A Conjecture Debunked - And Some More...

This is in continuation of the previous post here.

A speculative conjecture was made there: For *any* convex shape to be partitioned int N equal area pieces (positive integer N), so that the minimum total perimeter of the pieces is minimized, *all* pieces (they need not be identical in appearence to each other) must be necessarily *convex*.

This is invalid. A paper by Koutsoupias, Papadimitriou and Sideri titled "On the Optimal Bisection of a Polygon" (available for download at http://cgi.di.uoa.gr/~elias/publications/ ) has the following, rather straightforward counter-example: if an equilateral triangle is to be cut into two equal area pieces (N=2) with minimum cut-length, the cut is not a straight line but an *arc* centered on one of the vertices.

The above paper proves a result: If any polygon (not necessarily convex) has to be cut into two (not necessarily connected) equal area sets with least total cut-length, the cuts are arcs which meet the boundary of the polygon orthogonally.

However, if the polygon has to be cut into N (rather than 2) equal area pieces, the cuts do not necessarily begin and end at the boundary of the 'target'. In this general N>2 case, I have come to know that the cuts, if and when they meet in the interior of the polygon, do so at *120 degrees*. The source which revealed this has not specified if the cuts remain arcs.

I find this result (I do not know of a proof) mysterious and am searching for more details now. For a start, one can infer that the cuts meet three (and no more) at a meeting point in the interior, for *any* polygon being partitioned ...

Incidentally, after the previous conjecture was invalidated, I sort of 'pushed the envelope' and modified it thus: for any convex shape to be cut into N equal area pieces with least cut-length, for sufficiently large N (say, some function of the number of sides in the polygon), the cuts are straight lines. I am not clear if this 120 degree requirement and related facts invalidate this guess as well!

A 'Conjecture' - And A Quote From Arnol'd

I have been pondering this problem for a while:

Given any 2d convex shape (say a circular disc). We need to divide it into N pieces of equal area so that the length of cuts is least (in other words the sum of the perimeters of all the N equal area pieces is the least possible).

Here is a Claim (or a conjecture): For *any* convex shape to be partitioned and any positive integer N, *all* pieces (they need not be identical in appearence to each other) with minimum total perimeter must be necessarily *convex*.

If this claim is true, we could immediately have the following corollaries:

1. The lines which cut the target convex shape into the equal area
pieces with minimum total perimeter are all straight lines.

(even if the shape to be partitioned is not convex, I doubt if any of the cuts in the least total perimeter partitioning of it into N pieces can be 'non-straight'. at least I have not yet found any specific example)

2. For the case of dividing a *circular region* into two (ie. N=2) equal area pieces with minimum total perimeter, a diameter is the cut.

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I am told this problem could be related to Plateau's problem and other problems from variational calculus. It could also be related to concepts in computational geometry such as 'minimum ink partitioning'. I am yet to find any concrete leads though...

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To sign off this post, here is a quote from Vladimir Arnol'd:

-"Mathematics is a part of Physics. Physics is an Experimental Science, a part of Natural Science. Mathematics is that part of physics where experiments are cheap."

The article from which the above was extracted may be read here .

Pictures From Badami - And The 'Seated Vishnu'

This site has several photographs from Badami - the art and the landscape - and some general 'gyan' too. Here are Aihole and Pattadakal .

And here is the top-level index page with links to pages on several other sites.

Unfortunately, we did not know of these pages before our recent trip, on which I have written here , here and here .

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Here is a photo of the Seated Vishnu cave sculpture I had mentioned in the last post with a remark about its resemblance to the idol at the Tripunithura temple in Kerala ( here is an online picture). This particular 'royal' form of Vishnu is quite rare - indeed as far as I know, there is no other such sculpture or idol from Kerala. From Tamil Nadu, there seem to be only very few specimens - there, Vishnu is traditionally shown standing stiffly or reclining - hardly ever seated; and even if seated, not on a 'snake-throne'. Anyways, here is yet another example from Karnataka - of Hoysala origin - which resembles the Tripunithura idol even more.

Although Tripunithura lies deep within Kerala, the place seems to have had a long association with Karnataka - for instance the priests at the temple have traditionally been Tulu-speaking Brahmins from coastal Karnataka - so this seated Vishnu design could well have been borrowed from there.

While on Vishnu icons, one senses a strong Mahayana Buddhist influence in their evolution. The general depiction of Vishnu - as a handsome and heavily bejewelled youth - seems to hark back to paintings and statues of Bodhisatva Avalokiteswara - and it is conceivable that the idol of Badrinath, Vishnu sitting in meditation, originally 'belonged' to Avalokiteswara (and indeed, some forms of Siva too, 'Kalyanasundara' for instance, seem to betray Buddhist influence). And perhaps the reclining Vishnu image was an adaptation of the 'Sleeping Buddha'.

Update (Jan 2011): A sitting Vishnu relief that dates back to the Gupta period can be seen at the Dashavatara temple at Deogarh, UP (Wiki).

Impressions - Badami

This post had better be titled: "Badami Rocks!" or better, "Badami On The Rocks" (for reasons which will be presently revealed) but for the sake of continuity, I am going with the more prosaic "Impressions".

Koodalasangama, the meeting point of Krishna and Malaprabha rivers lies on a vast black soil flat that is typical of most of the old Bijapur district. From near here, we take a turn westward off the Chitradurga highway. As we approach Amingad, a blighted cattle-trading village, the landscape slowly changes - the soil turns red and the terrain begins to undulate; past Aihole and then Pattadakal, occasional hills appear, covered with scrub and studded with sandstone boulders; the odd boulders grow into imposing clusters and fantastically eroded cliffs and monoliths as we near Badami, our destination.

Badami is a large village strung out along a main road that runs towards bigger towns to the west and south. 'Development' also fans out a bit into the flat farmlands to the west. The eastern limits are set by two massive sandstone hillocks which rise abruptly a good 200 feet above the town. A largeish lake nestles between these hills. On approaching, one sees that each hillock has narrow gullies and gorges cut into it by erosion - these gullies also provide steep paths for reaching their flat tops, both of which have man-made fortifications and bastions (perhaps medieval, Islamic additions; an old mosque stands near the lake). Close to the top of the northern hill stands an ancient temple. There are more free-standing temples on the far (eastern) shore of the lake; but what most folks come to see are the 6th-7th century cave temples - these lie half-way up the southern hill.

The caves are reminiscent, stylistically, of those at Mahabalipuram (which were approximately excavated at the same time) and Ellora (which were done a little later). Many of the themes are also common; but the reddish brown of the sandstone that takes on a golden glow in the slanting late afternoon sunshine invests on the carvings here a very unique 'Badami flavor'. And the rock, at places, has striations in a lighter yellowish color - they have been used in the Jain temple (cave 4) to give an impression of of shafts of light radiating from the principal Tirthankara image. The 'Hindu' carvings which impressed me most are: a Nataraja image with an attending Ganapati figure mimicking the boss's pose; a placid 'Narasimha'; Vishnu as 'Trivikrama', the one who measured the whole world in three steps (a theme which one can see at Ellora also) - rendered somewhat unusual by the presence of a Buddha-like figure among the divinities watching the miracle; 'Vaikuntha Murti', Vishnu seated (not reclining) on the serpent Shesha who has arranged himself like a sofa for his master - by the way, this is almost the form of the Vishnu idol at Tripunithura temple in Kerala - and I have not seen such a form anywhere else.

In antiquity, Badami used to be called 'Vatapi' and apparently, it used to have a famous 'Ganapati' idol, celebrated in a Dikshitar 'kriti'. The Pallavas are said to have conquered the place and made off with this idol. I don't know its present whereabouts.

We came here sort of mentally prepared to be impressed by the art, but not for the interplay of light and shadow among the rocks. While working our way up a ravine that runs thru the northern hill, we see a towering rock face lit by the evening sunlight pouring in thru another gorge - the illuminated portion stands like a golden pillar among the darker rocks - a sight like none else I have seen before!

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Over dinner, we encounter an unusual traveler. He looks, racially, South Asian, is well-built, has matted locks and beard but is otherwise smartly turned out; he spikes his beer with what looks like coffee powder (a mix I am not adventurous enough to try!); and he is polite to a fault and very apologetic about intruding into our conversation with an "are you guys camping here long?" query.

The traveler says he has been to Pune ("or is it Poonah?" he asks with what sounds like an American accent) and has visited the Osho Ashram and found it 'strange'. When I say I work in Software he remarks: "at least you learned something that can help you earn a living; I did philosophy and it has not given me a job! At least in Canada, you don't get to do much with philosophy" (I resist the temptation to ask him more questions about his past; neither party asks the other's name).

He goes on to complain about Badami not having any internet cafes or proper money changers ("these guys, you know, charge ten rupees a dollar as their cut; and I have given away two thousand Indian rupees, you know, just like that!" he snaps his fingers).

Then he asks: "I shall be going further south. How are Belur, Halebid, Swana... what is that, I'm sorry, my pronunciation is no good!"; "they are impressive places, but if you ask me, Badami is better, this place really rocks, literally!" I say expansively, still under the spell of sandstone. "Oh yeah, the landscape!" he remarks and turns back to his drink. Not much more is said. He eventually rises, wishes us a "safe trip" with a polite bow and retires to his room with a fresh can of beer. We don't see him again.

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Daybreak. We drive around the town, looking for rock formations to snap, in their morning glory. Many of the rocks are bustling with life - sparrows, pigeons and parakeets dart around busily; their cries fill the crisp morning air. I also note happily that Badami does not seem to be under threat from quarrying; there also does not seem to be much competition among religions and political parties, which has led to silly slogans and fervent declarations 'uglifying' the rockfaces in so many other places.

As we leave, I regret not having planned for a longer stay here. Maybe one should come again, perhaps during the Monsoon when the lake is full and waterfalls are said to cascade down those cliffs into its limpid waters; or better still, during a full-moon in winter and go walkabout in moonlight - I am told there is a lonely trail winding past the lake and up among the boulders towards the sylvan precincts of the ancient Mahakuta temple...

Impressions - Pattadakal

Pattadakal is now a small hamlet on the banks of the Malaprabha. Unlike Aihole and Badami, which are both nearby, it is a 'world heritage site' and appears to be taken more seriously by the foreign tourists (hordes of school kids are also seen herded around the site by usually grumpy teacher-custodians).

Almost all the Chalukya temples here (about a dozen of them) stand in the same protected enclosure of a few acres; all appear to be Siva temples. As at Aihole, there are examples of both the northern (Nagara) and the southern style of temple design; the difference is that here, one begins to see the southern style gaining a clear ascendancy (the largest temples, Mallikarjuna, Virupaksha and Papanatha are all Dravidian) and some maturity. The sculpture here is of a higher order than elsewhere in this circuit.

Some of the details which caught the eye: in the three temples mentioned above, there are carvings of couples on many of the pillars and on the walls - they are not really erotic 'Mithunas' as at Khajuraho etc; they seem to be variations on the 'donor couples' which can be seen in Buddhist caves at Karla, Bedsa etc (these temples were done at least 600 years after the caves). Again, one senses a difference. The Buddhist couples are manifestly amorous and the female figures are buxom and voluptous. Here, things are generally subtler - the male and female figures usually stand with arms around each others' shoulders but where visible, their faces express affection and an understated love rather than open physical attraction; there is one couple which was particularly notable for the tenderness on the face of the man; and unfortunately, the woman has lost her head (and unfortunately, I didn't take a pic). Also among these is one strange specimen where the female of the pair has a tiger's (?) head and human body. Anyways, one can probably conclude that the Chalukyan art of Pattadakal provides a link between the Buddhist cave art and later temple art of Southern and central India.

Some of the other sculpture illustrate mythological episodes like the slaying of Hiranyakashipu and Ravana shaking Mount Kailas (both these themes seem to have been favored - and elaborately developed - by the artists of the later Ellora caves). Then there are unusual episodes like Bheeshma lying on the bed of arrows, Krishna slaying an elephant and so on, carved in smaller-scale friezes. One can also see a small-scale copy of the grand Mahabalipuram relief of the battle between Durga and Mahishasura.

There is also a Nataraja with a divine attendant figure playing an instrument very like a compactified 'Mizhavu' ( The Mizhavu is nowadays seen only in some Kerala temples. It is a large metallic pot with the diaphragm stretched over the mouth and is used primarily as accompaniment for Koodiyattam, the traditional enactment of Sanskrit plays). I recall seeing a (much later) Kerala style mural painting of Nataraja in Ettumanur where Vishnu(?) plays the Mizhavu.

Some of the designs, like a ceiling carving of Nagaraja, the serpent king - the lower half of his body is serpentine and forms a spiral around his head - are to be seen in Aihole as well and seem to be common Chalukyan motifs. The complex floral designs which decorate the niches on the walls of several of the temples here seem to have received the attention of experts as well.

Some questions suggest themselves: the Chalukyas seem to have worked liberally with both the northern and southern styles of temple building (and also a bit of the 'intermediate style', developed later in Orissa). Did the northern style actually come from the north and the southern from the south? In later times, how did these two styles get so clearly separated - hardly any northern-style temples in the South postdating these Chalukyan experiments and the southern style too seems not to have penetrated the north either (Thought: the Ellora Kailasnath temple (which was perhaps built a little after the Pattadakal temples) has a Southern feel about it).

Impressions - Aihole

Aihole is a village deep inside North Karnataka. It is home to perhaps a couple of thousand people; and it contains anything between a hundred and three hundred temples - depending on which guide book one reads. Most are ancient Chalukyan structural stone temples (600 to perhaps 1000 AD). A few of the prominent ones - Durga temple, which is said to have inspired the design of the Parliament building in Delhi, Lad Khan temple, ... - stand amidst lush lawns in a protected enclosure, to enter which Indians pay 10 rupees and the others (hordes of firangees come here) ten times that amount; a neat little museum stands nearby. The rest of the monuments are 'free for all'.

At a quick glance, much of the sculpture on the temple walls is not quite top-class; some look derived from the Pallava carvings at Mahabalipuram and the Kailsanatha temple at Kanchi, which were done at nearly the same time (the Pallava kings who patronized those sites and the Chalukya's of Aihole were rivals); yes, there are a couple of examples - Siva standing next to his bull and Durga slaying Mahishasura - which seem to be original designs - and superbly executed too. Anyways, the real importance of the place,as the book says, is in the range of architecture on display rather than sculptural decoration; one can look at specimens of both the northern Nagara style temple design and the southern Dravida style in close proximity - there are a few temples which vaguely resemble the Orissa style as well.

A straggling extension of the village snakes up a gentle hill - near the base is a cave temple called Ravana Phadi with a 10 armed, larger than life Nataraja image and on top is the plain Jain Meguti temple. From atop the hill one can take in the full sweep of the fields and scrubland stretching way beyond the village in all directions.

Somewhere I see, written wistfully, many of these temples had been converted into houses and cattle-sheds. The implication: the present day denizens of Aihole have no appreciation for their architectural heritage. The villagers do seem seem very religious - the majority have holy ash smeared across their foreheads. We also saw some modern places of worship built of brick and concrete and interiors lined with bathroom tiles (more Maharashtran than Dravidian).

Life here seems hard. Water appears to be a concern although the river Malaprabha is not far and definitely, the lawns around the main temples get plenty. Bullock-carts are the principal cargo-carriers and most roads connecting the villages, except those on the main 'tourist trail' have long since crumbled into stone-strewn mudpaths, threatened by the scrub.

As we stop the car in a space between the closely built concrete dwellings below the Meguti temple , a gaggle of 4-6 year old kids come running. One little boy keeps touching me, pointing a finger at the sky: "Is he asking for money?" I wonder; perhaps I am mistaken. No word is spoken and the kids soon run off towards a shed-like structure from where they seem to have come - it has a facade supported by ancient-looking carved pillars and the rest of it, though of stone, is plain and boxy and the interior is quite gloomy. And then one catches sight of a woman standing there with a short stick, apparently calling them back - so, this is the local school!

Back after visiting the hill temple, I feel tempted to take a snap of the school. "Hey photo, photo!" the kids scream. The teacher seems to be admonishing them to be quiet and to stand steady. The flash goes. We start up the car and they all - including the teacher - wave us goodbye and one senses a slight tightening in the throat.