A Pale View of BLUE Hills

Light (and its interplay with matter) is the flavor of these days and here is a bit more...

Hills, more so those with substantial green cover, appear blue to a distant observer. I have never quite figured out why. An answer I have heard often is 'for the same reason that the sky is blue'. I could not quite see the connection. My understanding is, thus, rather 'pale'.

Here is a page from the cluster of expository material maintained by John Baez which explains how the real reason behind the sky appearing blue was not really known until Einstein settled some key issues in the matter. It also has some stuff about Blue Hazes due to scattering of light by organic compounds called terpenes emitted by vegetation and the rare Blue Moon phenomenon.

One could theorize as follows: the hills emit light of two colors - overwhelming green from all the greenery and the much less of scattered blue light from the steady terpene emissions from the same greenery. If we see the hills from close, the green is predominant and overpowers the terpene blue. However, from a considerable distance, there is a thick intervening layer of atmospheric air which scatters blue very preferentially - it is as if we see the hills thru a 'blue film'. Now, a film with a particular color blocks out all other colors. The green of the hills gets cancelled by the intervening air and the much weaker terpene blue persists and that is that. One could also guess that without the terpene factor, hills would have appeared simply dark rather than blue from a distance.

But I am not sure if a thick layer of air, though it scatters blue light preferentially, can be conceived as a film that allows only blue light to pass. A colored film essentially ABSORBS all colors than the color it lets thru; on the other hand air does not seem to absorb the other visible colors than blue - these colors simply pass thru unscattered for much longer distances than blue. Then how does the green color from hills fail to reach a distant observer?

One more point that puzzles me: Clouds are either white or a deep violet ( I dont think even the darkest rainclouds are really black). One sees no shades in between. Why only these two colors? That white clouds appear perfectly white with no tinge of blue probably refutes the 'blue film' idea given above. But I have no better idea.

Why The Pond Appears Shallower - and some thoughts on Feynman

"Refraction of light - it bends away from normal when entering air from water - so..." This line of statements backed up by (variants of) the same picture appears in most basic explanations of this phenomenon.

What is often glossed over is that light bending away from normal on entering air obviously explains ONLY that the virtual image of an underwater object (including the pond bed) gets laterally shifted but NOT the image appearing LIFTED vertically. In elementary explanations, one 'cheats' in the picture by continuing the refracted ray back into water ONLY up to a point that lies vertically above the actual object. This begs the question: Why is the ray continued back into water only that much?
As for the rod dipped in water appearing bent, one may note that the rod appears bent at only the interface between air and water; the portion in water is STRAIGHT.

Feynman has answered much of the question in his Chapter 27, Vol I of his famous lectures. He derives from Fermat's principle of least time the result that the ratio between real depth and apparent depth is equal to the refractive index of the medium, when the surface of separation is a plane. This explains the 'image getting elevated'. One could add that, observationally, whatever be the ANGLE at which the line joining the observer and the virtual image (the refracted ray) emerges from the refracting surface, this ratio between real and apparent depth holds (otherwise, the rod dipped in water will not only be bent at the interface, the part within water will also be CURVED). I have not entirely been able to convince myself that this angle independence also is contained in Feynman's explanation (basically I have not yet spent adequate time contemplating what the Master has said). One intuitive way of saying it could be: "refraction lifts up everything by a constant ratio of the depth at which it lies. Then once this lifting is over, it does not matter from where in the air you look at it, you see the same lifted image; and the position of the image does not depend on where it is viewed from in any case, even in the case of a plane mirror".

Maybe that is all there is to it (somehow some doubt lingers!). One needs only to be precise about what an image is and calculate how refraction causes a virtual image and then things settle. Perhaps the 'elementary way' of explaining things has caused confusion when getting to the heart of the matter...

Winding up that 'stream of consciousness', I would like to add here that the flower of the Indian Physics community is NOT unanimous in its opinion of Feynman lectures. Along with fervent assertions that they 'surely rank among the classics of our times', the lectures also have invited (sometimes round) condemnations for (allegedly) misleading students by creating an illusive, lazily intuitive picture of physics, by hiding nitti-grities in glib talk and intellectual sleights of hand. 'Dont think you can do things the way Feynman does. He is a genius; you are not!' is a very common way of rebuking young votaries of Feynman.

The Bulbs Glow!

This post could alternatively be titled 'The Broken Bulbs, Cracked!' :)

Well, Umesh Nair (a big THANKS! to him) has kept online a very detailed solution to the 'Glass Bulbs' puzzle which was posted earlier here.

Do check out his solution here

From Verb Conjugation To Pickled Mangoes

There is one feature that makes Malayalam truly stand out among Indian languages: its Verb Conjugation depends ONLY on the tense. There is no dependency whatever on the gender, number, person or whatever of either the subject or the object of the verbs. There are some irregularities in the way the tense affects the verbs but the subject and object never come into the picture. For example, 'poyi' is the past tense for the root verb 'po' (go). This form is unaffected whether it is I, we, you, he, she, it or they who has 'done the going'. Transitive verbs show no dependence on the object either.

Malayalam is said to be a youngish language (dating back to only around 14th century) and has borrowed heavily from Tamil and Sanskrit (and perhaps Kannada) primarily and also plenty of vocabulary from English and a host of other Indian and foreign languages, all this mixing has a given a certain richness and also an under-construction feel to the language. In such a scenario, the above speciality of verb conjugation seems particularly intriguing since none of the primary source languages has anything like it.

A theory I heard attributes this speciality of malayalam to Chinese influence - Chinese apparently has such an 'insensitive' verb conjugation. Prima facie this explanation sounds far-fetched. For languages to have been influenced so fundamentally, there ought to be intimate interaction between peoples. There do not seem to have been any serious Chinese trading posts or 'factories' in Kerala - although their ships did reach south India and even Africa regularly in medieval times - let alone colonization; no hint can be seen in the Malayali physiognomy that could indicate the admixture of Chinese blood - on the other hand, at least scattered Arab and European ancestry shows at least on a few Malayali faces

But there is also some hard evidence of very strong Chinese influence and perhaps even presence in Kerala. Silk, porcelain, frying pans, (a type of) fishing nets, etc..are still referred to as 'Chinese' in Kerala. The word 'samprani' in Malayalam and also Tamil seems to have been an adjective applied to more than one thing of Chinese/East Asian origin, from incense sticks to a special class of sailing vessels (India might have copied the common ritual of burning incense sticks from China). 'samprani' is also an insult in both these languages, a racist slur perhaps! The 'chempavu' strain of rice, a traditional favorite with Keralites is almost certainly the 'champa', first bred in China or probably, Vietnam. And the temple (and residential) architecture of Kerala has almost as much of a Chinese flavor as the 'rest of Indian' (resemblance between Nepal and Kerala temples could be due a shared Chinese influence)....

Yet, the folk memory of Kerala has not preserved much on the visitors from the far east. I vaguely know of only one story - a visiting Chinese merchant leaves a set of sealed 'bharanis' (porcelain jars) for safe-keeping with a poor Kerala family and they turn out to be filled with gold coins (up to this point, the story line is 'universal') and oddly enough one of those jars (it had a slight manufacturing defect and was called the 'kodan bharani') later turns out to impart a uniquely delicious flavor to pickled mangoes (!) stored in it!

Note Added On April 27th 2005:
I know very little about languages such as Tulu, Kodava and Sanketi which are spoken by smallish numbers of people in Karnataka and in particular, whether verb conjugation in these languages is as 'insensitive' as in Malayalam. Tulu especially is said to have some affinity with Malayalam and there is a dialect called Byari spoken by Muslims of Coastal Karnataka which is said to be a mix of Malayalam and Tulu. So, the above claim about Malayalam being 'unique' in its verb conjugation may not quite stand up to facts.

I still think the intensity and importance of the 'Chinese connection' with Kerala and Southern India in general is often not adequately appreciated and THAT has been the main theme of this post.

'Good', 'Bad' and 'Ugly' - some issues with coin systems

The change making or coin changing problem is well-known to be NP-complete (a famous class of deceptively difficult problems but technicalities therein are not the immediate concern of THIS post). The focus here is on some basic issues of change making; these are very simple to state - and to appreciate :)

Given a coinage and a target amount. Give the amount using the given set of coins with the least number of coins used. If the denominations in the coinage are say {1,2,5,10,20,50,100}, a 'greedy procedure' will do FOR ANY TARGET. This is a simple and quick-working protocol: one just keeps picking up from the coins the largest coin possible at each stage and that will give the best answer at the end (ie uses the least number of coins to change the target amount). eg: if target were 87 in the above coinage, we just add coins 50, 20, 10, 5, 2. We cannot do better than that.

However for some coinages (we could call them 'bad' ones), greedy fails to give the best breakup. If in the above example we had the denomination of 40 in addition to what is given above, 87 is best broken into 40, 40, 5,2. The greedy procedure will start with a 50 and fails to find this best solution.

What follows is my understanding of the problem (thanks to Ramana Rao for sharing his insights):

1. Just like finding the best breakup of a target for a given coinage, determining whether a given coinage is good or bad appears to be an NP complete problem in itself,

2. There could probably be a further classification within bad coinages - there are some bad coinages which can be corrected by adding a few more denominations (with the additional coins, greedy algorithms starts giving tbe best breakup) and some which are 'worse than simply bad', meaning no number of additional denominations will turn it into a good coinage. Note that coins to be added to a coinage are all larger denominations than those already there.

eg: {1,2,4,5} is a bad coinage - it fails for target amount 8 - which can be made good by adding the denomination 8.

{1,2,3,5,6} is again a bad coinage - it fails for 10 - which can be 'goodified' by adding 9.

{1 2 4 5 6 8 10} fails for 13 and cannot be made good by adding any number of coins. Any coinage got by adding new coins to this will fail for some targets. So, the badness in it is actually more and we may call this an 'ugly' coinage.

The simpler coinage of {1,3,4} is again an ugly one.

Qn: Given a coinage that is known to be bad, how does one decide whether it is ugly?

3. Most standard sources talk about only two methods to solve change making - the greedy algorithm and a dynamic programming pseudopolynomial time method (given nicely in for instance Brassard and Bratley's textbook), which always gives an optimal breakup.

Qn: Are there other in-between methods which may not be as simple and fast as 'Greedy' but can give better results (need not be optimal; could be an approximation)? Naive Greedy can at times be way off the mark.

eg: With {1,100, 150} as coinage and 200 as target, 'greedy' gives a very poor answer 150 + 1+1 +1+.... whereas the optimal 100+100 is hugely better. An in-between approach might come in handy in such cases.

4. We have assumed above that the stocks of coins of each denomination is infinite. The problem gets more real-life and difficult when the stocks of the coins available are limited. The dynamic programming method mentioned above can be made to work with minor changes if only one denomination is in limited supply but that is not really getting far. Item 3 above of finding alternative algorithms to Greedy becomes really 'urgent' here.

Iste Bunu Seviyorum!

A few days back. I buy some junk at McDonalds. The pack
has their punchline printed in many world languages, several of them East Asian ( little more than clusters of slashes to my ill-informed eyes). Among the lot, the following line catches the eye: "Iste bunu seviyorum". The initial reaction is "Latin! but why??". Then one reasons: "Maybe during the 'interregnum' in Vatican a whiff of the supposedly dead classical language is in the air".

A little later, one remembers: Hey, 'y' as in 'seviyorum' is not there in Latin. Then, what could it be,... Romanian?

Having quickly run out of guesses, I turn to the all-knowing Google and it reveals it is (most probably).... Turkish! One could have smelt Turkish if the words had some 'glu' endings or some c's with little slashes hovering above; but certainly not when everything about the inscription has such a strong latin flavor - although perhaps not enough of a flavor to impress Benedict XVI (who appears to be against treating Turkey as culturally part of Europe).

Anyways, I'm Lovin It!

Note: While on Turkish, it has at least some words with a Dravidian feel. I'm told, 'Kara' means 'black' in Turkish, just like in Tamil and Malayalam; moreover to make nouns plural Turkish gives them an 'ar' ending - again very Dravidian!

Glass bulbs - a puzzle

The 'base version' of the puzzle:

You are standing next to a 100 floor building and are given two glass bulbs from a stock of identical bulbs and need to destructively measure their strength as follows - the strength of the bulb is defined as the lowest floor of the building from which it will break on being dropped. It is required to calibrate the bulb strength with least number of drop experiments in the worst case.

For instance if there were only one bulb, one necessarily has to drop it from floor number 1, 2, 3,.... till it breaks. In the worst case one might end up doing 100 drops. There is no scope for improvement.

What if two bulbs are given and you are allowed to break both?

The full version:

Same as before but you are given M bulbs from a large stock and an N floor building. How to minimize the worst case number of drop experiments?

From a computer science viewpoint, this can be viewed as a searching problem - We have a key that is unknown and are trying to find it in a given sorted list by comparisons with elements thereof with the comparisons giving only "less" or "greater" as answers. More precisely we are trying to find the least element in the sorted list that is greater than or equal to the key.

But there is a constraint. The number of comparisons between the given key and list elements larger than the key cannot be more than a certain number (corresponds to the number of bulbs available to break). Without this constraint, one could employ a free binary search or some such procedure.

Anamika

Hello World!

Today happens to be A new year's day - as good a day as any other to begin something...

This blog is kicking off after several false starts. For several weeks, finding a suitable name for it was a bother; and then 'Anamika' sort of 'hurled into view'... Literally meaning 'the nameless one' (feminine implied, probably), Anamika is also the proper name for the ring finger in sanskrit.

The finger ring is surely an ornament worn in many if not most cultures all over the world. But one can't be quite as sure whether the finger of choice was universal as well! Even in ancient India, it is not clear if the ring (for instance Shakuntala's fateful 'Abhijnana') was indeed worn on the Anamika - if it indeed were, the ring finger probably would be called something like 'ring finger' and not given such a mysteriously blank name. The names of other fingers in sanskrit are quite evocative - the index finger is called 'Tarjani' , the threatener. 'Anamika' might well indicate a certain lack of distinguishing attributes, like the ring for instance.

During Vedic ceremonies, I have often seen the 'Pavitram' being worn on the Anamika. But that really is not hard enough evidence that the anamika was indeed the ring finger in ancient India!

From a wider viewpoint, the history of Indian ornaments is not a very well documented subject, at least for a popular audience.

Many of our traditional ornaments are not only antiquated; even 'conceptually', they are weird. The anklet - 'cilampu' - of Kannagi in the Tamil classic Cilappatikaram was a hollow doughnut-shaped metallic object, filled with pearls (or were they rubies? Confusion about the 'filling' leads to the tragic climax of the story. But that is, well, another story!). A smart way to conceal/store precious gems perhaps; but aesthetically, a colossal waste!