BMAD Posted May 6, 2014 Report Share Posted May 6, 2014 What is the expected distance of two points in a unit square? What is the expected distance of two points in a regular unit pentagon? Does the expected distance between two points in a regular unit n-gon converge as n goes to infinity? Quote Link to comment Share on other sites More sharing options...

0 bonanova Posted May 6, 2014 Report Share Posted May 6, 2014 If you mean unit area, the mean distance between two interior points of a unit regular n-gon converges to that of a disk. If you mean unit side, then the mean distance diverges, since that type of unit regular n-gon becomes the entire plane. Quote Link to comment Share on other sites More sharing options...

0 m00li Posted May 6, 2014 Report Share Posted May 6, 2014 If you mean unit area, the mean distance between two interior points of a unit regular n-gon converges to that of a disk. If you mean unit side, then the mean distance diverges, since that type of unit regular n-gon becomes the entire plane. how do we find the mean distance for a square? and for a regular unit pentagon? seems very very tough to me Quote Link to comment Share on other sites More sharing options...

0 bonanova Posted May 7, 2014 Report Share Posted May 7, 2014 Empirically, by simulation. Or do some area integrals. For an equilateral triangle with unit sides it's 0.364791843300... For a square with unit side it's 0.521405433.... For a circle with unit radius it's 0.905414787... Quote Link to comment Share on other sites More sharing options...

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## BMAD

What is the expected distance of two points in a unit square?

What is the expected distance of two points in a regular unit pentagon?

Does the expected distance between two points in a regular unit n-gon converge as n goes to infinity?

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