mathemagicio.us

From Scientific American, about the recent bird attack against the United States:

Heading north without engine power, the pilot of the Airbus A320 changed course and — in what some are describing as a heroic but calm act — glided to a watery landing in full view of buildings on the west side of Manhattan. “The pilot got on and said, ‘You guys got to brace for a hard impact,’” passenger Jeff Kolodjay told the Times. “That’s when everyone started to say their prayers. I got to give it to the pilot, he did a hell of a landing.” When the pilot, Chesley Sullenberger, emerged from the plane, he had not even donned a life vest, according to the Times of London.

When parents name a dude Chesley, they clearly don’t expect him to have balls the size of Durian fruit, but score one for rising above expectations.

Uncategorized News

Despite rumors to the contrary, Barack Obama is not a secret bubble sorter.

Uncategorized CS, Google, Obama, Youtube

This is what passes for investigation on CNN: a Rob Riggle lookalike reads a printout from the internet and concludes that the Gaza conflict is complicated.

While the most trusted name in news struggles to get its “whats” and “whens” straight, I suggest following CNN’s example and getting your “whys” from the internet. Here’s some reading material I found illuminating, roughly-chronologically connecting the dots from the current crisis to its roots.

1. Israel strike on Gaza: a primer and Israel’s ghosts from the Middle East Strategy at Harvard institute lay out the recent causes and events leading up to the war, as well as a plea for diplomacy.
2. An article from Ynet news about the November 5th Israeli attack on Gaza that marked the increase in rocket attacks. Many bloggers fallaciously insinuated that the tunnels were to be used for smuggling humanitarian supplies into Gaza from Egypt, but the location of Al Bureji makes this impossible.
3. Wikipedia has a detailed list of rocket and mortar attacks in Israel in 2008, including 37 rockets and mortars launched during the six-month ceasefire in 2008.
4. Amnesty International reported earlier this year on the major humanitarian crisis in Gaza as a result of the Israeli blockade from June 2007 onward, but only hinted about the strategic purpose of the blockades. The New York Times was much more explicit as early as 2006, detailing plans to destabilize a Hamas-led Palestine through economic starvation.
5. Wikipedia has a detailed article on the June 2007 Battle for Gaza, chock-full of sources, lest you believe the misconception that Hamas took power of Gaza democratically and peacefully.
6. The 1988 Hamas Covenant sheds some light on why Israel so strongly resists a Hamas-controlled Gaza.

Yes, this conflict is complicated, and my sympathies seem to sway back and forth in the above list. What isn’t complicated, because it is invariably true in situations like this, is that innocent people continue to suffer while those comfortably in power make cold decisions about their fate.

There’s huge pressure to “pick a side” in the conflict, where the valid choices consist of one army or another. But I can’t fathom how anyone who truly believes the blessed are the poor, the hungry, the mourners and the persecuted can sit on the sidelines half a world away and choose between two armies that are none of the above. They are not divided from those who are by political or geographic lines, but by on which side of the suffering they sit.

Uncategorized News

For anyone that needs a new wallpaper, here’s a breathtaking photograph peering into the center of our galaxy, thanks to telescopes Hubble and Spitzer:

More information is available here, or click on the picture itself for links to larger versions, including a 6000-pixel wide JPEG.

The amount of detail in the high-resolution version is absolutely amazing, but nearly insignificant in context: the picture captures an area 300 light years wide, or more than 3000 AU per pixel. Given that our solar system is roughly 80 AU in diameter, each pixel reveals a square of space 40 times wider than our own solar system.

We share an amazing universe.

Uncategorized Science!

Seven years after the fact, it’s slightly surreal to read post-9/11 responses again. Time makes it easy to forget that everyone in the world had hysteria-inducing levels of adrenaline flowing through their blood for at least a month after the attacks, enough to make even the brightest people believe the proper response to be for the highest civilisation ever to invade all the Arabs and convert them to our way of life. (Surely this seemed tractable at the time.)

Meanwhile, there’s this article over at BBC News about (ahem) Abū ʿAlī al-Ḥasan ibn al-Ḥasan ibn al-Haytham, a tragically poorly-known, 10th-century scientist whose name the West (for reasons I cannot understand) abbreviated Alhacen. To name only a few of his discoveries, he

1. Invented the scientific method, and stressed the importance of demonstration by experiment rather than philosophical argument,
2. Showed that incandescent objects radiate light outward in straight lines, 
3. Discovered various properties of refraction and reflection,
4. Argued that the speed of light is very great but finite,
5. Invented and explained how the pinhole camera works,
6. Learned how to integrate polynomials 700 years before Newton did, and
7. Very nearly discovered non-Euclidean geometry.

We may in fact be the highest civilisation ever, but sometimes it’s a good idea to remember which of its parts we borrowed, and how they sprang from such lesser parts of the world.

Uncategorized Science!

Ruby is a wonderful language, largely deserving of the fanaticism surrounding it. There are a number of ways you can exploit its syntax to write concise, beautiful code. For example, to shuffle an array…

deck.sort_by{ rand }

…or to pick out certain elements of one…

deck.find_all{ |card| card.suit == Clubs }

…or to seamlessly cache computations.

def average_earnings
  @average_earnings ||= some_lengthy_computation
end

(Above, the ||= operator acts analogously to the familiar += operator. So if the instance variable @average_earnings already has a non-nil value, it is returned without any further computation. If on the other hand it is nil, then some_lengthy_computation is performed, @average_earnings is set to it, and returned.)

In addition, there are also a number of ridiculously short applications written in it, including a web server in 70 lines of code, a message board application in 500 lines, and its slightly more verbose successor.

In addition to these, I present a proof of concept of my own: Mathematica and Maple-like symbolic differentiation in about a hundred lines of code.

Read more…

Uncategorized CS, Math, Programming, Ruby

Einstein famously wrote that “the most incomprehensible thing about the world is that it is comprehensible.” By this benchmark, one of the most amazing properties of our universe is the amount of self-similarity it contains, how the same phenomena occur on various scales of size, time, and in various forms of matter and non-matter. The laws of Nature seem designed to allow complex behaviour to be approximated by fundamental laws that appear also on smaller scales.

Mechanical waves (of water, sound, and so on), which are incredibly complex systems made up of a billion billion billion molecules, each made up of atoms made up of electrons and protons and neutrons made up of quarks, propagate in a fashion remarkably similar to the way individual photons do in an electromagnetic wave.

Even the familiar formula for a photon’s momentum, p = E/c readily approximates the momentum that a mechanical wave carries, when we substitute the wave’s velocity v for the speed of light c.

The waves of water in a pool interfere exactly like electromagnetic radiation does, but by entirely different processes! Photons interfere even when there is only one photon according to the laws of quantum mechanics. Comparatively, mechanical waves propagate and interfere for classical reasons, by the momentum carried from atom to atom in an unimaginably long chain, in order to mimic the behaviour of electromagnetic waves.

The gravitational force exerted by a planet 12,000 kilometers wide made up again of an enormous number of particles is approximately that exerted by a single point of the same total mass. And it is exactly into this shape that gravity tends to mold the matter in the universe, as if to make its effects as simple to analyze.

Combining the above two observations, we may analyze the mechanics of one billiard ball striking another by applying a much simpler model than the one seemingly required by reality.

This kind of approximately self-similar behaviour in Nature brings Microsoft’s Photosynth technology to mind. At an incredibly zoomed-out perspective, complex interactions average out and allow the billiard ball to be seen as a lone object, striking another ball and instantly transferring its kinetic energy into it. Only as we zoom in do we begin to see the detailed interactions inside the balls, and how the kinetic energy is transferred between them by means of a wave propagating through their matter. Zooming in further we see the balls’ individual atoms wiggling about and interacting, the cause of what previously appeared to be a wave.

If mankind one day decides to engineer an efficient universe, our own seems like a good model.

Uncategorized Philosophy, Science!

As we all celebrate another revolution around the sun and come halfway closer to Epiphany, let’s explore a holiday-themed math problem.

The number of gifts one receives at the end of the Twelve Days Of Christmas is ironically 364—a gift for each day of the year except Christmas itself.

But what about the inhabitants of Gaia-n, who celebrate not 12 but n days of Christmas?

On the first day, such an inhabitant receives 1 present; on the second day, 1 + 2; on the third, 1 + 2 + 3; etc. Then, thanks to a theorem by some clever little kid, we can write the number of gifts he receives on day k as , the k^th triangular number, and thus the total number of presents he receives in all ndays as the sum of triangular numbers (where C is for Christmas.)

A geometric argument easily shows that , so (assuming n is even)

which is four times a pyramid number! Substituting the closed expression for the sum and simplifying gives,

.

We can easily verify that this expression also gives the correct answer when n is odd, and moreover as we hoped!

On other planets, inhabitants may celebrate a fraction of a number of days of Christmas, and they may wish to extend the expression to any real (or complex!) number, i.e.

which we naturally call the Christmas polynomial.

We may also wonder how many gifts someone will receive who has relatives on each of the planets Gaia-1, Gaia-2, …, up to Gaia-n. That is, what is the value of the sum ?

Because we may write and , it is a reasonable guess that . This indeed turns out to be the case, and so we may naturally define a generalized Christmas function

(where χ is for Xmas.) This results in the sequences

and so on, allowing us to calculate a great number of Christmas gift-giving scenarios.

Uncategorized Christmas, Math