Graham's Biggest Little Hexagon

Graham's biggest little hexagon is the largest possible (not necessarily regular) convex hexagon with polygon diameter 1 (i.e., for which no two of the vertices are more than unit distance apart). It is therefore the biggest little polygon for the case . The solution is given by the above figure (which is shown with incorrect proportions in Conway and Guy 1996, p. 207), in which the red lines are all diagonals of unit length.

Graham (1975, p. 169) proved that the maximizing hexagon is unique up to geometric congruence. Its diameter graph is a cycle graph on five vertices with one pendant edge attached.

To find the hexagon, set up coordinates as illustrated above, then the polygon area formula gives

(1)

Plugging in together with

			(2)
			(3)

and eliminating and then gives the formula for as

(4)

This function is plotted above. The parameter interval is , with and . The unique maximum occurs at a stationary point in (Graham 1975, pp. 168-169). The same algebraic optimization can be formulated using Lagrange multipliers for the unit-distance constraints and a Gröbner basis to eliminate the coordinates and multipliers (Trott 2006).

Maximizing gives the area of the hexagon as the second-largest real root of

(5)

approximately given by (OEIS A111969). Note that the sign of the is positive, not negative as erroneously given in Conway and Guy (1996). Also compare this with the area of the regular hexagon of polygon diameter 1 (and therefore having circumradius 1/2), which is given by

(6)

so the optimal solution is 3.9% larger.

The values of and corresponding to the maximal solution are given by

			(7)
			(8)
			(9)
			(10)

(OEIS A111970 and A111971).