Journal of Geometry

, Volume 99, Issue 1, pp 43-66

First online:

On a problem of Croft on optimally nested regular polygons

  • S. J. DilworthAffiliated withDepartment of Mathematics, University of South Carolina
  • , S. R. ManeAffiliated withConvergent Computing Inc. Email author 

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We present a solution for the largest regular m-gon contained in a regular n-gon. We find that the answer depends critically on the coprimality of m and n. We show that the optimal polygons are concentric if and only if gcd(m, n) > 1. Our principal result is a complete solution for the case where m and n share a common divisor. For the case of coprime m and n, we present partial results and a conjecture for the general solution. Our findings subsume some special cases which have previously been published on this problem.

Mathematics Subject Classification (2010)

Primary 51M20 Secondary 52C15


Polygon containment optimally nested polygons axes of symmetry