Tall, Dark, and Mysterious

7/15/2005

Dispatches from the library

File under: Sound And Fury, This One Time, At Mathcamp, Those Who Can't. Posted by Moebius Stripper at 5:03 pm.
  1. Apparently my computer has some hard-to-diagnose computer ailment. In a sequence of events that bears striking parallels to those often experienced in my country’s health care system for humans, this has resulted in the blasted machine being discharged - thrice - from the hospital for sick computers before it was fully cured, only to be sent back for more tests and treatment. Fortunately, my computer’s health care has another important feature in common with mine: it is free. Unfortunately, I don’t know how long the waiting list is before it can see the appropriate specialist. Blissfully, this whole thing has been handled by my father, who has advocated tirelessly on its behalf. And now that I’ve praised my father, I won’t feel as guilty when I post this hilarious story involving him making a really stupid bet with my brother back when we were kids. Stay tuned!
  2. I had a lovely, lovely week at Mathcamp. Briefly:

    • I taught two classes - a squishy, very visual version of projective geometry, and Calculus Without Calculus - to the best audience of ten that anyone has ever taught. Rather than having students bitch and moan about how I gave them homework that, like, was totally unfair because it made them think and shit, I had students request that I skip the routine calculations because there were only ten minutes left in class and they wanted to get to the cool stuff. I also had those same students construct a model of the real projective plane using carrots and toothpicks.
    • On the whole, our campers kick all kinds of ass in all kinds of ways. However, every now and again we encounter some unpleasant behaviours that need to be addressed. For instance, we often have a small but vocal contingent of campers who boast, at length, about their mathematical prowess. Last year, some of the staff had the idea of addressing these sorts of things by presenting humourous skits that parodied the unpleasant behaviour.

      Fellow Mathcamp staff member A had the idea of writing a skit in which one character, played by me, would list all of the insanely difficult (”for a beginner, I suppose”) math classes she was taking. In researching for the part, we decided we needed to come up with extremely technical-sounding class names - ones that appeared to be about math, but were actually nonsense. We spent some time trying to come up with such technical sounding gibberish, until A had an idea: “Go to the ArXiv,” he suggested, “and look up quantum algebra!”

      A simple permutation of preprint titles resulted in our fictitious braggart boasting about her exploits in hypercategory theory, m-difference representations, quantum affine algebroids, cohomology of semiregular twistor spectra, and quasi-coherent sheaves on Calabi-Yau manifolds, Moore Method.

      I think our message got through. And if any of those topics do exist, I’m not sure I want to know.

  3. And now I’m back at not-Mathcamp, still waiting the remaining two weeks for the hardworking bureaucrats at the Employment Insurance headquarters to finish transfering my file from one address to the next. Fortunately, as long as the good people at Texas Instruments have their claws in high school curricula, there will be work for unemployed math instructors willing to tutor high school math.
  4. The kid I’m working with is in grade n, but I can’t for the life of me figure out what possessed any of his math teachers in grades five through n-1 to promote him. He’s got a really good attitude about learning, and is on the whole quite pleasant to work with, though, so I shall lay off the snark. I’ll say only that if there is a Hell, I’d like to put in a suggestion to management that assign everyone who played a role in introducing calculators to the classroom to spend eternity watching eighteen year olds extract same from their backpacks, turn them on, and key in a sequence of commands in order to figure out what two times one half is.

7/7/2005

Not from the solutions manual of Calculus With Analytic Geometry

File under: This One Time, At Mathcamp, Queen of Sciences. Posted by Moebius Stripper at 1:46 pm.
  1. You’re in the middle of the desert, 2 km north of an east-west river. Your destination is 1 km north and 5 km east of where you stand. You need to stop at the river for a drink at some point in your journey, but it doesn’t matter when. What is the shortest path you can take?

    Solution. Let the river be represented by the x-axis, your starting point be P(0,2), and your destination be at Q(5,3). The shortest path between P and Q that touches the x-axis is equivalent to the shortest path from P to Q’(5,-3). This path is clearly a straight line; properties of similar triangles show that you arrive at the river at (2,0).

  2. A statue is 4 m tall and mounted on a 2 m pedestal. Your eye level is 1.5 m above ground. Where should you stand in order to get the view of the statue deemed “best” by the writers of calculus textbooks - that is, the view such that the angle subtended by the statue at your eye is a maximum?

    Solution. Because I’m lazy and don’t feel like creating pictures, let’s say that the base of the pedestal is at O(0,0), the base of the statue is at B(0,2), and the top of the statue is at T(0,6). Then your eye will be somewhere on the line y=1.5. Construct the circle tangent to y=1.5 that has the segment BT as a chord. Then the point of tangency, P, is such that the angle BPT lies in the circle, while BQT lies outside the circle for all other Q on the line y=1.5. Hence BPT>BQT for all such Q, so you should stand at the P. The Pythagorean Theorem quickly shows that you should stand 1.5 m from the base of the statue.

  3. When widgets are priced at $50 apiece, 100 people buy them. For every $5 increase in price, 2 fewer people buy widgets. How much should a widget cost in order to maximize the widget company’s revenue?

    Solution. Letting x be the number of $5 price increases, the company’s revenue, which we wish to maximize, is R=(50+5x)(100-2x). Equivalently, we may maximize 5/2*R=(50+5x)(250-5x). Applying the AM-GM inequality to this quantity, we have sqrt(5/2*R) < = 1/2*((50+5x)+(250-5x))=150. The revenue is thus maximized when 50+5x=250-5x, ie, x=20. So, twenty $5 price increases, for a price of $150 per widget, gives the maxmimum revenue.

  4. Find the largest possible perimeter of a rectangle whose sides are parallel to the axes and that can be inscribed in an ellipse with equation x^2+2y^2=1.

    Solution. Letting (x,y) by a point on the ellipse such that the rectangle’s vertices are at (x,y), (-x,y), (x,-y), and (-x,-y), we have a perimeter of P=4x+4y to maximize. Apply Cauchy-Schwartz to the vectors (x, sqrt(2)y) and (1,1/sqrt(2)) to obtain P/4=x+y< =sqrt(x^2+2y^2)sqrt(1+1/2)=sqrt(3/2). The perimeter is a maximum when x=2y. Substituting into the equation of the ellipse, x=2sqrt(6), y=sqrt(6), and so the largest perimeter possible is 12sqrt(6).

  5. Two hallways meet one another at right angles. One hallway is 27 inches wide, and the other is 64 inches wide. What is the length of the longest ladder that can be carried horizontally around the corner?

    Solution. The ladder’s length, L, is that of the shortest line segment that passes through the point (64, -27) and is bounded by the axes. The lines passing through (64,27) have equations of the form y=mx-64m-27. The intercepts of such a line are at (0,(-64m-27)) and ((64m+27)/m,0), so we need to minimize L^2=(64m+27)^2+((64m+27)^2)/m^2=(1/m^2+1)(64m+27)^2. Apply Holder’s Inequality with three 3-norms to the vectors (1/m^(2/3),1), (4m^(1/3), 3) and (4m^(1/3), 3). This gives L^(2/3) = ((1/m^2+1)(64m+27)^2)^(1/3) >= (64)^(2/3)+27^(2/3), that is, L>=25^(3/2)=125. So the longest ladder that can be carried horizontally around the corner is 125 inches long.

12/17/2004

Mathematical crochet

File under: 1000 Words, This One Time, At Mathcamp, Queen of Sciences, Hubris. Posted by Moebius Stripper at 7:02 pm.

A crochet pattern for a model of the Lorenz manifold is making the rounds - so I figure there’s never been a better time to show off the Sierpinski triangle shawl I crocheted a few years ago. I also have crocheted Moebius strips and models of the hyperbolic plane; the latter, in fact, was my first project. Can you tell I learned to crochet at Mathcamp?

Here’s some more on the Sierpinski triangle.

8/9/2004

In which I bring Ebola to Mathcamp

File under: This One Time, At Mathcamp. Posted by Moebius Stripper at 9:59 pm.

I’m totally getting my money’s worth out of my medical insurance this summer - I went to the doctor the other day when my neverending cold (16 days and counting) seemed to be turning into an ear infection, and tomorrow I’ll be getting medical attention for my fingers, which aren’t healing as well as they should be from the time on Friday when I got all eight of them stuck in a window. (Story forthcoming as soon as I get the pictures of my bandaged hands online.)

In fake medical news, on today’s field trip, I found a toy store that sold stuffed microbes. I bought Ebola - isn’t it cute? Another camper bought the flu and the common cold, the latter of which I’m thisclose to hurtling against the room in frustration.

I think my favourite is the flesh-eating disease. It has no mouth, but it comes with a knife and fork!

8/7/2004

Call for paper

File under: This One Time, At Mathcamp. Posted by Moebius Stripper at 9:25 am.

One of the other Mathcamp staff is starting a collection entitled “Great Napkins of Mathematics”, in recognition of the fact that the best math seems to be done on napkins. If you have any, send them to him - ask for his address at martin at math.umn.edu - instead of throwing them out - he’d really appreciate that.

8/2/2004

Sick, sick, sick

File under: Sound And Fury, This One Time, At Mathcamp, Know Thyself. Posted by Moebius Stripper at 4:23 pm.

One of the unforseen consequences of the policy that holds us responsible for administering our campers’ medication: every single one of our ill campers gets filtered through the office, which has turned that locale into something of a doctor’s examining room, but without the benefit of surgical masks for its employees. I’m sure this isn’t the only reason that that the Plague Currently Known as the Camp Sickness has afflicted a good half of us so far, but it’s not helping that we require, say, every camper with a mild cough to enter the office, cough all over us, swallow some cough syrup, hand his infected spoon over to us to clean, and then depart.

The worst affected, it seems, is me, and I swear to God if this continues one more day, I’m going to become one of those people who laments, at length, the grave injustice that we can do [take your pick] but we still don’t have a cure for the common cold. I’d be taking this assault on my immune system in stride, were it not for the fact that it’s been going on TEN DAYS as of this writing, ten days that have left me variously exhausted, congested, and hoarse, all during a week where the temperature’s been in the high twenties, thirties with the humidity factor, and Seventh Circle of Hell with the I-feel-like-crap factor. Before anyone tells me that low thirties isn’t hot, you don’t know hot, where I live it’s REALLY hot, keep in mind that I’m a wimpy Canadian who left Ontario for Vancouver, in part because I cannot handle Ontario summers. Frigid winters are fine - bundle me up in a few snowsuits and I’m good to go - but in the heat one can only get so naked, and past a certain point it’s illegal, and not worth it either, because it’s still hot.

Compounding this is the fact that my dorm room seems to have been designed by some master sadist who aims to keep sick people sick. One of my less convenient idiosyncracies is that I, without fail, wake up as soon as my room becomes light, and not a moment later, and in an south-facing room in which the entire south wall is a window, this means I’m up by 5:30 am every single goddamned day. Which means that I have to be in bed before 9:00 if there’s any possibility whatsoever that I’ll get the eight hours of sleep I so desperately require if I’m ever going to shake this thing. It doesn’t help that I have a bed whose design conforms to the twin theories that 1) everyone enjoys sleeping on plastic mattresses - EVERYONE; and that 2) you never want to sit up in bed, and just in case you ever forget that you never want to sit up in bed, we’ll make your pillow fall right out off of your bed if you ever try to prop it up, and we’ll sometimes even make it fall right out if you so much as shift around lengthwise while you’re sleeping. And as though this weren’t enough, my room temperature is automatically set to “bake”, so that even when the temperature outdoors is somewhat bearable, the temperature in my room is not - though the addition of a small fan keeps this issue under control, somewhat.

I was going to write about how well the other day’s Calculus Without Calculus class went, but when your body seems to be laying waste to certain hypotheses such as There’s Only So Much Mucus You Can Produce and You Can’t Possibly Be Tired After Resting For Ten Straight Hours, it’s hard to focus on much else. In the meantime, I’ve missed four field trips so far, including two hikes, and I’m feeling right useless the rest of the time.

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