# Four equidistant points on Earth

Are there four points on Earth such that the distance between any pair of them is always the same? Distance is measured as shortest distance on Earth "as the crow flies" so without requiring any roads or similar.

Note that this looks very similar to this question and was inspired by it but the solution is very different, hence a new puzzle.

• Atomic bonds solved this puzzle billions of years ago Commented Jan 10, 2023 at 18:07

If we are allowed to approximate the earth as a perfect sphere:

Yes, the vertices of an inscribed regular tetrahedron are all equidistant, whether you go direct (through the earth), or take a great circle path over the surface.

• I do wonder if it's possible to find four such points on land... Commented Jan 10, 2023 at 15:20
• @MichaelLugo the maths might be harder but I suspect you're more likely to find a positive outcome if you pick your first point in one of Hawaii, Midway, Wake Island. I'm failing to do this without a physical globe, but I suspect some pacific islands will allow you to pick the other three points as a near vertical line running close to South Africa. You might also find 3 points in Canada, Columbia and the South Pole with a non vertical line, which I think should put the final point in Russia maybe? Commented Jan 10, 2023 at 23:59
• See this Quora answer: qr.ae/prsjFe Commented Jan 11, 2023 at 1:02
• @EricDuminil If we consider the earth with all mountains then this surface doesn't have any symmetry anymore so the edge length of the tetrahedron is not an invariant of the surface but depends on how/ where you put it on the surface. But I'm not really sure whether this gives you one more degree of freedom or not. I think it does but this starts becoming a nasty math problem :-) Commented Jan 11, 2023 at 21:17
• In Greg Egan's novel Earth (1990), the protagonists have to find four points on Earth that are distributed in this way. They end up using spots in Greenland, New Guinea, South Africa, and Easter Island. Commented Jan 11, 2023 at 21:39

Yes.

Imagine three points A, B, C ...as a small equilateral triangle centered around a north pole. Now stretch the triangle larger and larger, keeping its center at north pole. It's no longer triangle but a set of three great-circle distance curves. That's OK. Keep stretching up to the point where distance A-B equals to distance A - south pole. Then the fourth point (D) is the south pole.

• More generally, if the earth is reasonably "round-ish" and "smooth" [I don't know how to describe the required conditions], one can draw any plane through the Earth that doesn't pass through Polaris, select any point, and find two other points which are equidistant from it, and then form a tetrahedron on the same side as Polaris. For cuts parallel to the chosen plane that are closest to Polaris, the added vertex will be above the surface. For those that are furthest way, it will be below. If the planet is smooth, there should be a plane where the added vertex lands precisely on the surface. Commented Jan 11, 2023 at 19:32
• I couldn't find 4 points on land starting at the south-pole. 86°S 56°E is pretty close, though. The 3 others are then on Asuncion Island, in Oaxaca (Mexico) and in Al-Jawf (Lybia). Commented Jan 11, 2023 at 21:11

* Pick a random location on Earth.
* Go there with 3 friends.
* Ask friend #1 to start walking/swimming in a random direction, and keep doing it for 12172.6 kilometers.
* Ask friend #2 to do the same, with a 120° offset compared to the direction chosen by friend #1.
* Ask friend #3 to do the same, with a -120° offset compared to the direction chosen by friend #1.

The 3 friends travel along the branches of a 3-pointed star, centered on yourself. When they finally arrive, you and your friends will all be equidistant.

This is basically the same method as with the inscribed tetrahedron, but it might be easier to understand.
It also makes it clear that there are 3 degrees of freedom (your latitude, your longitude, bearing of friend 1)

It isn't too hard to automate the search for 4 points on land with a script:

It's apparently a good idea to start in Argentina, New-Zealand, East Australia, British Columbia, East Africa, or the Philippines, among others.

Here's the corresponding code.

• Would you be willing to share your script, or at least an outline of it? I’m sure it’s “not too hard” as you say for someone who’s proficient with the right tools — but as someone who’s never worked with geographic data, I wouldn’t know where to start on this, and would be interested to learn! Commented Jan 12, 2023 at 14:34
• @PeterLeFanuLumsdaine I was wondering the same. In particular, I suspect I could work out the math but I'd have no idea how to know which points are on land! Commented Jan 12, 2023 at 14:41
• @PeterLeFanuLumsdaine thanks for the interest. I'll upload the whole python script, which isn't very long. It's too quick-n-dirty for my taste, though, so I'll clean it up a bit. Commented Jan 12, 2023 at 15:59
• @MichaelLugo I didn't know either, googled "python is on land", and found a cool website called stack overflow. :) I used geopy for calculating coordinates after a 12000km trip. Scipy to minimize the difference between edges of the tetrahedron, and folium to display markers in the map. A globe might have been cool too, for visualization, I'll try it with matplotlib. Commented Jan 12, 2023 at 16:03

Is it possible to find four such points on land?

Within the terms of the original question, which doesn't specify non-zero distance - yes it is. Just put all four points in the same place on any convenient bit of land. But whether you can do it the other way, with the vertices of an Earth-sized regular tetrahedron - I don't know.