Biggest Little Polygon

The biggest little polygon with n sides is the convex plane n-gon of unit polygon diameter having largest possible area.


Reinhardt (1922) showed that for n odd, the regular polygon on n sides is the biggest little n-gon. For n=4, the square with diagonal 1 has maximum area, but an infinite number of other 4-gons are equally large (Audet et al. 2002). The n=6 case was solved by Graham (1975) and is known as Graham's biggest little hexagon, and the n=8 case was solved by Audet et al. (2002). The following table summarizes these results, showing the percentage that the given polygon is larger than the regular n-gon.

narea% larger than regular n-gonreference
60.6749813.92%Graham (1975)
80.7268672.79%Audet et al. (2002)

The biggest little polygon graphs on n=6 and 8 nodes are implemented in the Wolfram Language as GraphData[{"BiggestLittlePolygon", n}].

See also

Graham's Biggest Little Hexagon, Minimum Diameter Polygon, Polygon Diameter

Explore with Wolfram|Alpha


Audet, C. "Optimisation globale structurée: propriétés, équivalences et résolution." Thèse de Doctorat. Montréal, Canada: École Polytechnique de Montréal, 1997., C.; Hansen, P.; Messine, F.; and Xiong, J. "The Largest Small Octagon." J. Combin. Th. Ser. A 98, 46-59, 2002.Graham, R. L. "The Largest Small Hexagon." J. Combin. Th. Ser. A 18, 165-170, 1975.Reinhardt, K. "Extremale Polygone gegebenen Durchmessers." Jahresber. Deutsch. Math. Verein 31, 251-270, 1922.

Cite this as:

Weisstein, Eric W. "Biggest Little Polygon." From MathWorld--A Wolfram Web Resource.

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