Squaring The Circle

Stupid Question ™
May 10, 2004
By John Ruch
© 2004

Q: I know that “squaring the circle” means doing something impossible, but what does it refer to?
—Beth C., from the Internet


A: It refers to the geometrical attempt—now known to be impossible—to create a square that has the exact same area as a given circle.

Specifically, it refers to one ancient Greek formulation of this problem that is, if you will, even more impossible than usual.

The irony is that it’s possible to come very, very close to squaring the circle—to an accuracy of several decimal places, in fact. So what’s the big deal?

Well, “close” only matters in practical applications. That’s apparently all they cared about in ancient Egypt, source of the first known reference to squaring the circle, which was just a rule-of-thumb attempt. (They just shrunk the circle a little and plopped a square around it, which produced a pretty decent estimate.)

Today, there are a variety of relatively simple pen-and-paper calculations that produce a really good nearly-squared circle. Great for laying kitchen tile or what have you.

But what the common phrase refers to is an ancient Greek theoretical formulation. It’s actually one of three conundrums from ancient Greek geometry, all highlighting different theoretical difficulties. Not surprisingly, it’s also the one with the catchiest name and involving the simplest shapes.

The other two are doubling or duplicating the cube and trisecting an angle.
Squaring the circle is also the “most impossible” of the three—the other two involve operations that are possible in certain specialized circumstances.

Now, the fact is, many Greek mathematicians explored other ways of squaring the circle, using marked points, dynamic curves, conical sections and other geometrical arcana. Such methods can get you pretty close, whereas using the arbitrary straight edge/compass method gets you nowhere. So why the drawing rules?

They were probably intended to create a thought experiment highlighting the impossibility of a truly exact squaring of the circle. Intentional or not, the thought experiment has worked—not in squaring the circle, but in developing all sorts of mathematical principles.

As a simplified thought experiment, it also emphasizes the key difficulty of the whole problem: defining the number pi, which is really what the whole squaring-the-circle stunt is all about.

A quick trip back to math class: Pi is the ratio between the circumference of any circle and its diameter. Pi is also, therefore, a key number in determining the area of a circle (area equals pi times the radius squared, you may recall).

Problem is, pi is not a whole-number ratio. It’s an endlessly repeating decimal monstrosity that begins 3.1416…. and continues on without pattern, apparently forever. (It has been calculated to many billions of decimal places.)

In short, a circle—a defined, geometric figure—contains a mathematical number that is literally infinitely imprecise. It is a grand mystery, and it means we can’t even say exactly what the area of a circle is, let alone what a square of the same area would look like.

Put in math lingo, pi (and its square root) can’t be expressed by any finite set of mathematical operations. That includes any series of geometrical straight lines, like a square.

Put in plain English, what squaring the circle really means is drawing a line that is the square root of pi in length, to use as the sides of the square. But the square root of pi is also an endlessly repeating decimal. You can’t draw a finite square with infinitely long sides, now, can you?

(Without going into detail, the same kind of fundamentally irrational problems lie at the core of the other two impossible challenges. Doubling the cube involves the cube root of 2, which can’t be expressed geometrically that way. Trisecting an angle involves the cosine of 1/3 of a given angle, which is geometrically expressible only for a tiny set of angles.)

Were the ancient Greeks idiots? Did they really think it would be fun to try to draw a finite line representing an infinite ratio? They were a bunch of lazy slaveholders with plenty of time to sit around and think up weird things, so knowing better probably wouldn’t have stopped them—but they didn’t know better. Squaring the circle was an attempt to trap pi and squeeze it for information.

Fact is, it wasn’t until the 1760s that pi was proven to be irrational, meaning it repeats endlessly. And it was only in 1880 that pi was conclusively proven to also be transcendental, which essentially means it can’t be expressed algebraically (and thus not as geometric lines, either).

The impossibility of squaring the circle has not stopped attempts by a parade of well-intentioned amateurs and religious kooks that is as endless and decreasingly significant as pi itself.

Even in ancient Greece, squaring the circle was a byword for attempting the impossible or fantastical. With our fuller understanding of its background, we can add some nuances to the modern meaning of the phrase.

For example, the Greeks made the impossible even more difficult by forcing arbitrary rules onto the squaring challenge, which can arguably be said for many “impossible” social and political ills today.

It’s quite likely the arbitrary rules were there, however, because squaring the circle was intended as a thought problem—a fruitful self-stumping. An Internet search shows that policy papers about social problems that use “Squaring the Circle” as a title often take this same approach—a hopeful declaration that good will come out of trying to solve an apparently impossible problem.