Abstract
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.
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References
Chazelle B.: The polygon containment problem. Adv. Comput. Res. 1, 1–33 (1983)
Daley D.J.: Optimally nested regular N-gons. J. Optim. Theory Appl. 41, 425–437 (1983)
Dureisseix D.: Folding optimal polygons from squares. Math. Mag. 79, 272–280 (2006)
Wetzel J.E.: An ancient elliptic locus. Am. Math. Mon. 117, 161–167 (2010)
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To Hallard Croft, teacher and friend
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Dilworth, S.J., Mane, S.R. On a problem of Croft on optimally nested regular polygons. J. Geom. 99, 43–66 (2010). https://doi.org/10.1007/s00022-011-0065-3
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DOI: https://doi.org/10.1007/s00022-011-0065-3