Abstract
A small polygon is a polygon of unit diameter. The question of finding the largest area of small n-gons has been answered for some values of n. Regular n-gons are optimal when n is odd and kites with unit length diagonals are optimal when \(n=4\). For \(n=6\), the largest area is a root of a degree 10 polynomial with integer coefficients and height 221360 (the height of a polynomial is the largest coefficient in absolute value). The present paper analyses the and octogonal cases, and under an axial symmetry conjecture, we propose a methodology that leads to a polynomial of degree 344 with integer coefficients that factorizes into a polynomial of degree 42 with height 23588130061203336356460301369344. A root of this last polynomial corresponds to the area of the largest small axially symmetrical octagon.
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Notes
Graham [10] claims that the optimal hexagon is axially symmetrical, but states that “the details of the proof are not particularly interesting and are omitted”.
These polynomials were obtained using the Maple symbolic calculation commands factor(resultant(PAz(u), Pz(u), u)) and factor(discrim(RA(z),z)).
References
Audet, C.: Maximal area of equilateral small polygons. Am. Math. Monthly 124(2), 175–178 (2017)
Audet, C., Hansen, P., Messine, F.: Extremal problems for convex polygons. J. Global Optim. 38(2), 163–179 (2007)
Audet, C., Hansen, P., Messine, F.: Extremal problems for convex polygons—an update. In: Pardalos, P.M., Coleman, T.F. (eds.) Lectures on Global Optimization, Volume 55 of Fields Institute Communications, pp. 1–16. American Mathematical Society, Providence (2009)
Audet, C., Hansen, P., Messine, F.: Ranking small regular polygons by area and by perimeter. J. Appl. Ind. Math. 3(1), 21–27 (2009). Original Russian text: Diskretnyi Analiz i Issledovanie Operatsii, 15(3), 65–73 (2008)
Audet, C., Hansen, P., Messine, F., Xiong, J.: The largest small octagon. J. Comb. Theory Appl. Ser. A 98(1), 46–59 (2002)
Borwein, P., Mossinghoff, M.J.: Polynomials with height 1 and prescribed vanishing at 1. Exp. Math. 9(3), 425–433 (2000)
Deaux, R.: Introduction to the Geometry of Complex Numbers. Ungar, New York (1954)
Foster, J., Szabo, T.: Diameter graphs of polygons and the proof of a conjecture of Graham. J. Combin. Theory Ser. A 114(8), 1515–1525 (2007)
Gelfand, I.M., Kapranov, M.M., Zelevinsky, A.V.: Discriminants, Resultants, and Multidimensional Determinants. Birkhäuser, Boston (2009)
Graham, R.L.: The largest small hexagon. J. Combin. Theory 18, 165–170 (1975)
Henrion, D., Messine, F.: Finding largest small polygons with GloptiPoly. J. Global Optim. 56(3), 1017–1028 (2013)
Johnson, S.C., Graham, R.L.: Problem #7. SIGSAM Bull. 8(1), 4–4 (1974)
Lazard, D.: Problem 7 and systems of algebraic equations. SIGSAM Bull. 14(2), 26–29 (1980)
Mossinghoff, M.J.: Isodiametric problems for polygons. Discrete Comput. Geom. 36(2), 363–379 (2006)
Reinhardt, K.: Extremale polygone gegebenen durchmessers. Jahresber. Deutsch. Math. Verein 31, 251–270 (1922)
Yuan, B.: The largest small hexagon. Master’s Thesis, Department of Mathematics, National University of Singapore, (2004)
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Audet, C., Hansen, P. & Svrtan, D. Using symbolic calculations to determine largest small polygons . J Glob Optim 81, 261–268 (2021). https://doi.org/10.1007/s10898-020-00908-w
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DOI: https://doi.org/10.1007/s10898-020-00908-w