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Squaring the circle

Finding Pi.

A Sunday morning coffee shop – away from home, telephones and distractions – is often a pleasant place to undertake reflections when there arise those situations in which reflection becomes necessary. There is something different about Sunday morning – even better than Saturday morning – due, possibly to the fact that most people have already had one day off and so they have allowed relaxation to sink in and become comfortable. Would that every morning could be Sunday morning but, unfortunately, we can't have everything.

So it was that on a particular Sunday morning, a pleasant fellow whose name is unimportant to our purposes here, sat with a large, steaming ceramic mug before him at a high, round table by the window. His eyes were in constant motion following the goings on in the busy city street before the subdued, peacefulness of the shop but, leaving no outward sign, his concentration was focused solely on an inner monologue.

'Sometimes I'm not sure if I'm the square peg trying to fit in the round hole or I'm the round hole trying to fit the world as a square peg – either way, something doesn't work,' said his inner voice as he gingerly took a sip from the white mug.

As is the case with some people, his mind began first with the comical visualisation of the square peg jammed ineffectually and awkwardly in the wrong space – become a light metaphor for anything which is poorly adapted but, that same vision quickly devolved into an analytical space. In it, he saw a circle: associated with the circle were two squares; one with the same diameter as the circle so its sides touched the outside of the circle and the other circumscribed by circle and with its vertices touching the circle's perimeter.

'If I was the small square, everything would be OK,' the voice of his mind jibed. 'But that's really just the ancient approach to finding pi and the perimeter of the circle, isn't it?'

Inwardly, he poked into some dusty and rarely used files of knowledge.

The Ancients had no concept of pi and so they couldn't calculate the perimeter of a circle. It took many centuries for humans to arrive even close to the rough approximation which we commonly use: 3.1415926. Pi is an irrational number – a constant – which relates the diameter of a circle to length of its perimeter. If a circle has a diameter of 1 unit, then the perimeter is exactly the value of pi.

In order to work around the lack of this knowledge in ancient times, the technique of squaring the circle would be used. The perimeter of the outer square is just 4 times the diameter of the circle. The smaller square has the diagonal which is the same as the diameter of the circle, so the perimeter can still be calculated. Knowing these two numbers just means that the perimeter of the circle is somewhere between the two and it is better than nothing at all.

He absently wiped a drop of spilt coffee from the table with a serviette and the process continued – leading... somewhere.

Later, it was discovered that the perimeter of a regular polygon – like an octagon – is also easy to calculate. The octagon is much more similar in shape to the circle and so more closely approximates its perimeter; the difference between the perimeters of the circumscribing and circumscribed octagons is a better estimate of the desired value.

He found himself slowly sinking into a tantra of shapes gradually growing more and more complex while looking for the elusive value of pi and its relationship to his own thoughts.

By creating polygons with ever greater numbers of sides – 4, 8, 16, 32, 64, 128 and so on – the estimation of the circle's perimeter becomes more exact from a very basic estimate of about 3 in the unit case to something with an increasing fractional precision. It is interesting too, that plotting together in a Cartesian space, the values of the polygon perimeters – one decreasing and one increasing toward the true value - the resulting curve is asymptotic and limits toward pi.

Since the complexity of pi was discovered, it has entranced many because, far from the infinite procession of decimal digits being random, they are not. There are patterns to be discerned in those billions of numbers. Some have gone so far as to suggest that pi is a representation of the mind of God.

'It's the process of arriving that is important though, isn't it? It's how you get there.'

He looked at the empty bottom of the mug, drawn back to reality by a loud, friendly conversation at a nearby table. The sun had, relatively speaking, shifted position.

'It's the same in finding a path and a personal truth in life. Sometimes you err in excess and sometimes you err in deficit but, somewhere between the two, lies that which you are seeking. There's not much chance of getting it spot on but you can always get closer.'

The mystic, geometric tantra faded from his mind. He smiled and stood, retrieving the empty and cooled mug and, leaving the table, he deposited it on the serving counter and nodded briefly to the waitress there.

Once outside, he lit a cigarette and watched a little girl bounce along holding the much larger hand of her father. Nearby, a young couple cooed at each other, enjoying their own little universe and the warm summer sun.

'I guess there's always 'finding pi' to guide us,' joked his mind.

He smiled at the obscure allusion and began to walk.

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