Erdős–Ulam problem

In mathematics, the Erdős–Ulam problem asks whether the plane contains a dense set of points whose Euclidean distances are all rational numbers. It is named after Paul Erdős and Stanislaw Ulam.

The Erdős–Anning theorem states that a set of points with integer distances must either be finite or lie on a single line. However, there are other infinite sets of points with rational distances. For instance, on the unit circle, let S be the set of points

where ${\displaystyle \theta }$ is restricted to values that cause ${\displaystyle \tan {\tfrac {\theta }{4}}}$ to be a rational number. For each such point, both ${\displaystyle \sin {\tfrac {\theta }{2}}}$ and ${\displaystyle \cos {\tfrac {\theta }{2}}}$ are themselves both rational, and if ${\displaystyle \theta }$ and ${\displaystyle \phi }$ define two points in S, then their distance is the rational number

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