Since the summer, I have been obsessed with the problem of squaring the circle—that is, finding the square whose area is equal to that of a given circle, with no tools other than a straight edge and compass.
I took interest in the problem when reading the ending of Dante’s Paradiso (in Allen Mandelbaum’s translation):
As the geometer intently seeks
to square the circle, but he cannot reach,
through thought on thought, the principle he needs,
so I searched that strange sight; I wished to see
the way in which our human effigy
suited the circle and found place in it—
and my own wings were far too weak for that.
But then my mind was struck by light that flashed
and, with this light, received what it had asked.
Here force failed my high fantasy; but my
desire and will were moved already—like
a wheel revolving uniformly—by
the Love that moves the sun and the other stars.
I, too, had difficulty reaching the principle—and it was precisely the principle that I needed to find. The difficulty lies in determining what the problem exactly is.
No one has found a way to square the circle with compass and straight edge alone; in fact, I believe it has been proved that it can’t be done. Yet I kept on trying, thinking that I would learn something from the attempt.
Yesterday morning, something close to a solution came to me—not a solution, mind you, but something that points in the direction of one. It’s probably old hat—or ancient hat—but it’s interesting, all the same.
We will consider the circle with radius r=1, since expanding or shrinking it (and the square) is then a trivial matter.
We know that if the circle has radius 1, its area is π. Thus, for a square to have that area, it must have sides of length √π. But how do you find length √π? There’s the puzzle.
When the square and circle have the same area, the ratio of the circle’s circumference to the square’s perimeter is a constant, 2π divided by 4√π ; that is, π/2 divided by √π; that is, √π/2.
In other words, the ratio of one-quarter of the circle’s circumference t0 one side of the square will always be √π/2 when the square and circle have the same area.
You could consider the ratio in terms of this figure:
|AO| is equal in length to the side of the square (√π when the circle has radius 1); |OP| is one-quarter of the circle’s circumference (1/4 * 2π = π/2), which you can measure with a string. (I know you’re only supposed to use a straight edge and compass, but this exception will prove helpful.)
Now, the ratio of |OP| to |AO| is equal to the ratio of|BC| to |AB|, where |BC| = √π = |AO|, and |AB| = 2. We already established this ratio a few paragraphs up.
(Again, the ratio of one-quarter of the circle’s circumference to the square’s side must be √π/2.)
So now the challenge is to tweak the figure until |AO|=|BC| (while keeping points A, P, and C on a single line and A, O, and B on a single line). When we get the two segments to equal length (without changing |OP| or |AB|), we have brought both |AO| and |BC| to √π. Now, you must do this by trial and error, by shifting the OP segment. If |BC| is greater than |AO|, then move OP closer to BC; if it is smaller than |AO|, then move OP closer to A. Your adjustments will be smaller and smaller until you either make |AO| and |BC| equal or come as close as you wish. The next step would be to find a mathematical way of doing this.
Once you have made |AO| and |BC| as close to equal as you wish, you then make AO the side of your square, which you superimpose on a circle of radius 1 (so that both have a common center). You will see that the square is mostly, but not entirely, contained within the circle. It will look like the first of the two figures above.
Although this is not a satisfactory solution, it results in a close approximation and seems to be on the right track.
Update (on a tangent, so to speak): The angle in the figure is the inverse tangent of √π/2, or 41.54822621257918513… Its common logarithm (1.6185524875…) is tantalizingly close to the golden ratio, but not close enough for the trumpets to sound.
Note: This is a reposting of a piece that originally appeared on October 19.