriculum ever. Don't waste your time with a textbook. Go straight to the heart of real mathematics and actually learn something. Please.
I also gave my talk on Latin squares. People actually came to watch it. It was astounding. People actually came into the room right before my talk started, to watch it. It was amazing. Who would've though that people actually would come? And the guy chairing the session memorized the number in my talk. That was nice of him, but I felt kind of bad because it was a really long and somewhat useless number. It was actually a pretty engaging and informative talk, if I do say so myself, though not as enlightening as talks that were of a higher level. And 500 people came to the conference. That was a lot of people. And it rained. And Kate Kraft was there. But she didn't come to my talk.
Oh, and the number in the title is the total number of 10x10 Latin squares. I assume you were wondering.
I got into Williams-Mystic! Yay! Take that, Williams-Exeter Programme in Oxford! Take your humanities majors and your stellar GPAs and leave all the engaged mathematical learners back in this country; I'm going to Mystic! (Williams-Mystic site)
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