# Locations on the Earth that are all separated by the same distance

What is the maximum number of distinct locations you can select on the surface of the Earth, such that the distance between every pair of locations is the same? Assume that the Earth is a perfect sphere and the distance between two locations is measured along the surface of the Earth.

Bonus question: Can you find actual locations of significance (such as cities) that have this property?

• Surface of the Earth. Updated the last statement to reflect this better. – Dmitry Kamenetsky Jan 22 at 1:41

4 - arranged as a tetrahedron

Explanation

In order to select any more than 4, we would have to satisfy 5. But this is impossible, as it would have to begin as the tetrahedral arrangement with an extra point, but there is no point on the surface of a sphere which satisfies this.
A tetrahedron is the only arrangement of four points in 3D space for which the vertices are equidistant (kind of the definition of a tetrahedron). And for any tetrahedron, there exists one sphere on which all vertices lie.
This can also be shown by adding points incrementally.
2 points is trivial, and will always be equidistant, so any two points on the globe will satisfy.
3 points must be arranged as an equilateral triangle to preserve the equidistance. For (almost) any two points, there are only 2 options for the third.
adding the 4th point will only work on certain sizes of triangle, so that it lies on the surface of the globe.

• It would seem the part about the Earth's surface is superfluous to the question. Good answer though. – Earlien Jan 22 at 1:58
• Can you provide more information? What are the edge lengths relative to the circumference? Perhaps a diagram. – Dmitry Kamenetsky Jan 22 at 2:45