# As many holes as you want

As many holes as you want

My morning headlines told me, a few days ago, that there were 6,000 potholes on Bombay’s roads. Adding flavour to this nugget was that a Ganesh mandal chief announced, after surveying 25 roads on which Ganesh idols would travel, that he found 2,500 potholes.

Now like most of you, I’m persuaded that Bombay is a pothole-ridden city, especially when the rains come. Ours not to reason why about that, at least in this space. But this is the space to consider numbers like 6,000 and 2,500. There are probably several thousand roads in Bombay. If all of them together have 6,000, how is it that 25 roads alone sport 2,500 potholes?

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Please don’t assume that potholes tend to form where Ganesh processions go. There is a more plausible explanation, and it goes like this: if you’re counting potholes, you can choose any number you want and it will be correct. One day last year Bangalore reported 33,772; one day last month Sonoma in the US reported 81,175.

I swear I am not making those numbers up. This is why I believe that of the city’s count of 6,000, my own road has 405,883 potholes.

Bear with me, I’ll explain.

If you’re counting potholes, the first thing you must establish is what is a pothole. Look at Bombay’s streets from a satellite—Google Maps, for example—and you’ll see no potholes. From a double-decker bus, you’ll see a few. From a car, more. Walking, even more. All because the closer you get to them, the more you see and count. What’s your total, then? That depends: do you count only holes a metre or more across? (What you might see from the double-decker). Half a metre? A couple of centimetres? Even smaller?

I haven’t surveyed my street, but given time and an inordinate amount of money for my pains, I have no doubt I’ll find 405,883 potholes. Nose to the ground, I’ll count every little depression in the surface, every gap between stones, as also larger cavities.

And considered that way, there’s a curious parallel between potholes and coastlines.

Seen from a satellite, India’s coast is about 7,500km long. You can approximate its shape using a few straight lines to form a rough “V".

Get a little closer and straight lines are no longer enough: you start noticing gulfs (Kutch) and estuaries (Sundarbans) and bays (Marine Drive), each of which add to the length. Close in more to see even more features: maybe an outcrop of rocks that jut into the sea, or a man-made jetty.

I mean, when you’re measuring the length of Marine Drive, do you treat it as one smooth curve? Or do you account for the greater length of the jagged form made by hundreds of clunky tetrapods that actually meet the water?

And there’s ever more detail as you get ever closer: go from tetrapods to individual stones, then grains of sand and you are still not finished because you can whip out your trusty microscope—you carry a microscope, don’t you?—and measure individual molecules and atoms and electrons…and the more you do this, the larger your estimate of the length of the coastline becomes.

India’s coast is at least 7,500km long, but it’s really as long as you want it to be. In effect, it’s infinitely long. I mean it.

Now I think that’s kind of cool. But it gets better. Our coast is infinitely long, but it forms the boundary (or part of the boundary) of a decidedly finite expanse of land, the country of India. To understand this, imagine you have a square tablecloth 5,000km on each side. (Large table, what can I say.) Lay it on the country, centring it somewhere near Bhopal. Your tablecloth will certainly cover all of India, meaning India’s area is less than the tablecloth’s area. Therefore, finite.

What we’re dealing with here is the phenomenon of fractals, a term coined by mathematician Benoît Mandelbrot in 1975. Again, this is not the place to explore them. I’ll say this: as Mandelbrot described them, fractals are remarkably beautiful mathematical creations. When I studied computer science in the 1980s, we students would vie with each other to produce ever more complex, ever more gorgeous fractals. And yet for me, their real beauty lay in the often simple equations that produce them.

I’ll also say this: it never occurred to me that I’d be thinking about fractals again all these years later, and that thinking prompted by potholes in my city. (Not so beautiful.) But that’s what a claim of 6,000 potholes will do.

And I’ll leave you with this thought. It’s true: depending on how you count them, there are just as many potholes in Bombay as you want there to be. In effect, our roads have an infinity of potholes.

I told you this was a pothole-ridden city. So’s Sonoma.

Once a computer scientist, Dilip D’Souza now lives in Mumbai and writes for his dinners. A Matter of Numbers will explore the joy of mathematics, with occasional forays into other sciences. Comments are welcome at dilip@livemint.com

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