You may have heard that famous phrase “The sun never sets on the British Empire.” While the statement is still true (as Randall Munroe explains in his excellent What If? blog), it originally referred to the Spanish Empire of the 16th and 17th centuries. In one of his theological tracts, *An Advertisement Touching a Holy War*, the estimable Francis Bacon (everybody loves Bacon, amiright?) says

“…both the East and the West Indies being met in the crown of Spain, it is come to pass, that, as one saith in a brave kind of expression, the sun never sets in the Spanish dominions, but ever shines upon one part or other of them…”

Regardless of which country the phrase originally referred to, there’s an interesting geography question here—how would you know if the sun never *did *set on your empire? It turns out that high school geometry is enough to solve the problem, at least approximately.

Let’s lay down some basic observations first:

*Your empire would need to have nonzero land area*: Let’s ignore undersea kingdoms (sorry, but Doggerland doesn’t count).*Your empire would need to have at least 3 territories*: Theoretically, if your empire consisted of two antipodal points, the sun would set in one territory at the same time that it rises in the other. But that’s boring.*The Earth is (nearly) a sphere*: For the purposes of this post, it*will*be a sphere. Otherwise, the math gets really messy).*On the Equinox**, it doesn’t matter what your territories’ latitudes are*: If the sun’s shining on 40° N, 88° W, then it’s also shining on 0° N, 88° W and 89° N, 88° W. And if your points don’t lie beyond the Arctic or Antarctic Circles, then you’re safe for most of the year.

Given these restrictions, we can make a key simplification: all three territories lie on the Equator, and consist of single points. Now, we can examine the problem two-dimensionally—imagine you’re looking down on the earth from above the North Pole, and that your three points are distributed around the Equator. It’d look something like this:

If we view the sun as being infinitely distant (apologies to Calvin’s dad), the sunlit side will be a semicircle that rotates around the disk of our 2-dimensional Earth. Looking at the picture above, it should be clear that the only way for your empire to be completely dark is for there to be a territory-free arc of more than 180°. Now here’s a cool fact: *your empire will be eternally-sunlit exactly when the circle’s center lies within that red triangle.* (Think about it: can you place a sunlit semicircle on the edge if the center isn’t inside the triangle?)

It turns out this fact relies on a 2,600-year-old geometry fact called *Thales’ theorem*. Here it is:

Thales’ Theorem: Given three points A, B and C on a circle with center O, if the line AC passes through O then ∠ABC is a right angle.

Legend has it that Thales celebrated the discovery of this theorem by sacrificing an ox (good times!). A generalization of the theorem appeared in Eulcid’s *Elements* as Proposition 33 of Book III. Now, here’s an easy generalization of Thales’ theorem and Proposition 33:

Theorem:Given three points A, B, C on a circle with center O, the triangle ABC contains O exactly when ∠AOC is less than 180°.

It gets even easier if we notice that the diagram above makes some unstated assumptions about the points: specifically, A, B, and C are given in order of increasing longitude, and the International Date Line lies on the arc AC. So, using some common sense, we get the following simple rule for our empire:

A nation is perpetually-sunlit if, and only if, it has three territories A, B, C (listed in increasing longitude) for which the longitude of C minus the longitude of A is greater than 180°.

Based on this formula, only two nations are perpetually-sunlit today: the United Kingdom (take Pitcairn Island, London, and Diego Garcia) and France (Martinique, Paris, and New Caledonia). The Dutch qualified until they gave up Netherlands New Guinea in 1962.

So, how could you go about building an eternally-sunlit empire of your own? Unfortunately, you don’t have many options—most of the world’s land has been claimed already. But there are a few scraps of territory you could snap up:

- The Bir Tawil (21.871° N, 33.737° E): A trapezoid of sand in the Sahara, the Bir Tawil is on the Egypt-Sudan border but is claimed by neither country. Go plant a flag there, and it’s yours.
- Russkaya Station (74.766° S, 136.803° W): This Antarctic station, closed in 1990, is located in Marie Byrd Land, which is not claimed by any country. You could probably buy it for the right price, and, if you felt like it, use it to film your own fanfic episodes of the X-Files. (And yes, there are certainly times of the year when it’d be in darkness, but beggars can’t be choosers.)
- After that, you need only rustle up some sovereign territory east of 33.73° E and west of 136.87° W. I’d suggest looking at Private Islands Magazine (my local library doesn’t carry it, I have no idea why) for ideas. Of course, you’ll still have to lead a revolt against the country that your new island is situated in.

And stay away from Yadua Island—I call dibs.