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).

Exclaves, Enclaves, and Doughnuts

In which we explore the strange paths taken by national borders around the worldAlso, doughnuts. Mmmm, doughnuts…

Most national borders that exist today are the result of many decades (or even centuries) of redrawing, and these redrawings were often entangled with the wars, treaties, or imperial ambitions of the time. Sometimes, the border is the result of a very clear principle—for example, the US-Canada border runs neatly along the 49th parallel for much of its length (though this article shows how it’s not quite that simple). Other times, it’s the result of a long an torturous process—compare this map of the Holy Roman Empire in 1786 with the simpler subdivisions of the modern state of Germany. [Side note: all maps from this point onward are taken from Google Maps.]

In this post, I’ll share some examples of unusual national borders—just to get the lay of the land. In the next post, I’ll examine some of the most complicated examples, and begin to apply a mathematical concept (the fundamental group) that, hopefully, will clarify the messy business of national borders.

Before going any further, though, some terminology is in order.

Enclave: an enclave is any portion of a state/province/country that is completely surrounded by another state/province/country. One good example of an enclave is the entire nation of Lesotho, which is “enclaved” within South Africa.

Exclave: an exclave is a portion of a state/province/country that is separated from the main part by multiple states/provinces/countries. A good example of this is the Kentucky Bend, a portion of the US state of Kentucky that is surrounded by Missouri and Tennessee.

An interesting side note: the western borders of Kentucky and Tennessee are defined by the Mississippi River, following the course it ran when originally the border was originally defined. Over the years, most rivers will change their course in multiple places, which means that many states have small chunks that lie on the opposite side of the Mississippi River (go explore the above map to see what I mean). These are called pene-exclaves, since the border doesn’t separate them from their state, but rather a geographic feature (in this case, the river).

Warning! These definitions are not mutually exclusive. Some, but not all, exclaves are also enclaves. To avoid confusion, I’ll just say exclave to mean one portion of a country that’s separated from the main part, and enclave to mean a country that’s completely surrounded by a single other country.

Another Warning! These definitions depend on which country you’re referring to. One example of this is the Spanish town of Llívia (see map below); it is enclaved within France, but it is an exclave of Spain.

Let’s see what else is out there! We continue our tour through Europe:

Enclaved Countries. There are a number of microstates in Europe, but only two of them are true enclaves: San Marino and Vatican City, both surrounded by the Italian Republic. These two, along with Lesotho, give us all of the world’s enclaved countries.

Alpine Villages. The Alps are host to two interesting exclaves, both surrounded by Switzerland. One is the Campione d’Italia; it’s surrounded by the southern Swiss canton of Ticino. The other is Büsingen am Hochrhein, a German town surrounded by Schaffhausen canton.


The Politics of Railroads. The Belgium/Germany border is host to a strange series of exclaves.

As you can see from the map, there are five German exclaves surrounded by Belgium (one of which is just a house & its yard), which are just barely separated from their homeland. There are two roads and a rail line running to the east of these exclaves, which are owned by Belgium. Apparently, the entire thread of territory was once a rail line (the Vennbahn) that Germany ceded to Belgium as part of the Treaty of Versailles. The roads intersect with German roads and highways, but are still Belgian. It’s really strange—go poke around the map.

Doughnuts. Lest we spend all of this post in Europe, the Arabian peninsula is host to a fascinating little enclave/exclave situation. The village of Nahwa strides the border of Oman and the United Arab Emirates (UAE), which itself isn’t all that unusual in itself. However, the UAE portion is part of an enclave, surrounded by Omani territory which is itself surrounded by UAE territory, thus creating a doughnut-shaped chunk of Oman inside of the UAE.

The innermost UAE territory is an example of a second-order exclave: an exclave within an exclave. This might seem to be the height of absurdity when it comes to national borders, but it turns out we’re only getting started. In my next post, we will hit the accelerator and see how complicated things can really get—including a look at the world’s only third-order exclave.

See ya next time!