Number of Distances Separating Points Has a New Bound

Scatter three points in a plane, then measure the distances between every pair of them. In all likelihood, you’ll find three different distances. But if you arrange the points in an equilateral triangle, then every distance is the same. In a plane, this is impossible to do with four points. The smallest number of distances you can engineer is 2 — the edges and diagonals of a square.

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