Monday, August 23, 2010

Who Killed Hugo?

A few weeks back, I lamented the death of my pet spider, Hugo, and I wondered what might have killed him. One of my thoughts was that he met another spider who was just a bit bigger and nastier than he was, and they had a bit of a spider-rumble. But Hugo was a pretty big guy, and I couldn’t imagine any spider hanging out in my house who was all that much bigger than Hugo.

Then I went away to the states for a few weeks. Upon my return I discovered that, not only was there a ginormous humungo Godzilla-spider hanging out in my bathtub, but he was acting like he owned the place!

I think I found Hugo’s killer!

I eyed him accusingly but he just snarled back. Well, this bathroom was not big enough for the both of us. So I left. I mean this guy was really big! Upon throwing a ruler near him I determined that his wing-span broke 6.5 inches. I did try to photograph him (with a telephoto from half a mile away. I didn’t want to get anywhere near this thing). I apologize that the photo is so bad that it is hard to see how damn large this thing was. But believe me, he could have eaten a large rodent.

I knew that I had to find a way to eliminate this demon-spawn. I thought about an exorcism, but not being religious – yet still being frightened to death when I watched the exorcist, I decided that this was not the right approach.

I armed myself with some serious heavy ammunition – all the shoes I owned. I stood out in the hallway, a good distance from the beast, and catapulted the shoes in his direction. After the first shell landed right next to him, he knew he was under attack. He scampered to the other side of the bathtub. Ha! Unfortunately he still had a spider-sized brain, and he did not realize that the other side of the bathtub was much easier for me to hit. With the second shell (my old hiking book) I fired and squashed him flat. I waited for a few minutes to see if he was planning on throwing the boot back at me. But after a few more minutes I determined that he really was deceased.

So it was that I avenged the death of Hugo.

Right now I am off in Ireland for a few more weeks. I'm wondering what new monster is going to be living in my house when I get back.
Saturday, August 21, 2010

The Girl Who Played with… Euclid’s perfect theorem:

Like most of the northern hemisphere, I’m in the middle of reading Stieg Larsson’s books “The Girl with the Dragon Tatoo” series. In the second volume, (“The Girl who Played with Fire”) our heroine, the punked out antisocial hacker, Lisbeth Salander, gets absorbed in recreational mathematics.

Now, in almost any popular novel, when the author starts delving into math, I am usually pretty familiar with whatever theorem, or unproven conjecture the author is mentioning -- frequently much more so than the author. I applaud the attempt to bring more science into popular fiction, but I don’t usually learn much from it.

However, on page 21 of the “The Girl who Played with Fire” the author mentions an ancient – and very beautiful! – theorem by Euclid, which rather shockingly I had never seen before. (A major gap in my education!) I scratched my head for a moment, then figured out Euclid’s proof (despite the fact that Lisbeth Salander sort of gets it wrong).

First of all: Here is the statement in the novel:

She was fascinated by Euclid’s discovery in about 300 B. C. that a perfect number is always a multiple of two numbers, in which one number is a power of 2 and the second consists of the different the difference between the next power of 2 and 1. This was a refinement of Pythagoras’ equation and she could see the endless combinations

6 = 2^1 ( 2^2 -1)

28 = 2^2 \times ( 2^3-1)

496 = 2^4 \times( 2^5-1)

8128 = 2^6 \times ( 2^7 -1)

She could go on indefinitely without finding any numbers that would break the rule.

(Yes, Larsson likes to write in Italics). OK, now the proper statement of the theorem.

Definition: A perfect number is a number where the sum of the number's factors adds up to the number itself. For example:

The factors of 6 are 1, 2 and 3. 1+2+3 = 6 so 6 is a perfect number.
The factors of 28 are 1,2,4,7,14. 1+2+4+7+14=28 so 28 is a perfect number
and so forth.

Euclid’s theorem: IF (2^k-1) is a prime number THEN 2^{k-1}\times (2^k-1) is a perfect number.

Salander does not mention the IF required of this theorem, but note that on her list of perfect numbers, she lists k=2,3,5,7 which are cases where (2^k-1) is prime. This type of prime is known as a Mersenne Prime]. How Euclid proved this theorem is beyond me. He was brilliant, but he did everything with geometry – and very little algebra (which is how I intend to prove it).

Almost 2000 years after Euclid’s proof, Euler proved that if an even number is perfect, then it is the form given by Euclid’s theorem. It is still not known if any odd perfect numbers exist --- although if they do exist they have to be ginormous since it has been proven that no odd perfect numbers exist less than 10^{300} . Assuming there are no odd perfect numbers, then there is exactly one perfect number for each Mersenne prime. It is not known how many of these there are.

OK, a quick proof of Euclid’s theorem: Consider the number 2^{k-1} (2^k-1) . If p = (2^k-1) is prime, then the only factors of (2^{k-1})p are

1,2,4, \ldots, 2^{k-1}


p,2p,4p, \ldots, 2^{k-2}p .

Now the sum of the series

1 + 2 + 4 +\ldots, + 2^{k-1} = 2^k -1

And similarly

p + 2p + 4 p + \ldots + 2^{k-2}p = p ( 1 + 2 + 4 + \ldots 2^{k-2}) = p(2^{k-1} -1)

So the sum of all the factors of the number gives

(2^k-1) + p(2^{k-1}-1) = (2^k-1) + (2^k-1)(2^{k-1}-1) = 2^k(2^{k-1}-1)

which is the number itself!
Thursday, August 12, 2010

Harvard is Easy – Almost

This year’s hiking/camping trip with Carissa and Lin was to Mount Harvard –- the third highest peak in Colorado at 14,420 feet (4,395 meters to those who insist on Système-International d'Unités). It is the fourth highest peak in the 48 states behind California’s Mount Whitney, and Colorado’s Mount Elbert and Mount Massive.

You can read about last year’s Hiking trip on this page and this page (I apologize that the pictures are now defunct). Given that we failed to reach a summit last year, we decided on a simpler climb this year. Harvard is listed on the web as a clearly marked trail with little, if any, scree to slog through, making it a high, but relatively easy climb.

Our intent was to hike a few miles into the park on Friday and set up camp in between Mount Harvard, Mount Yale, and Mount Columbia of the Collegiate Range of the Sawatch. Then on Saturday we would climb Harvard, and on Sunday we would do the slightly smaller Columbia, and also walk back out to the trailhead by nightfall.

As planned, our hike started on Friday with full packs. We walked a few miles up the trail (gently uphill) until roughly the place where the trail splits towards either Harvard or Columbia. We set up camp near the fork in the road. The campsite we found had clearly been used by many before – it was perhaps the softest and flattest dirt I’ve ever seen anywhere in any woods. Add to this a standard thermarest air mattress and a thick sleeping bag, and quite frankly my tent was better than the beds I suffer through in many a hotel. I had no complaints whatsoever about comfort.

This was the view from our campsite. For a while we thought that the mountain in this picture is Mount Harvard, but it isn’t. I think it is known as Birthday Peak.

Carissa and Lin were dead-set on not eating freeze-dried camping food this year. So the first night’s dinner was curry-cous-cous with tofu and raisins and cashews and some broccoli soup. It was terrific. We had enough left-over cous-cous for about two days worth of breakfast and then some. After dinner we dove into our respective tents to try to escape the evening feeding frenzy of mosquitoes.

Another observation about Colorado mosquitoes: Last year I noted that the Colorado breed of mosquitoes are a lot slower and dumber than the east coast variety. This year I will add the observation, that if they do manage to bite you, it itches far far less than in New Jersey. What sin did New Jersey commit in a past life to deserve fast, vicious, and itchy mosquitoes? Maybe they are a product of toxic waste --- sort of like Godzilla being created by nuclear waste.

Now back to hiking:

When hiking in Colorado, one should really be on the way down from the mountain by noon since thunderstorms tend to come in the afternoon and it is rather dangerous to be caught up high during a storm. Even given this restriction, it is actually possible to climb both up and down Harvard and then up and down Columbia all in one day, but only if you start insanely early in the morning. During the night we heard a few climbing parties slogging by in the wee hours of the morning, and we assumed this was their intent. We, on the other hand, slept luxuriously late, ate our cous-cous breakfast, drank some coffee, and started up the trail at around 7.

Having started the morning a fair distance up the trail already (and with no intent to do two mountains in one day) we did not feel pressed for time. As long as we kept slogging along at a slow pace, we felt we would make the summit easily by noon.

For several miles, the trail was an easy (albeit uphill) country walk. It was well marked and easy to follow and the terrain was about as smooth and clear as you could ask for. For this, we must thank all those people who don’t understand statistics very well. You see, upkeep of many trails in Colorado is paid for by the vast proceeds of the Colorado lottery. It makes me almost want to buy a ticket even though I do understand statistics more or less. In fact one of the topics of conversation on our hike was Baysian reasoning. (But don’t get the idea it was all intellectual conversation --- we had long discussions of which muppet videos were the best).

The smooth trail goes up the left of the picture above. Bear lake is on the right (Despite the name, we didn’t see any bears – but we were careful to hang all our food in bear bags nonetheless). In the background is Mount Yale. Just behind Bear lake is the little peak which we could see from our campsite (possibly Birthday peak).

A bit higher up, the trail became much more steep. But for most of the way (again thanking the lottery) it was an extremely good trail --- like walking up a very long flight of stairs. Of course, once we got above about 12,000 feet, we found ourselves huffing and puffing quite a bit due to the thin air --- about 40% less oxygen than at sea level. Step-breathe-step-breathe, we made progress slowly but surely. Aside from all the huffing and puffing, the climb was not really all that difficult for most of the way.

Along the way we saw a bunch of random cute animals, including Pikas and Marmots which are both fundamentally rats, but are also insanely cute. That's a pika above and a marmot below (the fat one). I still expect him to start singing "I'm all right... don't have to worry 'bout me".

Near the very top, the trail suddenly turned into a boulder field and finally stopped being so wonderful. In Colorado, hiking trails are ranked into “classes” describing how technical they are (See here for a description). Harvard is supposed to be a class-II climb, meaning that you rarely need to use your hands. However, over this boulder field (with some pretty exposed cliffs staring you in the face) it is probably class-III. Here's a picture of me a bit before the boulders really start getting serious.

Fortunately, the boulder field is quite short and we reached the summit with no real trouble – and well before noon. Here’s a picture of the three of us on the summit (Carissa is in front, Lin in the middle, and I’m hiding in back).

Some guy was on top smoking a cigarette and taking pictures with his iPhone --- and get this: he actually had reception on the top, so he was sending pictures as email. This photo was taken by his iPhone and sent directly to my Mom. Actually it is not such a bad picture. The picture would have been better if he could have backed up a few more feet to get more of a view, but then he would have fallen off a five hundred foot cliff.

In this final summit photo, I’m pointing to what I think is Mount Oxford. We had discussed climbing Oxford as well, but decided against because the trail is not supposed to be well marked – maybe I’ll try it next year. Actually it turns out that Oxford is the mountain further to the right in the photo (over my left shoulder). The mountain I’m pointing to is probably Mount Belford.

After a peanut butter lunch on top (jealously protecting our food from an eager Marmot), we started the long climb back down. From the top, the weather still looked clear so we had plenty of time to go slow downwards. (I climb down extremely slowly – Carissa and particularly Lin seem to have the genes of mountain goats and were able to go down at a much greater speed).

Despite my slower progress I nonetheless managed to twist my ankle on the way down. Crunch. Ugh. Fortunately it didn’t seem too bad and although it hurt a bit, it was no problem to keep walking on it.

As we got closer to the bottom, and the weather had still not turned foul, we felt free to linger even longer as we went along. Back down in the country fields we even took our time and sat by a nice creek for a while soaking our feet. At this point we had only about two relatively flat miles back to our campsite. Here I pose for a picture near the creek. Harvard is in the background.

Just as we were leaving the creek to head back to camp I commented “You know, Harvard was pretty easy!”. Well, unfortunately, it seems the mountain gods heard me. About a dozen steps after that my twisted ankle started making an unpleasant (and painful) crunching sound. Fortunately, Lin was carrying an Ace bandage which patched things up temporarily and I had little trouble getting back home. That will teach me not to insult the mountain gods.

So Harvard was easy, almost.

Back at camp, we started cooking dinner. This time it was mac-and-cheese with (dehydrated) tuna. Again yum. Although bringing real food is a bit more heavy than bringing all dehydrated hiking food, I’d vote that it is totally worth it.

While we had planned to climb Columbia the next day, considering the condition of my foot, this was not going to happen (at least for me). I was hoping that at the very least the ankle wouldn’t swell up too much, and I’d be able to hike out of the woods without too much problem the next day (being carried out would be rather humiliating). Carissa and Lin graciously decided that they would not climb Columbia without me (Actually it looked like a bit of a scree-fest anyway – the grapes were sour).

The next morning my foot was no better, but no worse either. So at least I would be able to walk out of the woods. We slept late, cooked breakfast (oatmeal), packed our tents and had a relatively easy hike back to the cars. Had a few beers by a river to celebrate our successful trip, and called it a day.
Tuesday, August 10, 2010

The Best Physics Lecture Ever

Out in Aspen Colorado, a few times each summer, there are physics lectures aimed at the general public, given in the memory of Heinz Pagels. Many years ago I heard one such lecture on the subject quantum mechanics given by a friend of mine named Shankar (one name only) who is a professor at Yale. The lecture was both entertaining and inspiring and accessible to all (not just to the physics cognoscenti). I remember how impressed I was, and I thought how great it would be to be able to give such a performance.

Last year I was honored to be asked to give one of these Pagels lectures. I spent literally weeks preparing it, trying to live up to the high standard set by Shankar. The lecture went very well, and maybe I got close to his level, but still I have to give credit where it is due: his lectures are still the yardstick by which other physics lectures should be measured. Perhaps we should declare a lecture to be a milli-Shankar if it is one one-thousandth as good as one of Shankar’s lectures.

This summer, Shankar gave yet another lecture --- this time on the subject of relativity – one of the most beautiful subjects in physics. This lecture was even better than his previous one. In fact, it was perhaps the best physics lecture, on any subject, that I have ever heard!

Fortunately, you can find all of these lectures streaming on the web. While sometimes a bit is lost in the translation to low-quality streaming video, nonetheless, I think they are all worth watching.

Here is Shankar’s lecture on relativity.

Here is his lecture on quantum mechanics.

... and in case you missed it last year, here is my lecture

Bobba Hannah Sandusky

My Great-Great-Grandmother, Hannah Sandusky, was a rather unusual woman. Having immigrated to Pittsburgh around 1860, she became known as “The Angel” to those she helped. To everyone else, she was known as “Bobba Hannah.” “Bobba”, or “Bubbe” is Yiddish for grandmother. It also means “Midwife”, which was her profession -– although she never accepted fees for her services. Over her lifetime she delivered many thousands of babies for the poor in the Pittsburgh area. Although she was horrible at keeping records, she managed to register over 3500 deliveries – and probably far more remained undocumented. Hannah also spent a lot of time doing other charity work, as well as the duties expected of Jewish grandmothers, such as matchmaking.

Thanks to some major legwork by several of my cousins (Kudos to Miriam Baker in particular) this last weekend in Pittsburgh we had the first ever reunion of the descendents of Bobba Hannah. Hannah and Louis Sandusky had eight children: three boys and five girls. Only four of the daughters had offspring, so the Sandusky name died out, and the clan became divided into Raphaels, Gordons, Schugars, and Simons. The clan is now spread out around the world, and until this last weekend, I had only met a tiny fraction of my extended family.

During this weekend I heard a number of interesting stories. There was the story of the funeral crashing cousin who ended up being locked in a closet by the undertaker; and there were many stories of my grandfather, who was frequently on the wrong side of the law but managed to always evade the feds one way or the other. But perhaps the most interesting story was how my grandmother met my grandfather.

In 1913, shortly after immigrating to Pittsburgh from Lithuania via South Africa, Eva Grabowski was told that an unusual event would be taking place: a funeral procession which was made up of both Jews and African Americans. Needless to say, this was Bobba Hannah’s funeral.

Here I quote from a transcript dictated by my grandmother a few years before she passed away:

I was standing on the verandah watching the funeral procession... On the other side of the road a young man was also standing on a balcony watching the procession where he spotted me. We became acquainted and his name was Ike Simon. He was a big man with a heart of gold and when he asked me to marry him I decided to accept, having in mind, if it didn’t work out, I would leave him and return to South Africa.

Well, it did work out, and I’m sure Bobba Hanna, the part-time matchmaker, would have been very pleased.

[Note: There is some confusion in the story as to whether they met at Hannah's funeral or some other funeral. It makes a better story this way].