Using symbolic calculations to determine largest small polygons

Charles Audet (), Pierre Hansen () and Dragutin Svrtan ()
Additional contact information
Charles Audet: Polytechnique Montréal
Pierre Hansen: HEC Montréal
Dragutin Svrtan: University of Zagreb

Journal of Global Optimization, 2021, vol. 81, issue 1, No 10, 268 pages

Abstract: 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 $$n=4$$ n = 4 . For $$n=6$$ 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.

Keywords: Small polygons; Planar geometry; Disciminant (search for similar items in EconPapers)
Date: 2021
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Citations: View citations in EconPapers (1)

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DOI: 10.1007/s10898-020-00908-w

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