February topics: the largest small hexagon

[ursula]

octahedrite suggested a post about "cool geometry topics". I waited until I stumbled on a cool geometry fact via work. This one's elementary but adorable: what's the largest small polygon? For the purposes of this riddle, "small" means that the diameter (biggest distance from one corner to another) is a fixed small number, which we might as well say is 1 unit. Then you try to find the largest area given that constraint.

It turns out that when the polygon has an odd number of sides, the largest small polygon is always a regular polygon. So the largest small triangle is an equilateral triangle, the largest small pentagon has five equal sides and angles, and so forth. But the largest small hexagon is not an equilateral hexagon! You can find a picture of it at MathWorld, see an animation of its rotations at the delightfully old-fashioned website Hall of Hexagons, or read Ron Graham's original paper, which involves an argument via the excellently named (by Conway, unsurprisingly) thrackleations.

The largest small octagons, 10-gons, and 12-gons have also been identified, but for even-sided polygons with 14 or more sides, finding the best one is still an open problem.

(You can suggest more topics here.)