The Life of a Mathematician

Yesterday I got a package in the mail from my family with birthday presents. They sent me two books: Great Expectations by Charles Dickens, which I've never read, and The Three Musketeers by Alexandre Dumas, of which I've only read a very abridged version, along with watching various movie renditions, plus the Wishbone episode. They also sent me a photo album, filled with, um, photos. A few are ones I took at summer camp and stuff, but most are various family pictures. I am very pleased to have them, as I don't often take a lot of photos myself.

The other package that arrived yesterday contained a copy of Geometer's Sketchpad, which will allow me to go back and fill in all the graphs I had to skip over in the solutions I'm writing. One thing Adobe Illustrator does not do is graph equations.

Well, I've got a system for counting the number of topologies on a given finite set of points that is much faster than listing them all, but it's still a long way from being a formula. However, it looks like it might eventually turn into a recursive formula. The trouble is that the recursion could be nested any number of levels deep.

Today was my birthday, which makes me think of last year when I ended up helping Mike with algebra homework on that day, then later played Star Wars: Epic Duels with Abe, Greg and Kyle after Mike told them it was my birthday and I was by myself. And, this line of thinking is going to make me homesick so we'll move on to another.

Class this morning was talking about the final for the first few minutes, then the rest of the lecture was a biographical sketch of Euler. Quite interesting. I didn't remember that he was contemporary to Goldbach and mentored by Gauss. Then again, I have a poor memory for historical details in general. Next Tuesday he plans to answer any study questions they have, then prove that e^(i*pi) + 1 = 0.

After class we did the last round of placement testing (until this fall) with the homeschoolers. Just two students this time, a Level 2 and a Level 3. The boy taking the Level 2 test also had a birthday today. He turned 14. It also looks like he did well on the test, based on a quick look-over. I'll actually grade them in the morning.

While they were taking the test I started writing up more formally some thoughts I'd had about a problem some of us attempted when we were taking combinatorics. The problem itself comes from point-set topology. Given a finite set of n points, how many topologies are there on the set (assuming the points are distinguishable)? I realized it had been a while since I'd written up a technical proof, but it came back pretty quickly. I haven't gotten as far as trying to do any counting yet, but I've got what might be a good system for describing the topologies.

On Friday I got a haircut. The barber shop also sells antiques and art. The guy that owns it does photography along with a little bit of painting. He used to be an engineer, and his favorite subjects are engineering and motorcycles. He was showing me a new technique they have for prints: There's a printer that actually puts paint onto canvas, so that the finished product looks like a painting. In a way I guess it is a painting, but it's done by machine. He showed me an example of the canvas print next to the photograph it was based on; the print looks a lot better because of the texture and depth.

Yesterday I helped move a large pile of broken drywall into a trailer to be hauled off. I was the wheelbarrow operator. It took about three hours. It's been hot recently, but we had cloud cover for a lot of that time.

Last night we had dinner with Mr. Nickel -- and his wife, who is finally back in town after being gone for about 4 months. We played Bocce Ball in the backyard. It's sort of a cross between croquet, bowling and horseshoes. One of the families had a ten-year-old named Nathaniel who was very enthusiastic about the game.

Okay, I've found a way to become alert sooner than 6:00, while having technically slept the previous night. Can't say I care for it. The trick is to have a headache that starts in as soon as I lie down, and continues despite the fact that I fall asleep almost immediately, so that I continue to have the headache even while dreaming, which makes the dreams restless and frenetic, then sleep continuously until 5:00, wake up, walk into the restroom, and have the headache dissipate almost entirely as soon as I stand up. The contrast, combined with what seems like a reasonable fear that the headache might return if I lie down again, is sufficient to wake me up entirely. The restless and frenetic dream is on my other blog.

On Friday, when I walked into the room before the first class with the homeschoolers, the teacher handed me a paper with a problem on it. It was the extra credit problem from the test that the algebra class (second hour) had just done, and he hadn't had time to find the solution for when they asked about it. Here's the problem:

The roots of a quadratic equation are 1 + sqrt(5) and 1 - sqrt(5). Find the equation.

There's a more difficult way to do it that finds an infinite family of solutions, and an easier way that finds just one solution. I demonstrated the more difficult way to the class, because that's what I thought of, but then a few minutes afterwards the easier method occurred to me so I showed them that as well.

My brain absolutely refuses to function before 6:00 in the morning. Having had enough sleep helps once 6:00 arrives, but before that it just means I'm well-rested and unable to process information. The only exception to this rule is if I stay awake all night. In that case, my reasoning ability starts a process of exponential decay, beginning at 22:00 with a half-life of 90 minutes, but leveling out at around 2:30 when it reaches 25%. It'll hover there for the rest of the night, and then increase again as high as 70%, maybe even 80%, when I tap into emergency energy reserves at dawn. This spike lasts until sometime between 9 and 11, at which point my body insists on a complete shutdown for maintenance and repairs. All of this is assuming there is no caffeine involved.

No, I haven't had any reason to pull an all-nighter in this job, but I do have a reason for trying to wake up and start functioning at 5:30 on Tuesday mornings. I've gotten as far as the "wake up" part. My last roommate has a quote on record of me saying, "righteousness begins at 6 am". I believe this was in response to his request that I define what I meant by an "ungodly hour".

On the bus this morning I met a kid who is in high school, but is also taking Calculus 3 at a local junior college. Furthermore, it sounds like he's doing well and enjoying it. He thought that my job sounded like fun.

Almost no one showed up to Bible study tonight; just me, the guy whose house it's at (and not his wife since she's out of town), they guy who leads the study, and one of his daughters. As a result, it ended up being just a prayer meeting and shorter than usual.

I've been thinking about the Rubic's Cube, viewed as a group on six generators. Rather hard to come up with a complete list of relations.

I finished reading War and Peace by Leo Tolstoy, and The Jericho Sanction by Oliver North. I'm now starting over in my abridged version of The Journals of Captain Cook (from Penguin Classics). One interesting log entry from before the main voyage started:

"Wednesday 14th. Winds Easterly. First part fine clear weather, remainder Clowdy with squalls from the land attended with showers of rain. In the night the bend of the Hawsers of the stream Anchor slip'd, owing to the carelesness of the person who made it fast. In the morning hove up the Anchor in the Boat and carried it out to the Southward, in heaving the Anchor out of the Boat Mr Weir Masters mate was carried over board by the Buoy-rope and to the bottom with the anchor. Hove up the anchor by the Ship as soon as possible and found his body intangled in the Buoy-rope. Moor'd the ship with the two Bowers in 22 fathom water, the Loo Rock W and the Brazen head E Saild his Majestys ship Rose. The Boats imploy'd carrying the casks ashore for Wine and the caulkers caulking the Ships sides."

It sprinkled a little when I was riding to church this morning and also on the way back, but already the clouds look like they might be moving on. It's been pretty hot recently.

I found out that the kids I was tutoring are pretty much done with math for the summer, so I don't have that job now. Maybe something else will turn up.

I've started a new blog for recording my dreams. It's linked at the bottom of the blog list to the left. Also, Sudoku puzzles rated at five stars of difficulty are challenging.

I haven't posted in a while, mostly because I've been too tired to compose a post when I've had time, or too busy catching up with other things when I've not been tired. But there really hasn't been that much going on. I write solutions, I sit in on classes, I attend church and Bible study, and I tutor. The bike ride has gotten easier so that it doesn't wear me out anymore, even in the coming back direction, which is slightly uphill and usually against the wind.

Recently we've been giving placement tests to a group of homeschoolers to assess what classes they'll be ready to take this fall. There's a local "homeschool co-op" group that meets every week and has once-a-week lectures in certain subjects. I helped administer the tests and then graded them; the results were disappointing but predictable. I also got some experience in "education diplomacy," talking with mothers whose children had not passed whichever test they were taking. Not my first taste of it though; I have a few semesters' experience in dealing with college students in the Math 108 course.