Polygons all of whose edges meet at right angles are called rectilinear polygons. I am interested in rectilinear polygons with integer distance between each pair of vertices. Such rectilinear polygons are trivial to obtain when restricting oneself to polygons with 4 edges.

Do rectilinear $n$-gons exist that have integer Euclidean distance between each of the $\frac12 n(n-1)$ pairs of vertices for $n>4$?

  • $\begingroup$ Are you requiring diagonal distances to be integers too? $\endgroup$ – Henry Feb 14 '15 at 13:58
  • $\begingroup$ @Henry - yes, all Euclidean distances (including the diagonal distances) need to be integer. Have modified the problem statement to stress this point. Thanks. $\endgroup$ – Johannes Feb 14 '15 at 14:13
  • $\begingroup$ It might be tricky to construct examples: "Robert Israel gives a nice proof (originally due to Erdös) of the fact that, in any non-colinear planar point set in which all distances are integers, there are only finitely many points. Infinite sets of points with rational distances are known, from which arbitrarily large finite sets of points with integer distances can be constructed; however it is open whether there are even seven points at integer distances in general position (no three in a line and no four on a circle)." -- from ics.uci.edu/~eppstein/junkyard/open.html $\endgroup$ – Barry Cipra Feb 14 '15 at 14:59

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.