Polymathematics

A Bone to the Math People

Since my recent posts have been political and personal, I thought I'd put up a mathy post so all you math people don't desert me.  Maybe you can answer a question for me.  I once heard a discussion of the following question:  "Is there anything special about the number 2?"

One answer, "It's the only even prime", was labeled not a real answer, since that specialness only resides in the fact that we have a name ("even") for numbers divisible by 2.  If we had a name for numbers divisible by 3 (say, "threeven"), then 3 would be special, too, since it would be the only threeven prime.

Other answers depended on some group theory that was beyond my limited understanding of that subject.  But I have another answer.  The only problem is, I'm not sure if my answer just boils down to "it's the smallest integer greater than 1".  If it boils down to something deeper, please tell me!

It stems from a problem that I like to give 7th- and 8th-grade students:

Crazy Al owns a dollar store that isn't like any other dollar store.  It's not that everything costs one dollar, it's that everything costs at most one dollar.  In fact, every possible price (from 1¢ to $1.00) is represented by at least one item in the store.  Furthermore, Crazy Al has a big sign behind the counter that says "I will not give change, EVER!!"  So if you want to buy an item and you don't want to get ripped off, you have to bring exact change.  One day you need somthing in Crazy Al's store, but you don't remember how much it costs.  So you have to bring enough coins to be sure that you can pay for any item in the store with exact change.  With (common) American coins of 1¢, 5¢, 10¢, and 25¢, what's the smallest number of coins you could bring to be sure you could have exact change for any item?  What about with EU coins of 1¢, 2¢, 5¢, 10¢, 20¢, 50¢?  What if you could design a coinage system just for this purpose that was the most efficient way to guarantee exact change for items up to $1.00?

The answer, of course, is to have coins of 1¢, 2¢, 4¢, 8¢, 16¢, 32¢, and 64¢.  The powers of 2.  This seems clearly related to the fact that if you're going to build/write integers, 2 is the smallest possible base (which is because it's the smallest integer larger than 1).  But is this greatest efficiency a sign of anything deeper?  Is 2 being any more special here?

By the way, this is even a great party problem for adults...when they get into "I don't understand how math could be fun" mode with you, this problem (in 2 stages:  smallest number of coins, then devising the best possible coinage system) can often demonstrate how math doesn't have to be abstract to be interesting.

Okay, I'm off to my grandmother's 90th birthday party in the morning.  I've heard that birthdays are good for you--the more you have, the longer you live.

Nauseating, fun, fun

Three links to keep you busy.  The first is truly nauseating, the second ones are fun.

For the first link, an analogy in the spirit of the old SAT questions:

Islam:Al Qaeda::Christianity: ?

In other words, what small subgroup of Christians has perverted the religion far beyond what was ever intended in a way that preaches hate?  Check out the link below.  I wrote out the punctuation because I don't want there to be any chance that they can trace where the link came from and then come back here and spam me or something.  You'll have to cut and paste and replace the punctuation:

(Warning:  This is so bad I really thought it was a joke at first.  It's not.)

http[colon][slash][slash]www[dot]godhatesamerica[dot]com
[slash]ghfmir[slash]main[slash]index[dot]html

Okay, now for the fun ones.  One for word-people, the other for puzzle-people (you can actually do the puzzles by clicking the links):

Word-people
Puzzle-people

Have fun...

A Different World

"A Different World" was the spin-off of "The Cosby Show" where one of the Cosby kids went to college.  The Different World in question was, I guess, two-fold:  living on his own, and being a Black man in the world instead of a Black kid in a familiar neighborhood.  But even that world is not as different as the world of Black kids in urban neighborhoods.

Yesterday and today (and several other times in the past few weeks), I took part in a program that my martial arts school is involved with.  We hold a weekly class for teenagers who have been convicted of drug-related offenses, but not violent offenses.  These kids are free in the sense that they are not confined, but their daily lives are scheduled to the minute, and there are always people keeping track of where they are.  And a few of them take our martial arts class for an hour a week.  For the most part, these are teenagers that are about 2 or 3 lucky breaks away from just being normal, trying-to-make-a-decent-life teenagers.  But because kids in those neighborhoods rarely get breaks, they were caught and sentenced to this program.  It's a triage principle, I guess.  The worst gang-bangers are just locked up.  The law abiding kids are left to fend for themselves.  These kids in the middle get a lot of attention so they won't end up in the first group.  All of the boys in the program are Black.  The girls are of several races*.

Only 5 of the boys stuck with the program for long enough to get to their first promotion test, which was today.  I've been helping teach them for a few weeks (since I have so much more time in the summer).  We practiced with them one last time yesterday.  While I was changing into my uniform in a different room, a large group of these kids was left alone for a while.  By the time I got back, it had degenerated into mass chaos.  They were immensely loud, and 2 pairs of kids were fighting;  they weren't angry, just playing--but the swings were real and could have seriously hurt someone if they had been good enough to actually connect.  I compare this to the Commons at my school where kids are regularly left alone.  Sure, they can get rowdy and there can be some play wrestling.  But nothing like the chaos I saw when I got back.  Society pushes all kids into non-kid behavior...that's the purpose of society, of course.  But a White-dominated society pushes these Black kids so hard that when they let loose, the outbreak is much more extreme.  It was not hard to see how, in a prison, this could turn into a riot.

Of course the 5 boys who tested did very well.  One of them, I'll call him Greg, is a big kid...maybe 5'11" and 260 lbs.  But he learned quickly that brute force doesn't work as well in martial arts as good technique, and when I thanked him for helping put away our mats, he said "no, thank you for coming out here."  None of the other kids had the guts and decency to thank us.  I will go to help out with that program whenever I can this summer, and Greg will probably still be in it by September.  I'll probably never find out what happens to him in the long run.  He's a really sweet guy, and I hope his guts and decency will bring him back from the brink.

Teaching those kids martial arts (even just an hour a week) is so drastically different from the teaching I do every day.  I don't think I could teach them math knowing the kind of apathy that's so prevalent in the classrooms in the city.  I like math, and I can overcome kids' math anxiety, but I have much more trouble with apathy.

Yeah, it's a different world.  But really, of course, it's not a different world.  I can drive there in less than half an hour.  So many White people live in fear of individual Black people they meet, and so many Black people live in fear of the overwhelming mass of White people who don't understand their frustrations.  We're both afraid of each other, and yet it's so easy to find a place (the martial arts mats, for instance) where race really just makes no difference at all.  At a recent weekend seminar for our martial art (drawing about 70 people from around the country--even a few from Japan and Israel--who study our branch of it) I was almost finished with it before I realized that this was really one place where nobody cared at all about race.  My roommate was Black, the instructors were Chinese-American, Japanese-American, White, Filipino, Latino.  I don't think this is just me failing to see racism because I don't want it;  I've been trained to see racism.  All it took was to take the focus off of race and put it onto something we all had in common.  It seems so easy.  But it's not easy.

Sad and encouraging at the same time.  A different world, yet not a different world.  Doesn't exactly follow the logic of math.

*I use the word "race" in its socially defined sense.  Genetically, there is more variability within a "race" than there is between "races".  So the word doesn't really describe any biological reality.  But it certainly describes a social reality.

Another part of Poly

So, the reason I named my site "Polymathematics" is that the things I do (my life, my hobbies, my politics) are so varied that I wanted to see all (or most of) the parts of it in one place.  I'm still not sure if it will work.  But here's another part of my polylife...a big one.

In about April or May of next year, my wife and I will be adopting a baby from China.  Let's get some common questions out of the way first:

1. So you're at the end of your fertility treatment options?  No, actually, we haven't tried conceiving;  adoption is our first choice.  Neither my wife nor I have any serious emotional attachment to passing our actual DNA to a new generation--my life is my loves, views, desires, interests, etc.  I can pass those on sans DNA.
2. Do you know if it will be a boy or a girl?  Due to the political situation in China, our baby will almost surely be a girl.  Chinese culture and law are asymmetrical in how they treat boy and girl children--girls are fine, but boys (supposedly) take care of you when you're old.  If the Chinese government says you can only have one (or two if you live in a rural area and your first one is a girl), you might have to give up a girl so you can have a boy for your old age.  Sad, but true.
3. Do you know how old she'll be?  Probably 9-12 months, no younger than 6 months, possibly as old as 2.
4. Why aren't you adopting an American kid who needs a home?  (Smart-ass answer for a lot of people:  Why aren't you?)  Real answer:  We (mostly my wife, to be honest) did the research, and this choice works for us.  And we kinda resent the implied rebuke in the question.  So please just accept our decision.

We're currently awaiting our final document from the U.S. government, which could arrive any day now.  We will "be DTC" (Dossier To China) by mid-August.

Amazingly, in the middle of writing that very sentence, my wife informed me that the document came in the mail!!!

So in addition to all the normal preparing-for-a-baby stuff (which we're doing a bunch of over the summer because it's less busy than during the school year), we've been talking a lot about where we want to live when she starts school, since the school systems around here are enormously varied.  Which brings up one of the central issues of this adoption:  Our child will be of a different race than us.  We are Caucasian, she will be Asian.

What's it like growing up Asian in America?  Well, I have Asian students and acquaintances and friends (although not especially close friends).  I've done some reading about growing up as an Asian adoptee.  I've spent time learning about how racism works.  I can't know for sure what it's like, but here's what I know:  Asian-Americans experience race very differently from African-Americans.  The stereotypes are not as blatantly negative, but they are at least as strong.

• Asian kids do well at school.
• Asian kids work really hard.
• Asians are non-confrontational.
• Asian families are male-dominated.
• Asian culture is "exotic".
• Asian women are highly sexualized, but submissive.
• Asian straight men are asexual.
• Asian gay men are very sexual.
• Chinese people are sneaky, inscrutable.

There are lots of them.  And they hurt.  Teachers might be less likely to help a struggling Asian student if they think it can't be a serious problem.  Standing up for beliefs might be more likely to be punished if it is viewed as unnatural.  Will our daughter ever be sure that romantic interest in her isn't a fetish?  And since many of these stereotypes contribute to the "model minority" image, the pressure on kids to conform to the stereotypes is huge (if you could be pretty sure that certain non-academic behaviors would cause people to look at you as brilliant and hard-working, wouldn't you behave that way?).

So....what does that mean for our daughter?  Should we find a school that has enough Asian kids to make her feel more at ease?  If many of the non-Asians are Black, would that help her learn to cope better with racism?  What about the reported anti-Asian sentiment in the Black communtity.  Is that a stereotype, too?  What if it's not?

I know, I know.  All you Asians and African-Americans and Latino/as reading this are thinking "welcome to my world, Mr. Oblivious White Guy."  Having been a teacher for a while now, I hope I'm more sensitive to the insidious effects of racism than most Oblivious White Guys.  And I'm painfully (and guiltily) aware of my White privilege--in particular, the fact that Whiteness is basically neutral in American culture.  My wife said it well when she pointed out that having an instant interracial family will force us to shed this racial neutrality.  Of course it would be good for a lot of people to be forced to do this.  Probably good for me, too.

I don't have answers.  You know how some people think that math is hard?  I'm sure it is for some people who's minds don't think of it in the right way.  But I think race is hard.  That's because America doesn't think of it in the right way.

Welcome to America, daughter.  We love you.  I'm afraid that won't always be enough.

Rove, Libby, Wilson, and "Political Capital"

I hate phrase "political capital".  The way I understand it is that when people do you (a politician) political favors or when you win an election with a large number of votes, you suddenly possess a "bank account" you can draw from to get your initiatives enacted.  As a concept in political science?  Sure.  I understand why you want a word for that.  But I don't like the implications of the term.  That's because I don't like how it implies that you (again the politician) use it like you use money.  And in the American version of capitalism, everything (for better or worse) has it's price--you can sell ad space on your forehead, or save money on some products by allowing people to sell your personal information for marketing purposes.

So...calling it "political capital" implies that everything has its price.  Any advantage you might be able to wield is legitimate in the name of preserving political power...the only drawback is how much "political captial" it costs you.

I think this is basic problem that Bush administration officials Rove and Libby have in their current political scandal about illegally outing the name of an undercover CIA operative (the most recent version of the story is here).  Whether or not Rove or Libby were the actual sources of the leak, their public statements have been to the effect of "Whoa there!  We've done nothing illegal!"  Not once have they said "We regret ruining the career of a CIA employee and putting at risk the lives of any informants that might have been associated with her.  Defending our president's questionable information about Saddam Hussein was not worth that lapse."  Jonathan Alter (whose opinions are almost always dead on) describes the price this way in this article:

The bigger question is what this scandal does to the CIA's ability to develop essential "humint" (human intelligence). Here's where the Iraq war comes in again. The sooner we beef up our intelligence, the sooner we crack the insurgency and get to bring our troops home. What does it say to the people doing the painstaking work of building those spy networks when the identity of one of their own becomes just another weapon in the partisan wars of Washington?

And this is the connection to the phrase "political capital".  Whoever leaked this information considered the balance between the damage caused by outing Plame and damage Wilson could cause by being vocal about his opinions that Bush's information was bad.  They decided Bush was more important than the law and the integrity of security clearance.

But IT"S NOT MONEY, PEOPLE!!!  If Bush was wrong, then he was wrong, and you don't get to decide to break the law to make him seem more right at some "cost".  The fact that everything in a capitalist country has its (possibly acceptable) monetary price does not translate into a political system, where every potential move has its (possibly acceptable) political price.

Illegal outing of a CIA agent and putting people's lives in danger is. not. acceptable. ever.  Not for saving Bush's reputation.  Not for getting back at a pesky Wilson.  Not for anything.  It's illegal, immoral, and in my little version of the world, unpatriotic.

Religious Right Funds Witchcraft

So, my first political-type entry.  I don't know why I picked school vouchers to write about, except that it might appeal to some of the people who have managed to show up here so far, and it might generate some discussion.  (Hmmmmm, do I really want discussion, or do I just want people to be reading my blog?  Probably both.)

On the face of it, school vouchers don't seem like such a bad idea.  Some opponents argue about how letting families use their tax dollars to go to a private school will decimate an already bad public school system.  I have to confess that I don't buy that argument, although I am opposed to vouchers for other reasons which will become clear in a moment.  I don't buy it because, well, encouraging people to abandon bad school systems (or any bad system for anything) is exactly what proponents think is best.  Our society is based in capitalism, after all, and if a system doesn't work, it doesn't deserve any infusions of cash.

But any time the Religious Right is in favor of an idea, I immediately get suspicious (they've had some good ones, but not many).  School vouchers weren't their idea, but they jumped on the bandwagon enthusiastically  once  it was suggested.   The two-fold benefit to them is clear.  First, they reap the financial windfall of the increase in students attending their schools.  And second, they get to work on bringing more families into their churches.

But I don't think the Religious Right has really thought this thing through.  Will Americans stand for paying our tax dollars to a strongly fundamentalist school?  What if they teach (or imply) that all Jews are going to hell?  What if a family wanted to send their kids to a blatantly racist church?  There are plenty of them around.  And on the other side, what if a family wanted to send their kids to a school founded by Wiccans (witches)?  Or to a fundamentalist madrassa (Islamic school)?  I don't know if any exist now, but with the prospect of funding from vouchers, it won't be long before they do.

Will we have to make a list?  "Schools associated with Churches A, B, and C are acceptable, but schools associated with Churches Y and Z are not."  I don't see any way that this could conform to the Establishment Clause of the constitution...the very first words of our Bill of Rights.  The alternative is to allow funding of any church at all (including racist, Wiccan, or Islamic*).  And I really don't believe the Religious Right is ready to allow that.

There's another problem, too.  I don't have all the details of the economic philosophy worked out, but this is something I've been thinking about for a long time...you'll have to tell me what you think.  If we really consider sufficient education to be a right in the U.S., we should not leave it to market forces.  Anything left to market forces is vulnerable to cost-cutting, and to the prospect of leaving some people out.  I am all for an open market, don't get me wrong.  But you can't provide for basic rights (think health-care, for example) through market forces, because then some people would be willing to do without those rights, which is contradictory to the whole notion of basic rights.  Right?  Hmmmm...I'm still working on that one.  Your ideas are welcome.

*I group these together because they each would have problems with segments of the population, not because I hold Wiccan or Islamic churches in the same contempt as racist ones.

Euler's Line

Just to see if I can post it right, I'm going to post one of my currently favorite proofs.  The theorem is the existence of the Euler Line of a triangle.  That is, I will prove that the orthocenter, centroid, and circumcenter of a triangle are all collinear.  I will not use the standard analytic proof, where you find equations for the various lines and show that the intersection of two of them lies on the third.  This proof is synthetic (non-analytical) and very simple and elegant.  I would dare say that this might be the proof from "the Book" (as defined by  Erdös).  I found this proof on the Internet several years ago, but I regret that I cannot recall the source.

Consider the diagram below (click for larger image in a new window):

The upper diagram illustrates two easy preparatory lemmas:

1. Since the segment connecting the midpoints of two sides of a triangle is half the length of the third side, the triangle formed by all three of those segments (the medial triangle) is similar to the original triangle in a 2:1 ratio.
2. The lines that serve as the perpendicular bisectors of the sides of the original triangle are clearly altitudes of the medial triangle.  Therefore, the circumcenter of the original triangle coincides with the orthocenter of the medial triangle.

In the lower diagram, AH is an altitude containing the orthocenter O.  AM is a median containing the centroid D.  MR is the perpendicular bisector of BC, and it contains the circumcenter R.  Connecting D to O and to R gives 2 segments that we want to prove to belong to the same line.  I aim to show that that they do so by proving angle ADO congruent to angle MDR.  (This could only happen if O, D, R were collinear, since O and R must fall on opposite sides of the median.)

To prove those angles congruent, I will show that the triangles AOD and MRD are similar:

1. Since the altitude and the perpendicular bisector are clearly parallel, angle DAO and angle DMR are congruent.
2. AD and MD stand in a ratio of 2:1, since that's one of the basic properties of a centroid.
3. If you consider the medial triangle of ABC in light of lemma 2, it is clear that AO and MR are precisely corresponding segments of the original and medial triangles (since R is the orthocenter of the medial triangle).  Thus by lemma 1, AO:MR = 2:1.

This proves the triangles similar, which means the desired angles are congruent, which means the  points O, D, R are collinear.  QED.

None of the members of my department had ever seen that proof before, nor had I.  It's simple enough to show to a class of bright 9th-graders (I've done it twice now, and most of them follow with almost no trouble).  In fact, it's simple enough that the 9th-graders barely realize the brilliance of the result.  I hope you enjoy it.

My First Blog Post

I think I want to maintain a blog, so I'm starting one just to see.  I know I have a lot of things to say, but I don't know if anyone really wants to hear them.

So...uhhh...how do you start one of these.  We'll pretend it's a diary and see what happens.  I'm doing just about nothing today except cooking dinner and working out in my Martial Art (which I won't name for the purposes of anonymity).  Of course, I can be this lazy because I'm on summer break.  This is supposed to be my reward for the intense effort I put forth during the school year.  I really do need the rest.  This last school year was difficult.  I had a lot of kids who signed up for courses too difficult for them (some a couple of years ago, most against their previous teachers' recommendations), and they had their predictable hard time.  Some kids at my school just can't reconcile "I hate getting C's and feeling like I don't know what I'm doing" with "I have to take the most advanced courses I can or else I'll disappoint everyone in my life who expects me to go to Prestige University".  And we teachers have to clean up their unreconciled mess.  I gave a higher percentage of D's than I have any other year.  This is considered not-quite-passing but not-quite-failing-either at my school.  Students will be on academic probation because of that.  From their point of view, it's because of me.  At least some of them (and some of their parents) think of it that way.  I actually had a parent tell me that he didn't think our school should give out D's, since all the kids in the school were so smart that they could get good grades at any public school.  Right, buddy, sure.  You pay tuition so that we maintain high standards--how dare you criticize us when we do.

Hmmmmm, sounds a little rant-y.  I think I may have a few rants in me at some point.  It might do me some good to get them out.

Uhhhhh, okay.  Time to cook dinner:

Hungarian Mushroom Soup

It's my first try at this one.  It sounds good, though.