Intolerable

I suppose it is normal, as you grow up, to develop "deal breakers" - things you can't easily tolerate in other people. I'm talking about the kinds of things that, if you saw them in someone on your first date, would keep you from having a second date. You'd just know that person wasn't right for you.

One of the things I find really difficult to deal with in other people is frequent anger and judgment of others. Sometimes I meet someone, or get to know them, and I learn that their head is filled up with lines that other people shouldn't cross - things that turn other people into idiots or assholes to them. And often these things are pretty trivial or common - faux pas, simple acts of inconsideration or inattention, that kind of thing.

Yesterday, a coworker drove a couple of us to lunch. The parking lot of our office building has a 4-way stop and another driver proceeded through it when my coworker thought it was his turn. Instead of just letting him go, my coworker cut the guy off and, though he didn't do anything directed at the other driver, he did rant about it a bit in the car, saying something like, "Fucking moron!"

My other coworker, in the backseat, said something like, "Getting a little angry at other drivers there, huh," and the driving coworker said, "I only get angry at the idiots."

Well, you know, everyone is an idiot driver sometimes. I mean, honestly, not being sure when it's your turn at a 4-way stop does not make you an idiot or a bad person. And lest it sound like I am giving my coworker a hard time over one incident, he proceeded to talk about how many idiots there are on the road, and kind of went on like that for another couple of minutes.

I guess in general I feel like if you find yourself ranting about things that are very common - for instance, if you say things like you hate to go to restaurants these days because so many people are rude in such-and-such ways - you are the one with the problem, not the world. Common events - slow drivers, people talking on their cellphones on the bus, teenage boys who wear their pants real low, people forgetting to RSVP, etc. - do not justify real anger and a bunch of complaining to whoever is stuck listening to you.

I guess I draw the line somewhere between annoyance and anger. "God, this person is slow, what the hell, DO NOT DRIVE 20 MILES PER HOUR," seems OK to me. "What a fucking idiot. GET OFF THE ROAD! I cannot STAND slow drivers. If people can't fucking drive they should just stay home," is over the line.

And yes, I do see the irony in being intolerant towards the intolerant. But I don't think they're all idiots or assholes who should be killed or forbidden to live in the same cities as normal people or anything - I just think it's an unfortunate personality trait that a person should work to improve. And I find myself pretty much unwilling to hang out with people who unabashedly display it.

Not My Problem

Ed and I are, honestly, sort of both control freaks - me more than him, probably. (Is Ed a control freak? Having said that, I'm not sure. Ed?)

The worst control freak I ever dated was David. Sometimes I would tell him I was going to do something - something unrelated to him, mind, like "write a letter to my grandfather" or "get a haircut" - and then I wouldn't do it, and I would get a lecture later. "It really bothers me when you don't do what you say you're going to do," he would say.

For me, what I find is that I kind of monitor Ed and when he deviates from what I think he should do, it bugs me, even if it's trivial or I had no reason to expect it in the first place. (For instance, it bothers me when he doesn't use a sponge in what I think is the optimal way while cleaning the counters.) And I worry a bit over all kinds of things that are actually not my problem.

So we've gotten into the habit lately of telling each other, "That is not my problem." And since we generally joke with each other a bit roughly, this turns into a funny, sort of mean thing to say as well.

"Damn it! I'm out of milk!"

"That is not my problem!"

It probably sounds kind of cold, but it's done playfully and I think, oddly, that it actually helps. I find I'm more able to remind myself (even silently) that things like what time Ed goes to bed, whether he runs out of groceries, his handling of his bills, and so on, are not my problem.

(I don't mean to imply that Ed's life is a wreck from which I have to disassociate myself. Not at all. It's just crazy-making to try to ensure that someone else has remembered to balance their checkbook like they said they needed to do, or other minutia of everyday life.)

Anyway, if you are a control freak, or if you are dating one, try it out. But don't worry - whether it works for you or not is not my problem.

Math Satori: Generalized Quadrangle

One thing I came across in various articles about Laguerre planes was references to their "well-known" relationship with the generalized quadrangle. I eventually decided to try to hunt this down and use it for my paper, and it's been a very hard slog, causing me to have to look up term after term after term and try to understand all kinds of things.

The first thing I had to try to understand was what the hell the generalized quadrangle is. I mean, I know what a quadrangle is - a four-sided figure (like a rectangle, square, diamond, etc.) - but there's no obvious relationship between that and a Laguerre plane.

So, it turns out that a generalized quadrangle is a kind of incidence structure, like a projective plane. An incidence structure is something that has two sets - like points and lines - and a relation describing which ones are on or contain each other. So, for instance, if your whole structure consisted of one line, m, with three points, A, B, and C, on it, and a fourth point, D, not on it, your structure would be something like this:

P: {A, B, C, D}
L: {m}
I: {(A, m), (B, m), (C, m)}

where P is the set of points, L is the set of lines, and I is the set of pairs of points and lines that are on each other.

The Euclidean plane is also an incidence structure, but it has an infinite number of points, so it looks more like:

P: all pairs (x, y) of real numbers
L: all pairs (m, b) of real numbers, plus vertical lines that just have an x-intercept (a)
I: a point (x, y) is on a line (m, b) if y = mx + b; also, a point is on a vertical line (a) if its x-coordinate is a

So, the generalized quadrangle is an incidence structure that satisfies these axioms:

1. Two points can't have more than one line in common. (Note: they can have no lines in common, unlike in Euclidean space, where any two points determine a line.)
2. If you have a point A and a point B that aren't on the same line, plus a line m that B is on, there is a unique point C and line n such that A is on n and C is on both n and m.

There will also be some other axioms to make sure the generalized quadrangle is big enough to be worth dealing with. (Under most systems, for instance, a quadrangle is not a generalized quadrangle.)

Now you may have gotten this right away, but I got these axioms after working through a giant example of a generalized quadrangle, and one thing I couldn't see was why it was called a generalized quadrangle. What did it have to do with quadrangles?

In pursuit of this answer, I came across a couple of places that commented that a projective plane is, of course, a generalized triangle. What? I never heard this before. What the...ohhhhh.

I got it.

A projective plane is a generalized triangle because any two points determine exactly one line, and every two lines cross at exactly one point. That's just like a triangle. (Go ahead, draw one and see, if you need to.)

And a generalized quadrangle has the same kind of relationship as a quadrangle:

1. Lines don't cross more than once.
2. If you pick opposite corners, plus a side, you can get from one corner to the other exactly one way, and that trip involves one additional side and one additional point.

Wow.

This may seem kind of trivial, I don't know, but when I got it, I was ridiculously excited. I wanted to race around and explain it to everyone.

The Laguerre Plane Paper

I spent several hours this weekend trying to track down, learn, assimilate, and decipher more math for the paper on Laguerre planes. I was very successful at learning about the generalized quadrangle, but only partially successful learning anything that was actually relevant to my paper. (Some of it was really exciting, though.) I also spent several hours producing most of the second draft of the paper, which has a radically different (improved, I hope) structure from the first draft.

Learning math is hard. Organizing math concepts the way you want to write them is challenging and enjoyable. Actually writing math is...astonishingly easy. I can't believe how much easier it is to put words on paper for math than when I'm writing something like an English essay or history paper. I think it may be that I have so much material, and math is so straightforward, that there just aren't nearly as many issues with wording. And the issues you do have are about trying to make things as clear and understandable as possible, which is something I enjoy.

I was really apprehensive about this project at the beginning of the semester. I traditionally produce good papers, but I have a hard time making myself do so, and I don't really like papers or projects. I was also worried that I wouldn't be able to figure out enough math to produce a whole five pages. (My paper is currently six pages, not counting figures, and if I can write the final section it will probably be seven or eight total.)

Instead of it being a horrible nightmare, I've enjoyed it immensely.

But what's funny about it is, had I known how much work I would be doing, and how hard it would be, I would have been even more scared than I actually was. I've done so much hard work for this. I've pursued (sometimes successfully, sometimes not) so much crazy math. I've pursued the same crazy math over and over until I got it. And I've spent many hours (20, perhaps, so far?) just writing the paper, making the figures, and getting Latex to do what I wanted.

So it is definitely not that it's been easier than I expected. It's that it has been much more satisfying and enjoyable, and easier to get myself to work on, than I ever hoped.

Project Excitement

Lectures are all finished in my Geometry class. Starting Monday, for the next two weeks it will all be project presentations. We each have to give a 20-minute (specifically, 18- to 22-minute) presentation on our project. (There are twelve of us, so this is possible.) I go a week from Monday.

In addition to having us present them, the prof intends to post our papers to the course website. We will also fill out evaluations on all of the presentations, which he will compile into a comments sheet for the author, and use as part of the basis of the presentation grade, since the presentation is supposed to be for each other and not for him. And he says he will put some questions on the final pertaining to some of the projects.

I find all of this pretty exciting. I have to admit, I'm not completely looking forward to sitting through eleven presentations, but I can't wait to look at other people's papers (at least, the best and worst of them). And I'm glad that the stakes on this assignment are kind of high, in that other people will see your work and get to give input about whether your presentation was a waste of their time.

Meanwhile, I have to figure out how I want to reconfigure my own paper, which is still in draft form. And, of course, I have to decide what I want to present and how to present it. It feels very tricky to come up with a presentation that is only boring or incomprehensible rather than both boring and incomprehensible.

The Joy of Mix-and-Match Cooking

As I've been preparing more meals for me and Ed at home, I've discovered (with surprise) that I prefer to make meals composed of several things (generally a meat, a vegetable, some kind of starch - hopefully a whole grain - and corn, beans, or rolls to bulk things up). One reason for this seems to be that my one-dish meals don't generally turn out very good; it's just easier to do several simple things well than it is to make a well-composed, properly cooked dish.

Another advantage of this kind of cooking is that it's very flexible when you go to the store. There aren't many combinations of meat/starch/vegetable that don't go together, so as long as you have one of each, you're good.

It's also very cheap. By not requiring any specific ingredients, you can buy whatever is on sale at the store and make use of it. This is especially notable in the produce department, where there tend to be what I assume are loss leaders among the common vegetables.

Since I often (perhaps usually) shop on the same day that I cook, I can also take advantage of meat and fish that has been marked down for immediate sale, which is often half the price of the stuff that isn't as close to expiring.

The other night, Ed and I had three boneless pork chops, a box of whole wheat macaroni and cheese, three zucchinis sauteed with crushed red pepper in olive oil, and a can of beans. The cost for the whole meal (not counting incidentals like the milk, olive oil, etc.) was about $6.50 for both of us, which compares extremely favorably to...well, just about anything.

I kind of enjoy the frugality aspect even though I am not very frugal in general.

Early Meaningful Songs

Sometime around when I was a teenager, the world of music exploded into meaning. Those were the days when you would lie around clutching the little sheet of lyrics that came with some albums (tapes in my case) and reading them while listening dreamily to the songs.

When I was younger, most songs were just music, or I understood them really superficially. They didn't touch me. (Of course, most of the songs I liked when I was little, e.g., Rod Stewart's "Do You Think I'm Sexy," didn't really merit careful listening anyway.)

But some songs always touched me. Here are some of the earliest ones I remember associating strong feelings with, in (subjectively) chronological order:

Song I Can't Remember!

There is some song from the late 70s that I think became popular again in a more recent decade (a cover version). I used to hear this as a little kid and I thought it was really creepy and it gave me chills, though I think it's a standard social movement kind of message. I want to say it has lyrics like "can you hear the sound?" or "see what's coming round." It's about something (spooky to me as a child) that is approaching (which I now think is just some kind of social change). Can someone remember/figure out what song this is?

"Total Eclipse of the Heart" by Bonnie Tyler

After moving away from New Orleans after the fourth grade, I associated this song really strongly with Mrs. LeBlanc, my teacher in the previous two grades, who I had no further contact with. And sometimes with my mom or other adults. It seemed to capture the longing that I sometimes felt to be loved and attended to and understood (seen). I can still feel those old associations pretty keenly, though they seem a bit objectively ridiculous when I pay attention to the actual lyrics.

"The Rose" by Bette Midler

I saw this movie with my mom when it came out (1979, apparently) but it was a few years later (early middle school) that I started listening to the soundtrack of it. This song just seemed really meaningful and beautiful to me. Another song that struck me the same way (and thus gets an honorable mention here) around the same time was "Memory" from Cats (another soundtrack I owned; I later saw the musical live and it was disappointing).

"Things Can Only Get Better" by Howard Jones

The line in here that goes, "And do you feel scared? I do, but I won't stop and falter," was incredibly meaningful to me around the middle school years. It still moves me and seems associated with my approach to relationships. (And wow, is that ever an 80's video!)

I want to say, I actually listened to a lot of meaningful songs as a little kid - Bob Dylan, Joan Baez, Simon & Garfunkel, and all of that kind of stuff - but most of the meanings didn't really speak to me even when I loved the songs. (Well, why would they, you know? "If you're offering me diamonds and rust, I've already paid" is not exactly something a kid would identify with.)

What are the earliest songs you remember having a lot of feeling about?

Someone Wrote This?

From Google News, I was reading this PC World article about a rejected iPhone application. I kept finding myself confused by the structure of various sentences in the article, until I finally I realized, "Hey, this isn't me; this article is terrible."

Seriously, someone wrote this for publication (even on the web)?

Excerpts, with comments:

A novelty, essentially harmless, iPhone and iPod app has been rejected from the Apple store...

This kind of stringing together of modifiers works in spoken but not (IMO) written English.

Bailout Bucks is a simple novelty app that lets you put your picture as the face of fake dollars bills, in large denominations, so not appear that you might be attempting a forgery.

Not only is the last bit missing a word or two, but the whole ending part of the sentence is...just not how you write English.

The work of The Codist, a computer programmer and iPhone app developer, Apple has rejected Bailout Bucks based on the view it "ridicules public figures" or could do if it every [sic] saw the light of day.

Apple is the work of The Codist? Wait, what? (Yes, this is a slightly pedantic point, but I still found the sentence more than averagely difficult to read.)

The Codist goes on to question how the approval process works, is rejection down to one individuals understanding of Apple's terms and conditions, does personal taste ultimately decide what is in or out, and what supervisory level, if any, checks the process is fair and consistent.

Did someone dictate this over the phone, intending it to be the outline of a story someone else would write, only to see it published as dictated? How does this type of sentence come to exist in an article?

You would think that, given the overabundance of people with degrees in journalism, English, etc., even a lowly job like this one would be staffed by someone who could write grammatically.

Money in the Relationship

The night I met Ed, we ended up having a late-night snack at Denny's, and I paid. He was, after all, a poor grad student.

"I felt weird about it even that very first night," he said recently.

Being as I am not a starving grad student, I have generally paid for most of our restaurant meals and other outings over the course of our relationship. I not only have the money, but am willing to spend it in this way, because I really do like going out.

When Ed first moved in with me, last September I believe, our arrangement was that he would continue paying me the amount he paid for rent + utilities at his previous place, which was less than half of my rent and utilities, though certainly more than he cost me by living with me. I also didn't generally charge him for household things like laundry detergent, toilet paper, etc. Once he got a full-time job, it was understood that he would start paying his half.

Now that he has the full-time job, he is indeed paying his half of the bills, which is awesome for me. And a couple of weeks ago I negotiated that, since I cook meals for us two or three times a week and also buy other items for the household, instead of divvying up all of those bills, perhaps he could just pay for a meal out for us once a week and we'd call it even.

It's an interesting situation that I thought was fair when I proposed it. Say I spent $30 preparing three meals for us, and he then spent $30 treating us to a meal out. Aren't we even? Well, yes, except that I'm still forcing him to participate in eating out, so that we're spending $60 on four meals altogether. And Ed is trying to pay off student loans and maintain his former thrifty lifestyle, so he doesn't like paying for meals out.

Yet going out is important to me. I re-negotiated: I will charge him for half of all the mutual household things (meals, paper towels, whatever - and of course he can charge me half if he buys things like that), and if I want us to go out (which is typically once or twice a week) I will pay for it.

This makes him uncomfortable. Now that he could afford to pay for half of our meals out, he feels like not doing so in favor of paying down student loans is practically the equivalent of taking my money and paying his loans with it. Why should he get treated when he has a good income? And I'm sure the fact that he's the man and I'm the woman doesn't help; our culture doesn't really have a good model of a woman taking a man out to dinner.

But I think we have to resort to some principles here. A partner in a relationship should have the right to be as thrifty as they wish. I want Ed to take the best care of himself financially that he can. Yet, a person with a thrifty partner should not be unduly limited by the partner's thrift, I think. I have a job that pays me reasonably well, I am putting money aside for retirement, I have almost no debt (< $1000 as of this month), and I think I should be able to eat in a restaurant with my boyfriend once or twice a week.

Imagine if I had a rich boyfriend who wanted to fly to Paris with me for the weekend, but I refused because I was unwilling (and, in the long-term, unable) to pay my half. It's no big deal to him - he's happy to pay my half! - and my refusal only thwarts things he'd like to do with his own money, like have an expensive romantic weekend with his girlfriend.

Of course, I would be uncomfortable if I had a boyfriend like that, at least until I gave in and decided to just enjoy the largesse, which I imagine I'd be able to do after a bit. (After all, free money and things is basically good and nice, if you don't let principles get in the way.)

I guess there are two ways you could decide to do this, while dating someone:

1. I refuse to participate in things without paying my half, or
2. I'm willing to let my partner treat me to things that aren't in my budget.

You can take a stand and go with #1 if you want, but I think #2 is generally a nicer way to be for your partner, if you can relax about it a little bit and also trust your partner to keep their own boundaries intact.

If Ed and I get married and decided to handle our money jointly, some of my excessive spending will probably have to go away; I know that not all of it is justifiable from the perspective of what our joint financial priorities should be. But I do think, under present circumstances, that I would press hard for us to be able to eat out more than occasionally.

Paper Draft Complete!

Yesterday was a snow day, and I had no work to do at home, so after lunch I sat down to try to start writing my paper on Laguerre planes (due in three weeks). I was hoping to get in two or three hours of solid work, because I've noticed I can't focus on this stuff for very long in one sitting.

I spent a while going over another of the references I have, and then started to write.

And was completely unable to stop.

I put in a good 10 hours of writing. Apparently I had accumulated so much Laguerre plane material, and so many ideas, that I just could not stop until I got most of it out on paper. And, of course, a fair bit of my day was spent fiddling with LaTex (the editor I'm using) as well, but I was patient and also chose my battles carefully.

Tonight I was able to get the first draft completely finished. With the caveat that it's typical to be dazzled by even one's most mediocre work immediately after finishing it, I will say I am really impressed by my own paper. I wasn't sure I could write a math paper at all, and here I've written one that I think is pretty cool. It's not original work (and wasn't supposed to be), but it's quite nice and I was able to include some nice proofs that I worked out for myself.

The paper is supposed to be 5 pages, single spaced, 12 point font, not counting figures. My draft is 8 pages long, about 6 pages without figures. So it's definitely long enough (and, if it weren't, there are all kinds of more things I could put in right this very minute.)

I wrote the paper with an audience of my fellow students in mind, though I'm sure none will read it. (I think the professor said he's going to post them for everyone to see, so they technically could; I just don't think they will choose to.) Hopefully this means that when I find the paper in my files in a few years I'll be able to understand it.

Anyway, way to go me!

Less Mysterious

I love this bit of the paper I'm reading:

Laguerre sowed the seeds for much of the material of this chapter in Sur La Géométrie de Direction [11]. The specific approach we shall be taking is, however, quite different from the approach in the original, or even subsequent, publications of Laguerre. In Chapter 3, much of this material is redeveloped in the Minkowski model. The results of the present chapter are much less mysterious in that context.

I don't recall having ever seen, in a paper, this close an equivalent of a professor's usual reassurance that "this will all make sense later."

Source: Robert Knight's 2000 doctoral dissertation "Using Laguerre Geometry to Discover Euclidean Theorems."

Math Is About Isomorphism

I previously argued that math is about definitions. I am now ready to revise my position to the one in the title: math is about isomorphism. I feel that in some ways this idea is more advanced, and in other ways more primitive.

To say that things are "isomorphic" is to say that they have the same structure (lit: shape) in some important and useful ways. On the primitive side, of course math is about this! Consider these situations:

Mary has three chandeliers. John would like to borrow them, and promises Mary that if she lets him borrow them for a few months, he'll give them back along with a fourth chandelier.

I put $3,000 in a CD. Several years later, the CD will pay out $4,000.

These situations are isomorphic (in my sense, at least) and even the most elementary math is about recognizing isomorphisms so that we can use mathematical procedures across a variety of situations that are superficially dissimilar. If I have 3 apples and someone gives me 4 more apples, then how many apples will I have? 3 + 4 = 7.

Ed and I were talking about the coordinates being used in my geometry class, and how some aspects of projective spaces are isomorphic to linear algebra, and he made the obvious remark, "Linear algebra is isomorphic to a huge number of things."

And then he made the less obvious remark, which is coming to seem more and more true to me over time (although still not so very true), "It's amazing how few different things there are in math."

So, there you have it. Math is about isomorphisms.

Lärabars

I've been trying to figure out something different for breakfast for a while now, because it turns out I am far too lazy to make breakfast at home (even cereal), so I've gotten into this rut of going to McDonald's every morning, which is both expensive (I spend either $2.15 or $4.65 depending on what I get) and unhealthy. Also, it takes time, though not very much at all.

So, I have discovered Lärabars. These are sold in the general power/snackbar section of the grocery store, but unlike a PowerBar or Luna bar, each variety is made of very few ingredients. All of them start with dates and are a soft, chewy bar with nuts and possibly other fruits inside. For instance, yesterday I had the "Cherry Pie" flavor, which is made of dates, almonds, and unsweetened cherries, with no other ingredients. Today I am having "Ginger Snap" which has dates, almonds, pecans, ginger, cinnamon, and cloves.

Unlike the other brands of bars I've had, Lärabars don't seem to have any kind of funky or weird taste - they really just taste like fruits and nuts. And I really like the soft, non-crumbly texture. You also get a full serving of fruit in each bar. They cost about $1.25 each.

I originally felt some qualms about the nutrition. The bar I'm eating today has 220 calories (a bit low for breakfast), 5 grams of protein (decent for a bar), 14 grams of healthy fats (nice), and 17 grams of sugars. The sugars bit is kind of scary - that's the equivalent of 4 teaspoons of sugar!

But then I thought of it this way. The sugar comes from the dates and other fruits; it doesn't come from sugar, corn syrup, or ingredients that are secretly actually just sugar (e.g., "concentrated pear juice"). This is really the sugar found in a single (medjool) date, which I certainly wouldn't hesitate to eat.

Also, while there is no doubt a more ideal breakfast to be had, what I've been having is McDonald's. And there is no way that if you showed me, on one side, an Egg McMuffin, and on the other side, a small handful of dates, nuts, and other fruits, that I'd think the Egg McMuffin was the healthier choice. Right? So, yeah.

I also do find these bars surprisingly filling. If I were a more mystically health foody type, I might speculate that my body recognizes when it's getting "real foods" and responds with satiety. Instead I just consider it basically a mystery.

Also, for having so few ingredients, the Cherry Pie bar really did taste like a tart cherry pie, and the Ginger Snap bar was really good and gingery. They do an excellent job with these flavors. I still have the following flavors in my desk waiting to be eaten: Banana Bread (almonds, dates, bananas); Pecan Pie (dates, pecans, almonds); Lemon Bar (dates, cashews, almonds, lemon juice concentrate, natural lemon flavor); Cocoa Mōlé (dates, almonds, walnuts, unsweetened cocoa powder, cinnamon, chili); Apple Pie (dates, almonds, apples, walnuts, raisins, cinnamon).

Anyway, Lärabars - I recommend 'em.

Geometry

Class tonight was amazing. I didn't really take anything away that would be fun to describe here, but wow. I was really "on" mentally and able to get a lot from it.

I love math so much.

Cute

All right, I just found this picture (from Getty Images) too cute not to share:


(Obama jogging along in the White House with new First Dog, Bo.)

When Inanimate Objects...Cooperate

Tonight, I had to clean the kitchen. It was pretty bad, both because we hadn't cleaned it all week and because Ed finally unpacked some ancient boxes and donated some more equipment that was all over the counters. Over the course of the day, we ran five (5) dishwasher loads, and in addition to putting in that fifth load, I had to handwash a few items and then clean the counters and stovetop. It's not really that huge of a deal, but I was feeling overwhelmed by it. I also felt that if I didn't do it, I would never want to come home again.

I decided to try listening to my iPod in case it would help. Music is generally motivational.

Now, I have to admit, I skipped the first couple of songs that came on. But then I got this amazing assortment of, for me, possibly the very finest collection of songs to get some shit done to:

1. "Rapture" by Blondie. A fantastic beat, plus the story it tells fascinated me as a kid.

2. "Touching the Untouchables" by Men at Work. All right, this is not a great song. But I had this album on tape as a kid, recorded with my tape recorder from my almost-brother Thad's record player, so it brought me warm childhood feelings.

3. "Disco Inferno" by the Trammps. I can hardly imagine a better song to work to. So catchy! So hot!

4. "No One Knows My Plan" by They Might Be Giants. Yet another energizing song with a recognizable dance beat. The fact that the narrator is probably a serial killer is just a bonus when you're stuck scrubbing counters at 9:00 at night...

(All links go to Youtube videos featuring the studio versions of the songs.)

Smooth Move

This morning, Ed and I went on a hike at Red Rocks. As we were leaving the house, I couldn't find my keys. Now this may sound commonplace, but it's not; I really don't misplace my keys. They should have been either on their hook or in my bag, and they weren't. I looked around all of the other obvious places, and they were not to be found.

I knew Ed had let us in the front door the previous night, so I thought perhaps I had left my keys in the car. That's conceivable - I was bringing a lot of things up. So I suggested we could go down to the car and see if they were there and, if not, I could borrow his car key and find my keys at home later.

They were on the hood of the car.

I left my keys, including my car key, overnight, on the hood of my goddamn car. And they were still there in the morning.

The Isomorphic Thrill

The other night, I decided to turn momentarily away from the Polster/Steinke line of thought on Laguerre Planes (covered in my previous post) and look at one of my other sources. The third author I have a lot of writing by, including his dissertation, is Robert Knight. I opened up an article by him...something about bundle forms and "ovoidal Laguerre planes."

My impression of Knight so far, from lightly perusing some of his stuff, is that he's mostly going kind of old school with the Laguerre business. And indeed, in this paper, as background, he presents a set of axioms that I think are pretty close to Laguerre's own.

In the original Laguerre system, instead of circles and points, we have cycles and spears. Cycles (in the original representation) can be either oriented circles or points, and spears are oriented lines. That's my understanding at least.

But when I started jotting down the axioms, I was very excited to find that they were more or less isomorphic to the Polster/Steinke set. I got that pure mathematical thrill as I confirmed that my octahedron "plane" satisfied all of them (treating the points as spheres, the faces as cycles, and with one other additional but obvious rule to account for something not present in the other system).

A phrase I've encountered in this course, but not before, is "up to isomorphism," as in, "There exists a unique field of order 2, up to isomorphism." What it means is that you might be able to make different ones, but they all have the exact same structure even if the elements have different names. And, in a more general sense as well, I've encountered a ton of isomorphism in this class.

The obvious approach to hearing about Laguerre planes, for instance, is to ask "What is a Laguerre plane?" And you expect to get an answer, maybe something like, "Well it's like a normal plane except it curves around," or, "It's the surface of a sphere," or, "They call it a plane but it's really an octahedron." But there just isn't actually an answer like that. What makes something a Laguerre plane is satisfying some set of axioms about Laguerre planes, from which a bunch of theorems will then hold. Anything that satisfies those axioms is a Laguerre plane.

I mean, I had that octahedron, but there's nothing special about that. The Polster book had another finite example with the same number of points and circles; I found the octahedron easier to represent for the blog post. But the two were (clearly) isomorphic.

And that's the way in which the Polster axioms and the Knight axioms are (more or less) isomorphic as well: the same conditions satisfy (or pertain to) both.

As I mentioned, this kind of stuff gives me a real mathematical high when I get it.

Laguerre Planes

I am required to do a project for my Geometry class, and my topic is Laguerre planes. Since I have finally (basically) figured out what a Laguerre plane is, I thought I'd blog about it, while it's still fresh. Hold on to your hats, folks! There's gonna be pictures!

A lot of projective geometry is about incidence, that is, what points are collinear, which lines intersect, and that kind of thing. We don't really seem to talk about the normal things you talk about in Euclidean geometry - things like lengths, angles, areas, or in other words, the particulars of shapes. It's just about which points are in which lines, really.

So, a Laguerre plane is a type of circle plane (also called a Benz plane in one source). A circle plane has points, and the circles are made up of points. Points can be considered parallel to each other by some kind of equivalence relation*. If you make up a system like this, then it's a circle plane if it satisfies these axioms:

1. Any three different points, no two of which are parallel to each other, determine a unique circle. (Note: In Euclidean space, any three different points determine a unique circle.)
2. Given a point P on a circle C, and another point Q that is not parallel to P, there is a unique circle that contains both P and Q and either touches C at exactly one point, or else is C.
3. Every circle contains exactly one point from every parallel class. (For instance, if A, B, and C are all parallel, and no other point is parallel to them, then every circle contains exactly one of A, B, or C. A, B, and C would be a "parallel class.")
4. If you have different parallel relations - that is, completely different standards about what makes points parallel or not - then if you choose two parallel classes, each from a different relation, they'll have exactly one point in common. (You're allowed to have up to 2 parallel relations.)
5. A circle has at least three points.

(* An "equivalence relation" is something that works like equals, or like parallelism typically does more or less, in that, if A is parallel to B, then B is parallel to A, and if A is parallel to B and B is parallel to C, A is also parallel to C. Points are considered parallel to themselves as well, which is the third thing that makes something an equivalence relation.)

Now, Laguerre planes are a subset of these circle planes - specifically, the subset that has only one parallel relation, or way of determining whether points are parallel.

An example of a Laguerre plane could start with the points that make up a vertically-oriented cylinder. We'll call points "parallel" if they are on the same vertical line on the cylinder. The circles will be non-vertical cross-sections of the cylinder, which can be straight across or at an angle (making them what we'd usually call ovals).

Does this satisfy the axioms?

1. Given three points around the cylinder, no two of which are straight up or down from each other, can we make a cut of the cylinder that contains these points? Since three points determine a unique plane, and the intersection of that plane with the cylinder will be a circle or oval, yes, we can.
   
2. Given a point on a circle, and another point that isn't vertical from the first point, can we find exactly one circle that contains both and touches the first circle at exactly one point? I think we can.
   
3. It's pretty clear that any non-vertical cut across the cylinder crosses every vertical line exactly once.
4. This axiom doesn't apply since we only have one parallel relation (the one about the vertical lines).
5. Does every circle have at least three points? Yeah - we're doing this in the reals, so every circle has an infinite number of points.

That's (assuming I'm not mistaken) an example of a Laguerre plane with an infinite number of points, but we can make a finite Laguerre plane. In fact, an example of the smallest kind of Laguerre plane can start with an octahedron. The points are the vertices, and the "circles" are the faces. Points are parallel if they are not joined by an edge. Here we have only six points, and eight circles.

The parallel classes in this example are the three pairs of opposing points of the octahedron (for instance, the topmost and bottommost points in this picture make up a parallel class). You can see that each circle (i.e., face of the octahedron) touches exactly one of each pair of opposing points.

I really enjoy the elegance of this, especially the finite example (stolen directly from A Geometrical Picture Book).

Sources

B. Polster and G. F. Steinke, "Criteria for Two-Dimensional Circle Planes", Contributions to Algebra and Geometry, Volume 35, 1994.

B. Polster, A Geometrical Picture Book. Springer-Verlag, 1998.

The Adventure of the Large Cat

There is a very large tortoiseshell cat that I often see around my apartment complex. He (I think it is a he, though I could be mistaken) isn't fat at all - just tall and lanky, almost the size of a fox. I admire him often.

Well, yesterday as I was coming up the stairs to my apartment, he was coming down the stairs, which was mysterious - I'm fairly certain he doesn't live in any of the apartments on that stairway. So, naturally, I spoke to him, and he turned out to be quite friendly, and followed me up.

I intended to give him some food outside of the door, but when I opened my front door, he went right in! And showed no signs of wanting to do anything other than explore my entire apartment.

Sammy immediately backed away near the hall, but Tigerlily (!) went over to investigate this strange, large cat. Soon her tail puffed out to a gigantic size, and she and the other cat talked at each other a little bit as she cornered him behind the two armchairs. I was standing nearby and keeping them separated by walking towards Tigerlily when she got too close.

When Ed came out, he asked, "Are you sure that cat is safe?" I feel, honestly, that most cats are "safe" and I wasn't particularly worried about this big friendly fellow. He wasn't showing any signs of wanting to attack Tigerlily or anything, but I decided to go ahead and herd my two cats down the hall into my bedroom so that we wouldn't have too much ruckus.

Tigerlily hissed at Sammy the whole way down the hall! Every time she saw him she was freaked out! And Sammy was sidling sideways, scrunched low to the ground, clearly worried. I did pretty easily herd them both into my room, and then shut my door.

A couple of minutes later I shut Ed's door, because the big cat was exploring everywhere and I really didn't want him to stay too too long. He checked out the cat box and returned to the living room, where he was quite comfortable jumping up on tables and furniture for some pets.

It's always hard to know if a cat will tolerate being picked up, but after a few nice strokes I decided to try it. I let him know that I was going to pick him up, but not to worry, that if he didn't like it he should just let me know and I would stop right away. I gingerly picked him up, and he seemed slightly nervous but not too distressed. I set him down outside the door (where I had already put some food, hoping to lure him out; he never did eat the food), said, "Thanks for visiting. Stop by any time!" and closed the door behind him.

The cats came back out once I opened my bedroom door. Tigerlily went looking all over for signs of the big cat, but Sammy was crouchy and nervous for a while and hoped that the big cat would NOT reappear.

I think I am going to call him "Lopez" in honor of his interloper status.

Anne of Green Gables

Anne of Green Gables (by Lucy Maud Montgomery) was first published around 1908. It is the story of an orphan girl who comes to live with a middle-aged brother and sister in rural Prince Edward Island. Anne is 11 at the beginning of the book, and 16 at the end of it, and there are several sequels that follow her through the various stages of life.

I'd read the first book before, but lately I've been reading the whole series on my Kindle, and reflecting on how remarkable it is.

The books are kind of simple - this isn't Dickens, but more like children's literature. Anne is feisty and quite imaginative, but she's a very good girl despite all of the scrapes she gets into. The books assume and promote a very basic Protestant morality. To the extent that there is moral conflict in the books, it's between a sort of fuddy-duddy outlook and a more free-spirited one.

But I find two things fairly remarkable. One is that the setting and the way the books progress is kind of historically amazing. There are no slaves, no serfs, and very little in the way of class differences, and Anne is free to pursue her education, vocation, and romantic aspirations as best she can in a friendly world full of opportunities. Oh, people still die of consumption, and ride in horse buggies, but otherwise it's kind of amazingly modern in some sense.

It's also rather astonishing how feminist the books are on a kind of basic level. The patriarchal features of the day are assumed, of course, but so is the idea that women can be intellectuals and academics. When Anne seeks her B.A., some of the adult women of her town disapprove of women going to college, but the surrounding culture is supportive of her doing so, and her years at college are filled with the kind of witty intellectual repartee that I remember from, well, my years at college.

The way that Anne's romantic life is worked out is also pretty cool. If you have any experience reading older novels at all, you'll have a guess within the first 50 pages of the first book about who Anne might end up with, but the way that it works out - the motivations and learning processes of the characters - is a bit interesting and, well, somewhat modern.

Watching Anne mature through the books is really fun. She starts off as a sort of comically exaggerated version of a girl you might imagine reading the book, but becomes really recognizably a person, a young woman, and an adult, as she ages.

All in all, if you're in the mood for some rather wholesome light reading, I can recommend these books heartily.

Homogeneous Coordinates in the Extended Euclidean Plane

I've discussed the Extended Euclidean Plane ("EEP") before. Now I want to talk about homogeneous coordinates, because they are just very cool and convenient. They are used in all field planes, but the EEP works well as something to blog about without having to go into a whole bunch of other "stuff." (But in particular, we use them in a lot of finite field planes - that is, planes defined over a field that has a finite number of members rather than a field like the real numbers.)

So, the main distinguishing feature of the EEP is that there are no parallel lines. Lines that are parallel in the Euclidean plane meet at a "point at infinity" associated with their slope. These different points at infinity (one for each possible slope of line) make up the "line at infinity."

Normally, when we talk about coordinates in the Euclidean Plane, we use two numbers, so for instance, (2,3) might be the coordinates of a point. But for homogeneous coordinates, we are going to use three numbers - a 3-dimensional vector. What makes them "homogeneous" is that every scalar multiple of this vector refers to the same point, so that, for instance,

(1, 1, 1)

and

(2, 2, 2)

refer to the same point.

As a convention, we will make these coordinates from the usual Euclidean coordinates by making using the x and y values unchanged, and using 1 as the third coordinate. So the point formerly referred to as (2, 3) is now (2, 3, 1). It can also be called any scalar multiple of that, like (4, 6, 2) or (2/3, 1, 1/3), but there is no reason not to reduce any of those to (2, 3, 1).

Lines will also have three coordinates, which will come from the usual definition of a line:

y = 2x + 4

is the same as

2x - y + 4 = 0

and gives rise to homogeneous coordinates: [2, -1, 4]

A point is on a line iff the dot product of the vectors of the point and line is 0. So let's say we have the line y = 2x + 4, and the point (1, 6). The homogeneous coordinates for the line are [2, -1, 4] and for the point we have (1, 6, 1).

(1, 6, 1) (dot) [2, -1, 4] = 1 * 2 + 6 * -1 + 1 * 4 = 0

so the point is on the line as we expect. Note that this is the same thing as saying that a line with coordinates [a, b, c] has the equation

ax + by + cz = 0

and a point with coordinates (x, y, z) is on the line iff it satisfies that equation.

But what's with the three dimensions, yo? That doesn't make any sense.

The deal here is that we're kind of analogizing points (which usually have two coordinates) to lines (which usually have three), and lines to planes. Specifically, points become lines through the origin, and lines become planes through the origin. A point is on a line if the corresponding line is in the corresponding plane. Since two distinct lines through the origin determine a unique plane through the origin, and two distinct planes through the origin intersect in a unique line through the origin, this corresponds to our desired relationship between points and lines.

Now then. I've said that two parallel lines intersect at a point at infinity. How does that work? Well, let's look at these two parallel lines:

y = 2x + 1
y = 2x + 3

and, using the homogeneous coordinates, see them as a system of equations:

2x - y + z = 0
2x - y + 3z = 0

Using a bit of linear algebra (or regular algebra, if you prefer), we find that in order to solve these equations, a point (x, y, z) has to have the form (t, 2t, 0) for some t. So we can say that the homogeneous coordinates for this point are (1, 2, 0).

Using coordinates this way, in the EEP we will always get 0 as the final coordinate of a point on the line at infinity. The line at infinity, meanwhile, has homogeneous coordinates [0, 0, 1], as you can confirm if you like.

The linear algebraic qualities of this sytem mean that solving for the point of intersection of two lines, and for the line determined by two points, are exactly the same operation. In fact, given two line or point coordinates (a, b, c) and (d, e, f) we can solve for the point of intersection or the common line, respectively, by solving this determinant:

| x y z |
| a b c | = 0
| d e f |

i.e.,

(bf - ce)x - (af - cd)y + (ae -bd)z = 0

so the coordinates of the desired point or line are (bf - ce, cd - af, ae - bd).

So that's what I've been up to lately...