Thinking about,… Squaring the circle

It’s one of the great philosophical puzzles of all time: given a square of a certain area, how do you construct from it a circle of equal area using only compass and line? The truth is you can’t. This was proven by Ferdinand von Lindeman in 1882, but it hasn’t stopped people from trying, even to the present day, the reason being it’s more than a geometrical puzzle.

Philosophically, we think of the earth and all things in it as the square, or the plane of existence, while the circle represents the whole, the unity, or heaven. And while geometry can measure out most things, the one thing it cannot do is provide a construction that derives heaven from the profane dimensions of the earth. The nub of the problem lies in the strangeness of Pi.

At first glance Pi is a beguilingly simple number, the ratio of a circle’s circumference to its diameter. Divide the circumference by the diameter and you get 3.14159,… etc. The problem is “etc” is currently up to thirteen trillion places, and counting. The more we chase it, the further away it gets from us.

We assume from the evidence so far that Pi is an infinite and non repeating series, a transcendental number. It cannot be derived exactly by ratio, or by formula which means you can’t get to a circle equal in area to a given square, either by line and compass, or by supercomputer.

Or can you? Is there no intersection of line that gives us the radius of that circle? Can we not project it from the walls of the square? Might it not lie in inscribed circles? Can we get at it by projecting points of tangency? How about introducing other squares formed in harmonic series from the root square? Can we get projections from them?

Well,… no. Not exactly.

There are some very good approximations which the old philosophers must have had high hopes for when working within the accuracies permitted by actual line and compass. It’s only when we use computer aided design we get to zoom in and these constructions reveal their flaws. Then we stare through the gaps, not into the infinite void of Heaven, but at our own imperfections.

In my current work in progress, a guy drives himself nuts trying to square the circle. He represents the worst of our egoic tendencies, and he just can’t let it go even though it reduces him from a respected intellectual to the level of a suicidal crank. But he’s since had his revenge on me by having me fall under the spell of the conundrum myself. Yes, I know it’s impossible, but there’s still something beguiling about those approximations, at least to someone who grew up on Euclid and worked for a time on a drawing board with compass and line. I still have those compasses, forty years old now, and rendered obsolete some time around 1985 but there’s a definite beauty to them, also a creative potential, so I like to keep them clean out of respect, even though I rarely actually use them.

I stumbled across the approximation shown above, not with compasses, but with LibreCAD. I’m sure other would-be philosophers have found it too – I mean it’s hardly subtle. Indeed it’s a very simple and elegant construction that gives an answer accurate to within 0.13% of area. That sounds pretty good, but if you play the sums backwards, it yields an equivalent for Pi that’s only correct to the first two decimal places, so we’re a long way from attaining philosophical, spiritual or even just plain old mathematical transcendence.

What all this has to say about the human condition is that, at our worst, we can be self destructively pedantic in our quest for perfection, while at our pragmatic best we recognise a good approximation serves equally well.

This one’s a bit better – 3 decimal places: