## Proof Without Words VII

From here (PDF link). Mouse-over the image for a verbal description of the proof.

Also check out Proof Without Words IIIIIIIVV, and VI.

Filed under Numbers

## Measuring a Year

Here’s a puzzle:

You find yourself on an unknown uninhabited planet. You look up the sky and notice a giant star that looks like our sun. It rises in one particular direction and sets in the opposite, so you figure that this planet (that you’re on) orbits around this particular star. Now it’s easy to define a day on this planet: it’s the time between two subsequent sunrises. The question is: how would you define a year on this planet? (Assuming, of course, that your survival on this planet is not an issue.)

Someone posted this question on Quora, which got me thinking about the definition of a year. One earth-year is defined as 365 days, or 12 moon cycles, and these measures are not arbitrary. They (approximately) represent the time it takes for the earth to complete one orbit around the sun. With that in mind, the answer to the above puzzle is rather straight-forward for a planet that has an axial tilt.

If you landed on a planet with an axial tilt then the sun’s trajectory in the sky will change over time and you can track it to measure one year. The following approach can be used to measure one complete orbit: Mark the point at the horizon where the sun rises the next morning. Every subsequent morning the sun will rise at a slightly different point. (The amount of shift depends on the degree of axial tilt and how fast the planet is orbiting its sun.) For a while, the sunrise points will shift away from your initial mark in one particular direction (either left or right). And after a certain period, like a pendulum, they will start moving back towards the initial mark. (This U-turn marks a solstice on this planet.) It will keep moving and bypass the initial mark, move in the opposite direction, and then take another U-turn to come back to the initial mark. The morning when sun rises again from the original reference point would mark the completion of one year on this planet. (Note that this description changes slightly if you started observing the sunrises exactly on the day of one of the two solstices on this planet.)

Here’s how the sun’s trajectory looks like from Earth (looking southward from the Northern hemisphere):

In our solar system, most planets have an axial tilt. Earth’s current axial tilt of 23.4° is responsible for seasons, rain, and consequently, the existence of life on Earth. The most curious among these are Venus and Uranus; they are the only planets that rotate clockwise (while looking at the solar system from the top). Venus has flipped almost completely and is upside down as compared to other planets. While Uranus, with an obliquity of 98°, would look like a tilted rolling ball — as opposed to all other planets that look like tilted spinning tops.

Back to the question of measuring one year, if you landed on a planet without an axial tilt, I have no idea how you would measure a year.

Filed under Astronomy, Puzzles

## Aaja Meri Jaan

This song has been constantly playing in my head for the past several days. It’s from the movie I Love New Year, and is remixed by DJ Phukan. The original tune is by R D Burman who first used it as a background score in the movie Saagar, and then refurbished it for Aaja Meri Jaan. SP Balasubramaniam rendered this kick-ass song in his inimitable style. The new version pales in comparison, but one of the reasons why I like it is because it reminds me of RD’s phenomenal tune and SP’s haunting rendition. The song in Aaja Meri Jaan had more of a jazz flavor to it, while the I Love New Year version relies on simple acoustic guitar riffs supplemented nicely by accordion. Mauli Dave’s singing is okay, and the lyrics border on mediocre (except for that one line that goes ‘Meelon chalen phir humne jaana, koi thikana ghar sa naheen’) but I think overall it’s a decent track. Enjoy!

And here’s the original, terrific track by R D Burman:

[Hat tip: Milliblog!]

Filed under Music n Lyrics

## Fingered Speech

It’s easy to nod your head in full agreement when you hear someone bemoaning the corruption of (English) language — the decline of literacy — caused by texting, especially by those teenagers obsessed with smartphones. It’s quite common to scoff at their incessant initialism that seems to crop up a new “word” — like LOL and BRB — at a rate faster than our ability to keep up with them. Their utter disregard for capitalization and punctuation induce ridicule and even derision among the cultural pessimists who are quick to deplore anything and everything that is new and different than how it used to be.

While we grapple with this overcoming intuitive sensation that language (in specific), and culture (in general) is deteriorating, we conveniently overlook the fact that virtually every generation believed that people are not up to the standards of their parents and grandparents. Texting is yet another example, apparently, which exposes our pessimistic bias. In a thought-provoking TED talk (linked below) linguist John McWhorter discusses how – in spite of what our gut feeling suggests – texting is not a modern scourge. In fact, it might just be “a linguistic miracle happening right under our noses.” To comprehend why this might be the case, to grasp the reasons behind the emergent complexity of texting, you have to zoom out a bit, and start with understanding the difference between written and spoken language. [Here's a fun fact: If humanity has existed for 24 hours, writing came about at 11:07 pm.]

I recommend you watch this short (13-minutes) video for more details, but here’s the gist of it: McWhorter offers a novel point of view that doesn’t constrain texting as purely a written form of language. Instead, texting is a case of ‘writing as we speak’. It’s fingered speech, if you will. Just like how we don’t bother about capitalization and punctuation when we talk with each other in person, texting is also a form of spoken language with a very casual make-up, with its own rules, structure and vocabulary. While texting may be violating existing grammar rules, it’s developing its own set of complex, dynamic, and rich linguistic repertoire. And it’s happening at such a fast rate that not only we struggle to keep up with it, we fail to recognize the profundity of this possibly miraculous linguistic development.

Filed under Language

## Proof Without Words VI

Proof that the area of a circle is equal to π × r2.

***

Previously on this blog: Proof Without Words I, II, III, IV, and V.

Filed under Numbers

## Ant Zombie

You watch an ant in a meadow, laboriously climbing up a blade of grass, higher and higher until it falls, then climbs again, and again, like Sisyphus rolling his rock, always striving to reach the top. Why is the ant doing this? What benefit is it seeking for itself in this strenuous and unlikely activity? Wrong question, as it turns out. No biological benefit accrues to the ant. It is not trying to get a better view of the territory or seeking food or showing off to a potential mate, for instance. Its brain has been commandeered by a tiny parasite, a lancet fluke (Dicrocelium dendriticum), that needs to get itself into the stomach of a sheep or a cow in order to complete its reproductive cycle. This little brain worm is driving the ant into position to benefit its progeny, not the ant’s. This is not an isolated phenomenon. Similarly manipulative parasites infect fish, and mice, among other species. These hitchhikers cause their hosts to behave in unlikely—even suicidal—ways, all for the benefit of the guest, not the host.

Does anything like this ever happen with human beings? Yes indeed. We often find human beings setting aside their personal interests, their health, their chances to have children, and devoting their entire lives to furthering the interests of an idea that has lodged in their brains. The Arabic word islam means “submission,” and every good Muslim bears witness, prays five times a day, gives alms, fasts during Ramadan, and tries to make the pilgrimage, or hajj, to Mecca, all on behalf of an idea of Allah [...]. Christians and Jews do likewise, of course, devoting their lives to spreading the Word, making huge sacrifices, suffering bravely, risking their lives for an idea. So do Sikhs and Hindus and Buddhists. And don’t forget the many thousands of secular humanists who have given their lives to Democracy, or Justice, or just plain Truth. There are many ideas to die for.

[...]

The comparison of the Word of God to a lancet fluke is unsettling, but the idea of comparing an idea to a living thing is not new. I have a page of music, written in parchment in the mid-sixteenth century []. The text (in Latin) recounts the moral of the parable of the Sower (Matthew 13): Semen est verbum Dei; sator autem Christus. The Word of of God is a seed, and the sower of the seed is Christ. These seeds take roots in individual human beings, it seems, and get those human beings to spread them, far and wide (and in return, the human hosts get eternal life — eum qui audit manebit in sternum.)

Excerpt from the opening chapter of Breaking the Spell.