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# Biggest Little Polygon

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

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

 area % larger than regular -gon reference 6 0.674981 3.92% Graham (1975) 8 0.726867 2.79% Audet et al. (2002)

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

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

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## References

Audet, C. "Optimisation globale structurée: propriétés, équivalences et résolution." Thèse de Doctorat. Montréal, Canada: École Polytechnique de Montréal, 1997. http://www.gerad.ca/Charles.Audet.Audet, 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. https://mathworld.wolfram.com/BiggestLittlePolygon.html