e to the i pi plus one equals zero

One of the best teachers I ever had was Mr. Arrigo, for 11th grade pre-calculus. He was young, funny, hip, and energetic. It was almost incongruous that he was a math instructor. He seemed more like the big brother who’d already gone off to college. Of course he wouldn’t be one of my best teachers if he wasn’t also excellent as a teacher, which he was.

Throughout the year we covered topics in trigonometry, complex numbers, transcendentals, and logarithms. Little did we know that Mr. Arrigo was working up to a unification of all four.

One day in class he was particularly animated. We had been discussing the Euler formula, which gives this equivalence:

cosθ + isinθ = eiθ

(Here, e is the natural logarithm constant ≅ 2.718 and i is the imaginary number √−1.) He then asked us to work out the special case where θ is π (the ratio of a circle’s circumference to its diameter ≅ 3.14159). The cosine of π is −1 and the sine of π is 0, so Euler’s formula gives this amazing relation:

eiπ = −1

or, stated just a bit differently,

eiπ + 1 = 0

When Mr. Arrigo derived this result on the blackboard, he literally jumped up and down as he exclaimed, “One simple equation relating the five most important constants in all of mathematics!”

An xkcd comic

His excitement was infectious. Of course the result above is breathtaking. (If you’re not mathematically inclined, you’ll just have to take my word for it. Imagine being a world traveler collecting random antique curios from cities around the globe — then discovering one day that five of your favorite ones just happen to fit together perfectly to make an exquisite and accurate pocket watch. It’s kind of like that.) But Mr. Arrigo’s passionate presentation made it something more — an emotional highlight of my academic career.

A few years later, in college, I shared an office with my engineering friend Steve. One afternoon his actress girlfriend Amy stopped by and, one way or another, the conversation turned to mathematics. Taking turns scribbling on the whiteboard, Steve and I explained to a willing Amy how it’s possible to derive all the familiar rules of numbers and arithmetic from a tiny kernel of laws called Peano’s postulates. Amy seemed interested, so we pressed on into other topics such as geometry and its cognates, trigonometry and set theory. Thrilled that her interest didn’t flag, I mustered Mr. Arrigo’s passion and derived for her the amazing relationship between e, i, π, 1, and 0.

Soon after that, Amy’s drama-major mind finally had as much math as it could handle and our memorable pedagogical session petered out. But I could have kept going all day. I suspect Mr. Arrigo planted in me the desire to teach. I think Jonah and Archer are getting the benefit today of Mr. Arrigo’s passion way back then, and who knows? I may yet heed the urging of friends and family and become a teacher myself some day.

Mucoshave update

The latest addition to the Mucoshave oeuvre:

His beard grew bigger
He looked like a beggar
He couldn’t bag ‘er
For want of a booger

(Bat|Super)man Returns

Thanks to Netflix, I saw Superman Returns a few days ago. (It was pretty good, but I had some problems with it. Maybe in another blog post.) In the story, Superman has been absent from Earth for some years; the “Returns” in the title refers to the fact that he’s back.

This made me think of 1992’s Batman Returns. What “returns” in that movie? Nothing; Batman hasn’t been away since the events in 1989’s Batman. On the contrary, in that title, “Returns” refers to the fact that it’s been three years for the audience since the last Batman movie. Batman “returns” to moviegoers. (Or perhaps, more cynically, Batman provides “returns” on the studio’s investment. Hard to be too cynical about the studio [Warner Brothers] and the franchise that famously required director Joel Schumacher to make Batman & Robin “more toyetic.”)

It’s an annoying case of breaking the fourth wall with the film’s title — a trend begun, ironically, with 1978’s Superman: The Movie, closely followed by Star Trek: The Motion Picture in 1979. Hey — we know they’re movies. (Or, excuse me, motion pictures, as the case may be.) Tell me what happens in the movie. Fight Club, 12 Angry Men, Run Lola Run, those are movie titles. In a movie called Superman: The Movie, I’d expect to see Superman busting film-industry crooks on a studio backlot in Tinseltown.

The Batman and Star Trek franchises continued offending with Batman Forever (which can be interpreted no other way than as the producers thumping their chests in an “I’m king of the world!” moment) and Star Trek: The Next Generation (“next” after what? after the last time you folks watched a Star Trek TV show, that’s what). I’ve always found it strange that none of the gatekeepers of pop culture ever raised aesthetic objections to badly chosen titles like these (until now, of course, if we can agree to call me “gatekeeper”), whose conflation of the real and imaginary threatens to take the viewer out of the fictional world even before they step in.

I was pleased to see that the [Superhero] Returns title template has been redeemed by having “Returns” refer to events in the story. Maybe now we can work on redeeming …The Next Generation by having the story be about the actual children of characters from a prior story. Ugh, maybe not.

The armory

We did not encourage swords in our house, I swear. (Nor did we enhance their allure by making a big deal out of prohibiting them.) But then Jonah won a plastic katana at the Marin County Fair (really, he did, all on his own, by popping balloons with thrown darts), so then of course Archer had to have one too. One thing led to another and… well, now look.

Yes, every object in that picture is used as a sword (and plenty of others, ad hoc), even the ones that don’t look like swords.

Every time we go to the supermarket (which is near the toy store), it’s:

Jonah: Can we go to the toy store?
Me: What do you want to get?
Jonah: Swords. [Archer nods vigorously in agreement.]
Me: [exasperated] Don’t you have enough swords?!
Jonah: Just one more. Pleeeeeease?

Well, at least the “just one more Thomas the Tank Engine train, pleeeeeease” phase is over. Maybe this one will end too. Meanwhile, guess what Archer’s Christmas wish list was? In its entirety, quote:

One little sword.

What brings you here?

Herewith, a selection of search-engine queries that resulted in hits on this blog, according to my server logs.

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Jonah 1, parental instruction 0

Jonah has learned to ride a two-wheel bicycle entirely on his own! And here I was looking forward to teaching him using a novel (to me) technique suggested by my friend Bart.

When I learned to ride a bike, it was after hours and hours, spread acrosss days and days, of coaching from my dad, who jogged along behind me holding my bike upright, like a human pair of training wheels. One day as I was finally getting the feel of it, I asked him a question and got no reply. I turned around to see him receding into the distance. He had let go, and I was riding the bike on my own! In a panic I literally leaped off the bike onto concrete, setting back my bike-riding efforts by a few more days.

I remember my dad’s help with great fondness, but Bart’s method made much more sense when he described it to me. The student straddles the top bar of the bike, feet flat on the ground, hands on the handlebars, and walks the bike around for five minutes to get a feel for it. The student then stands on one pedal while kicking off the ground with the other foot, like riding a scooter. After ten or fifteen minutes of that, the student is ready to get both feet off the ground and start pedaling.

But Jonah’s gone and learned to ride on his own initiative, using the small learning bikes that his preschool has in its schoolyard. I am proud! …but just a little disappointed. That’ll pass, though, come springtime, when we light out for the park and ride together!

Last Lost

I have been catching up on season two of Lost courtesy of Netflix (since I no longer have cable). When I added season two to my Netflix queue it amounted to seven discs’ worth of episodes. As I finished each one I eagerly looked forward to the arrival of the next. Most episodes in season two ended with a “HOLY CRAP” cliffhanger.

I finished disc six just last night and HOLY CRAP! I got it in the mail to Netflix first thing this morning hoping for disc seven to arrive by Wednesday. I logged in to Netflix to see the episode titles on the next disc… and discovered that disc seven is merely the “special features” disc with no new episodes on it. I finished season two without even realizing it, and now I can’t see season three until it comes out on DVD. Bogus!


A few years ago we moved into our current house, in a neighborhood surrounded on three sides by wooded hills dotted with houses. In those first days, when Alex woke me up as usual for her morning walk around 6am, I was amazed and delighted to hear a dramatic dawn chorus of neighborhood songbirds. It was like nothing I’d ever heard, loud and irresistibly cheerful. At 6am you wouldn’t expect to hear much else, and yet the chorus competed with another sound. At first I guessed it must be a neighbor three or four blocks away testing a jet engine mounted on a rig in their backyard. At 6am. Soon I came to recognize it as the unceasing whoosh of combustion engines, rolling rubber, and steel slicing through air: the freeway, more than half a mile away. That sound is omnipresent on the otherwise sleepy residential street in front of my house. Perhaps the hills-on-three-sides shape of the neighborhood acts as a waveguide, channeling the noise and making it more prominent than it ought to be. In any event, having once noticed it, I now cannot escape it. About the only time it’s really quiet seems to be around 3:30am on Sunday mornings. (Alex is old and her schedule is less regular than it once was.)

My life has always been noisy. I grew up in an apartment in Queens right under an approach route for LaGuardia Airport. Most days of the week I spend two or more hours commuting in a poorly soundproofed economy car. And of course I am almost continuously sitting at high-powered computers and the constant drone of their cooling fans.

One day I decided to try to find a quiet spot. A really quiet spot, where I could hear no trucks rumbling by, no gas station air compressors, no high school football team; no crashing surf or gurgling brook; preferably not even any wind, or the rush of my own blood in my ears (as when they’re underwater or stuffed with earplugs). I didn’t want to not hear anything; I wanted to hear nothing. I wanted to listen to silence. Obviously no place in or near the Bay Area would suffice, so I located Pine Mountain Lake airport on the San Francisco sectional chart. Of all the places within easy flying distance, Pine Mountain Lake appeared to be the most remote and the most likely to be quiet (once I shut down the Cessna’s engine). My friend Steve came along for the trip. We crammed our bikes into the back of the plane just in case we had to put a little distance between us and the airport in order to find silence. But even on the mountainous roads of Pine Mountain Lake, the whoosh of cars and clatter of trucks are inescapable.

If I were more intrepid and more persistent in this quest I’m sure I would eventually have found something to satisfy me — in the desert, perhaps, or out on a calm sea. But this goal languishes way, way down my priority list. I was glad to see recently that others are more dedicated to the cause, and have been more successful, to wit: One Square Inch of Silence.

Archer 1, parental authority 0

Archer was home from preschool all last week with a fever. Andrea and I took turns missing work to care for him. Happily, his fever broke on Friday night and he was fine all weekend.

Having lots to catch up on at work, Andrea and I were eager to get the kids off to preschool this morning. But Archer had gotten used to lots of one-on-one parent-child time and was determined to stay home again. He refused to allow me to dress him. I coaxed him gently for a while and promised some fun family activities after school and work, but to no avail. Then I ratcheted up the sternness and started to tell him that certain privileges would be unavailable later if he continued to resist me now. When that didn’t result in improved cooperation, I resorted to, “Do we do this the easy way or the hard way?” The kids know that the hard way is no fun, so this threat almost always works — but not this time.

So, the hard way. I confiscated the items that Archer had been carrying around and pinned him to the changing table while wrestling his pajama top off and then his shirt on over his head. After lots of struggle, and plenty of crying from Archer, I managed to get his clothes on.

My victory was short-lived. Archer still held the trump card — the one sure way to make me remove the clothes I had just forcibly caused him to wear. With an assist from his convulsive crying and a belly full of Malt-O-Meal, he barfed all over them.

A cautionary tale for all who believe their authority is absolute.