The biggest little polygon with sides is the convex plane -gon of unit polygon diameter
having largest possible area. The biggest little polygons
for ,
8, and 10 are illustrated above. In the figures, diagonals shown in red have unit
length.
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 biggest little polygon graphs on and 8 nodes are implemented in the Wolfram
Language as GraphData["BiggestLittlePolygon",
n]
for ,
8, and (in a future version of the Wolfram
Language) 10.
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. A98, 46-59, 2002.Audet, C.; Hansen,
P.; and Svrtan, D. "Using Symbolic Calculations to Determine Largest Small Polygons."
J. Global Opt. 81, 261-268. 2021.Audet, C.; Hansen, P.; and Svrtan,
D. "A New Algorithm for Finding Largest Small Polygons Using Symbolic Computations."
2018. https://www.grad.hr/crocodays/pres_2018/2018_svrtan.pdf.Foster,
J. abd Szaba, T. "Diameter Graphs of Polygons and the Proof of a Conjecture
of Graham." J. Combin. Th., Ser. A114, 1515-1525, 2007.Graham,
R. L. "The Largest Small Hexagon." J. Combin. Th. Ser. A18,
165-170, 1975.Hurst, G. "A Closed Form Expression for the Area
of the 'Biggest Little' Octagon and Decagon." Apr. 14, 2025. https://community.wolfram.com/groups/-/m/t/3444306.Mulansky,
B. and Potschka, A. "A Zonogon Approach for Computing Small Polygons of Maximum
Perimeter." 2 Apr 2024. https://arxiv.org/pdf/2404.01841.Reinhardt,
K. "Extremale Polygone gegebenen Durchmessers." Jahresber. Deutsch.
Math. Verein31, 251-270, 1922.Sloane, N. J. A.
Sequences A111969, A111970,
A111971, A381252,
and A383173 in "The On-Line Encyclopedia
of Integer Sequences."
sides is the convex plane 
-gon of unit polygon diameter
having largest possible area. The biggest little polygons
for 
,
8, and 10 are illustrated above. In the figures, diagonals shown in red have unit
length.
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).
-gon.
and 8 nodes are implemented in the Wolfram
Language as GraphData[
"BiggestLittlePolygon",
n
]
for 
,
8, and (in a future version of the Wolfram
Language) 10.
