Journal of Geometry

, Volume 99, Issue 1, pp 43–66

On a problem of Croft on optimally nested regular polygons


  • S. J. Dilworth
    • Department of MathematicsUniversity of South Carolina
    • Convergent Computing Inc.

DOI: 10.1007/s00022-011-0065-3

Cite this article as:
Dilworth, S.J. & Mane, S.R. J. Geom. (2010) 99: 43. doi:10.1007/s00022-011-0065-3


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 51M20Secondary 52C15


Polygon containmentoptimally nested polygonsaxes of symmetry
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© Springer Basel AG 2011