Journal of Geometry

, Volume 99, Issue 1–2, pp 43–66

On a problem of Croft on optimally nested regular polygons



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 


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Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of South CarolinaColumbiaUSA
  2. 2.Convergent Computing Inc.ShorehamUSA

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