A hexagon of unit diameter is one which has a maximum distance of one unit between any vertices. It can be rotated between two parallel lines one unit apart. What is the largest area unit diameter hexagon?

Intuitively we might well think that the regular hexagon fits the bill. But in fact in 1975 R. L. Graham discovered a strange hexagon that has a larger area. He proved this was the largest possible and the hexagon now bears his name.

In this figure, each red line has unit length. The other dimensions are approximately:
a = 0.40235..
b = 0.53670..
c = 0.06095..
d = 0.34377..
e = 0.15623..

A good reference for some of the calculations involved is:
Eric W. Weisstein. "Graham's Biggest Little Hexagon." From MathWorld--A Wolfram Web Resource.

Graham's original paper is:
Graham, R. L. "The Largest Small Hexagon." J. Combin. Th. Ser. A 18, 165-170, 1975.

The figure below shows Graham's hexagon rotating in a unit box. Note that vertices do not always touch the box.

One way of looking at this is to say that Graham's hexagon is the largest hexagon that can be rotated in a unit square. Of course we then have to ask... what is the largest hexagon that can be rotated in a regular hexagon. We'll look at a regular hexagon with unit width between its sides. It turns out to be impossible to rotate Graham's hexagon in the regular hexagon - just.

In this orientation, the top and bottom vertices of Graham's hexagon touch the sides of the outer hexagon. The next two lowest points are approximately 0.03 vertically below the outer hexagon sides - hard to see in the figure, but I've calculated it.

So which is the largest hexagon to rotate in the regular hexagon...?

If it is a matter of maximising the top figure, where all the red lines are of unit length, (and I think it is - subject to confirmation), then I've calculated that the hexagon with the largest area is this:-

Again, each red line has unit length. The other dimensions are approximately:
a = 0.37374..
b = 0.574078..
c = 0.052177..
d = 0.3188
e = 0.1812

It rotates in the hexagon thus:-

It's beginning to look slightly like a pentagon - so doubtless you will wonder whether a unit width pentagon will rotate in the hexagon. The answer is yes - the Reuleanx pentagon is a unit width pentagon but with curved sides - demonstration here.


page date: 14Nov05.      I enjoy correspondence stimulated by this site. You can contact me here.