On the Fundamental Groups of Nations (Part I)

In which we apply mathematical topology techniques to national borders.

Welcome back! In my last post, we toured the globe to find some quirky exclaves and enclaves, including a doughnut-shaped enclave on the Arabian peninsula. I darkly hinted that these examples were only the tip of the iceberg. Now it’s time to see exactly how complicated the borders can get!

Before getting into any of the math, let’s look at the most infamous examples. First, we have the Baarle-Hertog enclaves, exclaves of Belgium that are enclaved within the Netherlands. They’re a mess.

However, this mess doesn’t hold a candle to the haphazard jumble of exclaves that occur along the India-Bangladesh border:

Perhaps unsurprisingly, there is a rather convoluted history to go along with the convoluted border. The Economist had a nice article on the matter in 2011: apparently, the confusion dates back three centuries, though how exactly it got started is up for debate. The best part of this: there’s a piece of India inside of a piece of Bangladesh, inside of a piece of India, which is itself inside of Bangladesh! This is a double-doughnut hole, and it happens at Dahala Khagrabari. Check it out!

Now, my main interest is the notion of path-connectedness. Here’s a mathematical definition (paraphrased from Kosniowski):

A space X is path-connected if, given two points a and b in X, there is a continuous mapping f from the interval [0, 1] into X for which f(0) = a and f(1) = b.

And now a more colloquial definition, with geography in mind:

A region is path-connected if it’s always possible to travel between any two points in the region without traveling through foreign territory.

This gives us a way to assess whether or not an object has been broken into pieces—in geographical terms, all the exclaves of a country are just the path-connected chunks of that country. (Note that according to this definition, even the “main” part of the country counts as an exclave.)

To get a handle on enclaves, we need a mathematical way to represent the holes within a country (or more specifically, the allowable path-types in each exclave). There’s a mathematical object that can do this, called the fundamental group. With apologies to the topologists, I will be using drastically simplified language for it. (For the curious: read more about it here.)

For our purposes, the fundamental group of a region can be written as a list of non-negative integers; the number in position n (starting at n=0) tells you how many exclaves have n holes in them. For example, a country made up of two exclaves, one having no holes and the other having 2 holes, would have fundamental group (1, 0, 1, 0, 0, …). We know that, eventually, we’ll run through all the exclaves, so this list can be truncated to (1, 0, 1). Notice that if you add up all the numbers in the list, you get the total number of exclaves.

So, this mathematical analysis can be summed up nicely as follows:

  • Find the number of exclaves (including the main body of the country).
  • Classify each of the exclaves according to the number of enclaves that they surround.
  • Write the country’s fundamental group as a list, where the nth number in the list (beginning with n=0) gives the number of exclaves with n holes in them.

Here are a couple of simple examples.

South Africa is a single piece, and surrounds one enclave (the nation of Lesotho).

Its fundamental group can be written as (0, 1). Incidentally, since Lesotho is made up of a single piece that surrounds no enclaves, its fundamental group is (1). Clearly, most countries will have a fundamental group of (1).

The United Arab Emirates is broken into two pieces: the main body of the country, and the second-order enclave contained inside the doughnut-shaped piece of Oman.

Since that doughnut-shaped piece of Oman is enclaved within the UAE, we end up with a fundamental group of (1, 1). Oman itself has a small exclave on the Persian Gulf, in addition to the doughnut-shaped piece and the main part of the country, so its fundamental group is (2, 1).

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