Fibonacci’s flowers

I love the sight of flowering cosmo. You find them growing in gardens, but they often escape and grow wild. As you can see, these are typically eight-petaled. On the other hand, all Himalayan wild flowers that I photographed on a trip a couple of months ago turned out to have five petals. Eight is the number that follows five in the Fibonacci sequence of numbers: 1, 1, 2, 3, 5, 8, 13, 21, and so on. Each number after the first two is obtained by adding up the two previous numbers. Works on aesthetics are full of the mystical properties of these numbers, and the relation they bear to the Golden Ratio, which is the ratio (1+√5):2. If you are interested, I can point you to one or two.

Is the number of petal in a flower always a Fibonacci sequence? Of course not. I’ve not seen any claims that four or six petaled flowers are less beautiful than five or eight petaled varieties. So it is surprising to find web sites on popular mathematics which make the unprovable claim that most flowers have a Fibonacci’s number of petals. This is a hollow claim because we don’t know what “most” means: is it 90 out of hundred or 51 out of a hundred. Should we count the number of flowers, or the number of species of plants? “Most” is a weasel word. Even so there are some impressive attempts to debunk this claim.

The most impressive amongst the scant evidence for Fibonacci’s flowers is the sunflower, which has 21 petals. There is a missing number, as you may have noticed: the even more mystical 13. Looking through the collection of flowers which I photographed, I can offer the example of a thirteen petaled gazania in the photo below.

I don’t have the legendary patience of a botanist, so I have never managed to count petals up to the next number in Fibonnaci’s sequence, which would be 34. But it seems that I don’t need to play along with this myth. All the photos that you see here are of compound flowers. Each one of the structures that we think of as petals is a separate flower and the cluster of rods at the center are also separate flowers. The central ones are called ray flowers and the ones we think of as petals are called disk flowers. You can easily look at the gazania in the last photo more closely and see that the ray flowers have five petals each. The cosmo also seems to have five petaled ray flowers. The disk flowers are also five petaled, but you have to look at the underside of the flower to see how they have fused together.

Contrary to mystics, botanists find that the number of petals is three, four, five or six. Some counts say that 70% of all flowering plants have five petals. I don’t know how precise this census is. Such percentages depends again on what you count: flowers, species or genus. But one thing is certain. Fibonacci numbers and the golden ratio may be nice stories, which have little to do with the budding of a flower.

Now, there is an escape route for the Fibonacci fans. Perhaps the number of disk flowers in a compound flower is a Fibonacci number. After all that is the number of “petals” most of us count. It seems that this story also fails. In the photo above I show you a spectacular water lily. If you count the petals, you will find a number between 21 and 34: definitely not a Fibonacci number. In fact, if you count the petals of the gazania in the photo I showed, maybe you can catch me out on a fib.


Closing time

I had dinner in a small town in Odisha. The restaurant was very full when we walked in, but emptied out before we finished. One of my colleagues had to catch a train later. So we took a short walk through the town. It was about nine, and most shops were shutting down. The wide street was lit by dim lights placed along the divider. This meant that most of the illumination came from shops, and once they started closing, the street began to get much darker. It was enchanting in its own way. We saw a line of bright yellow doors separated by blue walls; little kiosks which had closed.

Further along the row one kiosk was open for business. The vegetables on display were extremely fresh. A long cold chain is not needed to bring these to the town. The chopping boards are big and solid pieces of wood; seemingly cross sections of the trunk of a small tree. Banana flowers and jackfruit are two of the things that I would not normally find in a shop in Mumbai. There were also some long beans which I’d not seen before.

Next to the fresh vegetable stall was this tiny “supermarket”. Any grocery store with open access to the merchandise now calls itself a supermarket no matter how small it is. This was an interesting contrast to the vegetables. While that only stocked food which was fresh and had to be cooked, this had nothing which was fresh. Everything came inside a plastic package and was ready to eat. This was also much costlier per helping than fresh food, but the very price makes them an aspirational thing in these small towns.

Nearby a roadside eatery had served its last customers and was busy shutting down. The helper was carting the last chairs from the pavement into the shop. The counter was definitely closed. As I took this photo, I heard the sound of running water from the kitchen. I suppose the cookware and plates were being cleaned. There was much animated consersation between the cook and the helper, as I quietly took this photo and moved off.

I don’t know what this interesting looking stall was for. The man you see in the photo looked at us curiously as I took the photo. He spoke only Odiya, and none of us counted that as a language we could express ourselves in. My best guess is that this stall serves ready-made food of some kind. Perhaps things that supplement the food made in the home kitchen, but maybe largely aimed at labourers and other immigrants who have been drawn to this region by the large amount of construction under way. If this were so I think the stall would also have some of the wonderful local sweets.

This was the second of the odd juxtapositions. Right next to that temporary stall was this more permanent structure with ice cream and cakes. Even as the town wound up for the night there were customers here. A savvy sweet shop does not miss a trick. There was also a counter with the local Odiya sweets; that’s what the lady in the sari is looking at. The cold drinks and the toffees in bottles in the front counter would have attracted children a couple of hours before I took the photo. I liked the expectant look of this shop.

This junction of two major local roads was more typical of the time in this town. Once the lights went off, the town was generally dark. A couple of open shops provide the only light in the place. Everyone hasn’t gone to sleep yet. Parked motorbikes show that there are some people who still come to the shops at this hour. After all a major railway station is less than ten minutes away. We went there to drop my colleague, and that place was brightly lit and bustling with traffic.


The monsoon changes character in July. The storms of June with their constant whistling winds, and the occasional thunder and lightning have passed over the land. A curtain of heavy rain clouds follow. On Saturday I dared to go out several times to finish various chores and got drenched each time. I was cooped up in my flat all day on Sunday. On Monday I took a change of clothes to work. It is still raining today. July’s rain is a simple fact of nature: if there is open sky above you, there will be hard rain coming down.

The paddy field is flooded with the fresh water. Frogs are beginning to croak, and on the banks the grass grows whiter than snails’ eggs.
-Yogeswara (poem collected in the 12th century CE in Subhasita-ratnakosa by Vidyakara, translated from Sanskrit by Kosambi and Gokhale)

I wonder whether the experience of people living in this land a millennium ago was very different from mine. Even in this city with a population larger than some countries, I look out of the window and see the rain water pooling under a banyan tree, flooding the garden. Yogeswara would recognize this. The parking lot below my apartment will be full of earthworms which have crawled out of the slush. The roads will be taken over by snails. We were woken in the morning by mynas which found shelter on window ledges. Crows are too big to fit there. Wet large-beaked crows sit in long lines along parapets, descending to flooded lawns to check out the tasty treats which float up in the water.

Most trees have shed their flowers. The flaming red treetops outside our windows have been washed into a clean glistening green by the days of rain. A few hardy flowers will hang on till the next heavy rain. In our balcony there are a few flowers still. The Madagascar periwinkle in our balcony will remain in bloom through the monsoon, presenting such a pretty picture that I take photos of the blooms at least once in this season.

A midmorning snack

We didn’t manage to get much birding done on our aborted walk from Gushaini to the gates of the Great Himalayan National Park at Ropa. It was a bit too late in the morning, and I, for one, was too busy panting during the steep uphill sections to do much looking. So we climbed back down, on a “shortcut” which locals take. This is essentially just short of rolling downhill, until we came to a bridge under construction across the Tirthan river. We clambered across this, and climbed up to a motorable road on the other side. Why? Sanjay, our guide for the day, said that we could possibly see some birds about a kilometer higher up. By the time we decided that it was too late for birdwatching, it was midmorning.

Sanjay said there was a tea shop nearby. I didn’t mind some tea, so we walked down there. The pleasant young person running the shop (featured photo) was happy to make us some. In one corner of his shop was a kadhai full of oil and another full of sugar syrup. Sanjay took a look at these and decided that he wanted jalebis. The mix was ready, but the shop owner did not know how to make them. “Another person comes here and makes them in the morning,” he said. Sanjay decided that he was an expert. We sipped our tea while the stove was lit, and the oil warmed up. Experimental jalebis were made. The Young Niece started laughing when she saw the plate (photo above). They looked nothing like jalebis, but I notice that she ate them all right. They were crisp and sweet and tasted like jalebis.

Cranes of Mumbai

The two-day long monsoon shower ended around the middle of Sunday. I took this as an opportunity to walk through the back streets around the stock market. Even on a Sunday there’s usually something interesting going on here. I walked past the very busy street vendors and looked up at the tower of the stock exchange. I’d not noticed before how many data cables cross at all angles above the street. It looks like a safety net against the eventuality of a stock market crash. Poking up through this street-wide-web was a crane.

One of these old buildings had been pulled down and a new and unexciting concrete box was coming up in its place. The crane was parked right in front of it. The young man operating it looked really relaxed, feet up on his seat, moving the crane with a delicate touch of his left hand. I’ve seen people play arcade games with the same nonchalant elan. The cabin door was open and I thought that bright blue splash against the yellow cabin made a good picture.

This was a duplex crane! In the rear cabin someone was asleep, head resting on the disabled steering. I guess the operators were internal immigrants. Their work place is where they live most of the time. Usually workers who immigrate to the city share a rented room where they sleep in shifts. This man was either too tired to go back, or had decided that it was more efficient to sleep at work. I hope he is properly rested by the time his shift starts. A little inattention while manipulating the crane probably would not damage the neighbouring buildings, but could play havoc with the overhead cables.

That’s something new

When I saw this motorcyclist on the highway from a distance I thought he was carrying balloons. I tried to get shots as we closed in on him. The Young Niece craned her neck around Soni to look, and she was the first to say “They aren’t balloons, they are balls.” Indeed they are, and also buckets, wastebaskets, laundry hampers and other plastic things which you might need in the house. The Family was also involved in this little story by now. “Is he a door-to-door salesman?” she asked. We overtook him as he puttered on behind trucks.

Later, on this highway cutting through Punjab we passed this other man, motorbike loaded with brooms and woven cane baskets. The Young Niece was dozing, and The Family glanced out of the window without making a comment. I thought he was not less interesting than the previous guy. The sights nagged at me until I remembered a sight I’d seen next to a highway in Assam, at the other end of the country. That’s the photograph below: a motorbike laden with packets of food making deliveries to roadside stalls.

All these motorbikes are delivery men, just like the Amazon delivery men on their motorbikes in cities. As rural India becomes more prosperous, a new delivery chain is growing to satisfy the needs of small shops. While big companies and their ancillary services target the prosperous middle class in the compact markets of cities, the far-flung countryside of India is also going through a consumer revolution. Their demands are too small for the giant trucks which ferry goods to cities on the same highways. These motorbikes are part of this new story.


The distance between Jalori pass and Gushaini is not large, but the roads are narrow and hug the mountains above the winding courses of the Tirthan and Banjar rivers. We drove quickly through the town of Banjar. We had a glimpse of shops fronting narrow roads. A few turns, and the road had left the town behind. We wanted to stop for chai, and Soni decided that Shoja is the best place for a morning’s cuppa.

Shoja turned out to have all the charm that Gushaini and Banjar don’t. The Young Niece oohed and aahed about the view, so I walked with her a hundred meters along the road to the end of the town and took the featured photo. We’d been seeing these terraced fields near every village, but this was the nicest view of that we’d had. The clumps of chir pine (Pinus roxburghi) and banj oak (Quercus leucotrichophora) salted through the fields, clouds descending from the mountain tops, and the beautiful light were something to enjoy in silence.

When we walked back our chai was ready. I gulped down my glassful and wandered up the road to take a few photos of the town. The tailor was already at work at his pedal operated sewing machine. The booth behind him must be the trial room. I was surprised that the village is large enough to support a tailor full time. I guess its main earnings are from farms and orchards. Tourism may bring in a little money, since there are possibly some home-stays and a hotel in this village. It looked very clean and more prosperous than Gushaini, but that may just be because it hasn’t grown haphazardly.

By now everyone had finished their tea, and we all walked back down the road to take another look at the farms around the village. I got another shot of the slopes and the farmhouses nestled in the fields. There are several interesting small walks around this village, but I had one planned at a higher altitude. We piled back into the car and drove on to Jalori pass.

A nameless village

Dilsher gave us driving directions. The route took us to Gushaini, where we crossed the Tirthan and took a road which wound high above the Falachan river. About a kilometer from the village which was our destination we found that the road was under repair. We parked the car and walked over the broken stones in the road bed. With the directions, and GPS, I know exactly where the village is; I can find it on maps and satellite photos; but I cannot find a name for it. It was definitely worth a trip, just to see what traditional houses and village layouts are in this area.

As you can see from the photo above (and detail in the featured photo), some of the houses are elaborate. This one had thick mud walls at the lowest level, and steep stairs to climb up two levels to the living space. The lowest level holds livestock, the next is space for feed, and the topmost level is for people. Wood is used extensively only in the area meant for people. I’d seen this kind of overhanging wooden box also in the divisional town of Banjar, where the lower floors were given over to shops. Now I saw where this construction comes from.

Not all houses are equally elaborate. You need space for livestock only if you own some; so the big house belonged to someone who was rich in this village economy. Other houses were smaller wattle-and-daub constructions, as you can see in the photo above. Mats are fixed on to wooden frames, and then covered with clay and painted. Interestingly, the house has two stories, and a balcony running across the front of the upper story. The two nearer structures are sheds to hold hay. Since this was lower, I could easily see the admirable slate roof. I was also quite impressed by the solar panel on top of the pole. Is the need for electricity small enough that a panel like this suffices?

I saw no signs of air conditioning; in this place you don’t really need it. Heating for winter would be an issue, but it wouldn’t be electrical heating. I didn’t see refrigerators, but satellite TV had arrived, as you can see from the dish in the photo above. All in all, the amount of electrical power needed per household would be small. I liked these two houses in completely different styles standing next to each other. The one with the upper floor box is unpainted; the one with the balcony is painted in bright colours. The man who you see here was friendly and curious about us. He gave us a little tour of his neighbourhood, trading question for question.

We did not go inside anyone’s house, so I never found out how the rooms are organized: is it one large living and bedroom? Or are these internal divisions separating kitchen, eating and sleeping spaces? The larger organization was clear. The center of the village was a large open square with an enigmatic temple which I have written about earlier. The village is laid out in a series of linked squares with houses around the open centers. The largest houses stand on the square with the temple; the further you go, the smaller the houses become. In a square adjoining the one with the temple I found a well and two children playing. In this photo they lean over the well.

You can see that the social organization is changing. There are new and fairly large houses at the periphery of the village; they climb up the surrounding slopes. The photo above shows one such house. The house itself looks different: the lower floor has windows, which means it is not used for cattle or feed. The roof is made of corrugated metal sheets, and the wood is painted. But perhaps the most striking non-traditional addition is the brick and mortar outhouse with a plastic tank full of water resting on its cast concrete slab of roof. This seemed like it was part of the government’s worthy push to add toilets to every house. The toilet blocks look identical everywhere in this country; they have been designed in Delhi. Fortunately availability of water is still not an issue here; I saw taps outside many of the houses.

The newer houses often use non-traditional materials. This one stood close to the entrance to the village. Again, I saw windows on the lower floor, indicating that all floors are used by people. This house was elaborate: it had balconies and a box just below the roof. I was struck by the use of corrugated metal sheets for the sides. They can’t possibly bear the load of the upper floors, so there must be another structural part of the wall. Is that a wooden frame or a thick mud wall? My reluctance to knock on the door of a stranger meant that I never found out. Of the many things which I did not know about this village, this was perhaps the least. The main thing that I should have asked the curious guide we picked up was the name of the village.

Break of journey

After a long drive from Chandigarh to Delhi, The Family and The Young Niece flew back to Mumbai. I had to take a flight to work. After reaching Lucknow, I had a three hour drive to Kanpur. A break for tea was very welcome. The fact that the tea came in a kulhar was a bonus. The hot tea released the aroma of damp earth, a memory of rain, of deserted railway stations at night.

Luggage lift

When you are walking on a mountain path you do not expect cylinders of cooking gas or other kitchen essentials to go sailing over your head (featured photo). But that is exactly what happens in Falachan valley. The whole valley is criss-crossed by overhead wires. I initially thought that there were a huge number of power lines here, but realized soon that most of the cables are luggage lifts.

Sitting a few hundred meters above the river at the turning point of a climb I saw pine cones around me. Once I noticed the cables they were resting on I hesitated to pick them up; the cables could be live, and this could be a fire hazard, I thought. Later it struck me that the most likely source of these cables were the luggage lifts. Usually cable faults of this kind are attended to reasonably quickly (which could be a day up at these heights).

Walking along the road we came across a family back from the market busy sending their stuff up to their home. I liked that loading station: at my head height, off the road. The cylinder of gas was already loaded into the cage which had come down from the village. Something must have been sent up already, and this cage was the counterweight. It was loaded with jerry cans of water. As we watched the young man poured the water down into the trees. The Family gasped. “Do they waste so much water?” she asked. Indeed, in many towns in hills water is scarce. But we saw lots of springs and glacier fed streams up here. Little villages are probably not short of water. Yet. The empty jerry cans go back up in the cage, along with heavy goods filling the rest of the cage.

We watched the two men load up the cage. They made sure that things were properly placed and would not fall off. Then this young lady sent a message on her phone. Soon the cage was winched up. We could see the counterweight descending. “Is there a road all the way to your village?” The Family asked the girl. “Yes, it is about half an hour’s walk away,” she replied. Then she added, “Maybe two hours for you.”

We weren’t the only spectators. An old man with a load on his back stood with us watching this family. These lifts are an innovation. Although this valley was dense with them, I didn’t see them much elsewhere. I guess the locals have figured out a way to string the lines between mountains, and that technique will take time to diffuse into the neighbouring valleys. It took me some time to puzzle through these thoughts. By the time I realized that there was something special about the Falachan valley, it was too late to ask someone how they string cables between hills.