1/28/08 - 2/3/08 Mud In The Toes (415 minutes, 2 tempo runs)

Sunday, February 3 (35 minutes)
My pleasant jaunt with Ian and Jasper was graced from its beginning with a beautiful and enigmatic rainbow, which repeated itself, not as a double rainbow (although that too was present) but just repeating BIV part of ROYGBIV.
What caused this repetition? Multiple cooexisting raindrop geometries? A heretofore unobserved double sun? An unexpected refraction of some percentage of the light between the source rainbow and me? I may never know. But then again, I probably will know, eventually, at some point in my life. I suspect that with a concerted internet search I could figure it out right now. But I don't feel like it at the moment.
Another one I never really understood was why clouds don't fall down, if they're made of water droplets. One interesting explanation was that they do fall down, but the drops at the bottom keep dissolving and new ones form at the top. Also, they're very small and so wind resistance makes them fall slowly. Don't believe this though. I don't, really.

Also, here is a comic that was not suitable for print in the school paper:


UPDATE: 10 minutes later
I'm a liar. I couldn't stop myself from trying to learn about that ROYGBIVBIV rainbow. It works in a way somewhat similar to the colors in oil slicks and soap bubbles. The closest analog I was familiar with is actually Newton's Rings.
In a normal rainbow, light enters a raindrop and experiences a refraction on entering, reflection at the back, and refraction again on leaving. Different colors are refracted differently, causing the differentiation of colors you see in a rainbow.
But there's another key element - the reflection. The reflected light is polarized, and therefore susceptible to interference. What you'll find is that at a given angle between you, the raindrops, and the sun, light can actually follow a few different paths to get to you. Entering different parts of the raindrop and being reflected and refracted differently, but in such a way that the ultimate angle formed is the same.
So you've got light coming to you over a range of pathlengths. Depending on what that range of pathlengths is, you'll get either constructive or destructive inference of any one color.
The colors also have different wavelengths, so some colors will interfere destructively while others interfere constructively, resulting in striped patterns as you trace through angles.


Saturday, February 2 (70 minutes)
I have a fantasy about the Super Bowl. It'll be the halftime, and the speakers will boom over sixty thousand screaming fans "BUDWEISER, PEPSI, AND AMERICAN FOUNDATION FOR BETTER WRITING IN UNDERGRADUATE PHYSICS TEXTS ARE PROUD TO PRESENT THE SUPER BOWL XLII HALFTIME SHOW. FEATURING... CHARLES SIMONYI PROFESSOR FOR THE PUBLIC UNDERSTANDING OF SCIENCE AT OXFORD, RICHARD DAWKINS AND... TENZIN GYATSO, HIS HOLINESS THE DALAI LAMA...IN CONVERSATION MODERATED BY A RANDOM UNDERGRADUATE FROM CALTECH"

The crowd will erupt in a communal paroxysm of unbridled exuberance. Shouts of "Yeah, secular humanism rules!!!" and "Go Lama! Contemplate the shit out of that divinity" will spring from every mouth.

The three of us will emerge from under the grandstands accompanied by a fireworks display which, viewed from the north or south looks like Einstein's field equations of general relativity, and viewed from the east or west looks like the profile of Bart Simpson.
We'll take our seats, I'll hand up one hand and a sudden silence will descend swiftly on the expectant crowd.

The Dalai Lama will scooch forward in his chair, visibly a bit nervous, tap once at the microphone, and then say in a halting voice, "The key to happiness is first to love yourself, then to love others, then at last to understand it is the same."

Dawkins will be visibly annoyed, scowling with supreme rationality, and gesticulating emphatically he will declare in his refined British accent, "Yes, well that's all jolly good, but take a look at it this way..."

But I'll cut him off with a simple raised finger. "Excuse me just one second, Dick," I'll tell him politely. Then I'll stand up and expose my nipple to the cameras.
The End.

Also, I ran 70 minutes from my house to Caltech, then around some side streets in a fairly arbitrary manner. I noticed that the preponderance of heckling could be mapped out quite easily to correspond inversely to the median income of a residential neighborhood.

Friday, February 1 (90 minutes)
I realized just now that I forgot to describe my actual workout on Monday, being too busy talking about math. Monday I did 4x2mile with a lap jog recovery, going 10:53, 10:44, 10:43, 10:34. For the last two I took no splits, checking only the finish time. I think this was a good idea and I'm proud I had the discipline to pull it off.
Today I did 90 on the north field, picking up the pace through the last ten minutes or so. This is about as long of a run as I'm interested in pursuing right now, and to make it any more challenging I'll just start making the last 15 minutes up tempo, then the last 20, etc.

Thursday, January 31 (20 minute tempo, 4xPatton)
20 minutes in Lacy, going about 5:30/2 laps. Felt fine, but not completely locked in to tempo intensity. Afterwards I did "Renato Canova" hill sprints, just 4 x 10 seconds all out on the steepest hill convenient, Patton Way. Google Earth indicates an average slope of 11 degrees over the portion I ran.

Wednesday, January 30 (75 minutes)
Poked my way around the infield for 75. It was windy. That's all.

Tuesday, January 29 (30 minutes)
Today I completed the task Kangway assigned me of 10 reps of 100lbs on the bench. I even adhered to Ian's admonitions not to arch the back at all, and was completely conscientious of my form throughout the exercise. Afterwards I felt a slight smugness when I watched big muscly guys with lots of large weights on the bar grunting their way through a set - with arched backs. "Poor fools. They'll never minimize their chances of injury to the middle third of their bodies now," I thought.
After I did an easy recovery jog on the infield.

memoir: Summer, 1992

Monday, January 28 (4x2mile)
Yesterday, a calculus student I tutor was upset because she couldn't answer the test question:
"What is the largest possible area of a rectangle inscribed in a circle of radius 4cm?"
There are many ways to do the problem, but one of my agendas when I'm tutoring, and really the reason I'm in it at all (other than the, uh, money) is that I want to get people to forget the formulas for a minute and take a look at what is really going on.
I told her she didn't need to use calculus if she didn't want to. It was okay - however she wanted to solve the problem, if the method was valid, it doesn't matter whether it's the same one the teacher had in mind.
I was thinking she would then just do this:



But she couldn't, and I didn't know why, until I realized: when she read the problem, she read the word "rectangle". She didn't know that the appropriate rectangle, inscribed in that paragon of symmetry the circle, was inevitably a square.

Here is the problem with tutoring: you can't think about it. Someone asks you a question, you need to tell them the answer immediately. So I told her the part she was missing - that the rectangle had to be a square, by symmetry. She asked why.
I wanted to say, "if you switch the role of x and y, the circle remains the same, and so the rectangle must as well. Therefore the rectangle must have the same length and width" But just before it slipped from my mouth I realized that even as an intuitive suggestion it's not true.

Just because the circle has a symmetry doesn't mean the inscribed rectangle of maximum area must have it, too. The circle has a rotational symmetry, and no rectangle has that.

Instead, I shirked it. I took the problem another way, and showed her how to set up a parametrization and use the calculus techniques we had practiced, like this:



Then, halfway through, I saw a more direct way, so I made her work through it like this:



I try to make people solve the same problem over a few times when I can, so they might understand the formula is only a crutch which eventually they will drop. But what I really wanted to do was show how you could skip right to the shape being a square, which makes the problem almost trivial. It's a great approach because you should avoid complicated analytical deductions where intuitive arguments will do. Having that ability makes to a much better problem solver than drilling your way through hundreds of equations.
I wasn't sure how to do it, though.

What I was thinking was essentially this:
Suppose it is not a square. Then you can draw it to look like A. But by rotating the picture, it also looks like B.



But wait:



If you were to solve that problem, keeping the assumption that the correct rectangle is NOT a square, you'd get two answers - one for the rectangle that looks like A and another for the rectangle that looks like B. Two different values of l solve the blue problem, if the rectangle is not a square.

That seems preposterous to me. How could there be TWO solutions to that blue problem, where you're just varying how far from the center of the circle you draw a cord, then inscribing a rectangle from it, and maximizing the area of the rectangle. Obviously there can only be one solution, and so the pictures "A" and "B" must look the same. The rectangle must be a square.

But what if it isn't obvious to the student that the blue problem can't have two solutions? Why do I believe that's true? This is where it gets hard. I know it's true. I can prove it by actually doing the problem, but I want to prove it by a simple, direct, intuitive means with no equations. That solution has to exist, but I don't know quite what it is. Not without long hours of hard though, anyway. This makes me wonder - if something is so obvious, but I don't know why, then I don't really understand it. How much, then, of what I thought I understood very well, would prove to be equally murky in my mind if I were hard-pressed about it. This is kind of disturbing to me. If you know a simple, direct solution to how to show by the symmetry of the problem that the rectangle must be a square, please post and put my mind at ease.