Proof of the Pi

The proof of the pudding, they say, is in the eating, but what about the proof of the pi?

The ancient Egyptians, the Babylonians, Archimedes and blogger Bob Brague will tell you that we need pi (π) for circle geometry, and that it is roughly 3.14159. Blogger Yorkshire Pudding will also tell you that we need pie, lots of it, but he would be referring to the kind he makes from minced meat topped with mashed potato and baked. Yorkshire Pudding is right. There is no need for mathematical constants and strange symbols. We only need to know that the distance around the circle of a shepherds pie, as near as dammit, is three and one-seventh (22/7) times the distance across. You can use this to ensure you are baking enough for everyone. 

I took for granted what they said about pi at school, without any real understanding. If understanding is the ability to think of the same thing in different ways, and to be able to switch between them, this is my attempt to do that. 

So, this is another mathematical post, like the one in January about the Pythagoras theorem. I wondered whether the same technique could be used to illustrate similar concepts; such as pi.  

Here is a circle of 14 units across, zoomed in on the top right-hand quarter to make it easier to see and count. The quarter-circle is 7 units high, and it takes 11 units to go around its edge. So to go all the way round the full circle would take 44 units, which is three and one-seventh times the distance across the whole circle (14 x 22/7 = 44). 

Does it also work for area? Can it show that the area of a circle is three and one seventh times the area of a square fitted from the centre to the edge (American: Area = πr²)? 

Here is the circle again, with a square drawn from the centre to the edge, zoomed in on one quarter. 

If the square is divided into a 7 by 7 grid of 49 smaller squares, then most of the smaller squares are inside the circle, but some are outside. Of those outside, some are complete squares while others are part-squares. Counting them, I reckon that a total equivalent of around 10½ smaller squared are outside the circle, leaving 38½ inside. I have tried to show how I counted 10½ by putting numbers on the quarter-circle. Those with the same numbers make up one square. 

Multiplying this by 4, it would need 154 (38½ x 4) of the smaller squares to completely fill the full circle. This is equal to three and one-seventh times the 49 in the square on the radius (49 x 22/7 = 154). 

To prove this visually, I used three larger squares to cover three-quarters of the circle. Then I moved the parts that were outside the circle (shown in grey) into the fourth quarter. So far, in all, this has used 3 larger squares, a total of 147 smaller squares. 

But it does not quite cover all the fourth quarter of the circle. We need an extra 7 smaller squares (shown in yellow), in other words, one-seventh of a larger square.  

So, the area of a circle is equal to three and one-seventh times that of a square drawn from the centre to the edge. (Area = πr²)

Arithmetically, it takes 38½ smaller squares to fill the fourth quarter, but there are only 3 x 10½, or 31½, available to move. We are 7 short. 

I get it. At least I think I do. 

Hand Signals and Semaphor Indicators

Amongst the audiotapes I mentioned towards the end of last year, is one recorded by my aunt and cousins in the early nineteen-sixties. My uncle had taken a job in Germany, but they had yet to join him. They mention near the beginning that I had brought my recording machine so they could wish him a happy birthday. My own thirteen-year-old voice is heard briefly at the end of the tape, but the less said about that, the better.

What forgotten memories it brings back!  

After the usual birthday song, they talk about what they have been doing. My youngest cousin says: “Here is a song we learnt at school”, and begins to sing:

Sides together right,
Sides together left,
Sides together right left,
Sides together both.

We did that one in my year too. It was a dance in which you moved your arms about like a boy scout semaphore signaller. It then moves on to your toes: “Sides together point, sides together point ...”. Dear Miss Cowling: how you loved to join in. Remind me how to point both toes at the same time.

Then my aunt mentions she is about to take her driving test. Our town was a great place for it. It is completely flat with no hills. To test your hill start, you either did your three-point turn on a street with a particularly high camber, or went through a T-junction where the road rises a few inches due to the spoil dug out from the docks. There were also no traffic lights, no roundabouts, and only one zebra crossing. It limited what you could fail on. A few years later, I passed first time, four months after my seventeenth birthday. The test centre there closed years ago.

Even so, my aunt was anxious about the test. She took it in a Fiat 600 shipped back from a previous overseas stint in Aden. The Fiat was fine there, but a bit tinny and unsuited to the Yorkshire weather. There was always something wrong with it.

Things did not begin well. She told the story many times. To say she was a nervous driver, lacking in confidence, would be understatement. The examiner made no attempt to put her at ease, staring blank-faced ahead throughout, giving strict instructions in a stern voice. 

In those days, you had to be able to use hand signals. Remember those? Sides together right for a right turn, a kind of circling movement for left, and a wave like a sea gull to slow down. There were also special signals for white-gloved policemen on point duty. It was not easy through the tiny windows of the Fiat, especially if it was throwing it down with rain. The longer it went on, the surer my aunt became that she had failed.

It was a relief to finish the hand signals and be allowed to use the electronic indicators. However, the Fiat did not have the modern self-cancelling flashing lights we have now. They were the old semaphore type. A little orange-tipped arm, about six inches long, flipped out from the side of the wing. You had to remember to put it back in again after you had turned.

So when one of the semaphore indicators flipped out but refused to flip back in again, my aunt lost all remaining hope of success. She pulled up, got out, and tried to push it back in by hand, but it was firmly stuck.  

“Well, that’s it now,” she sighed hopelessly. “I’ve failed. Drive me back to the test centre and I can go home.”

The examiner was stolidly unsympathetic.  

“Get back in woman,” he barked.

She meekly did as told and completed the rest of the test using hand signals.

When they got back, my aunt answered the obligatory questions about road signs, braking distances, and the Highway Code, certain it was futile. The examiner completed his paperwork in stony silence.

“I am pleased to inform you that you have passed,” he announced. He had to repeat it.

“Thank you. Oh thank you,” she stuttered in disbelief. “I promise I won’t let you down.” 

1957 Fiat 600