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

Breasts

What a title to grab attention! I wonder what the hit rate will be. However, those here for salacious reasons (you know who you are) may be disappointed. This is not what you are looking for. It is about embarrassing side-effects of the Tepotinib medicine I take. 

And they truly can be embarrassing. It messes with your proteins and hormones to strange effect. In an earlier post I mentioned scrotal oedema (14th November). It has you rolling round like a bow-legged sailor. Fortunately, this has now subsided and I can go back to sea; well, walk around the village and do the gardening, at least.

But there is a still more embarrassing side-effect, which I would not be mentioning at all had it not been sorted: gynaecomastia. It translates from the Greek as “female breast”: man boobs.

I am not talking about a bit too much flab and fat in the chest department (you also know who you are; we think we do too), but something more uncomfortable. It took a month or two to pluck up the courage to tell the consultant I was a little sore around the nipples. A month later it was becoming painful. A hug from my wife had me crying out, and bumping against a door frame made me writhe in agony. I don’t know how you women manage. Breast feeding must be a nightmare. There were hard circular lumps under the skin and they were growing bigger. I began to worry it might show.

The consultant said it was not something he had come across with Tepotinib, but he did an additional blood test. My testosterone levels were right down. Both men and women produce testosterone and oestrogen in different proportions. My testosterone was around the female level.

Four months and four jabs in the bum later, I am relieved to report that it has gone completely. The jabs could have been at shorter intervals, but I went for a more careful approach.  I didn’t want to start acting like Rambo.

No further jabs needed. I now have a gel you rub on - no, not there - you rub it on your shoulders. 

“Testogel”, would you believe? Two pumps per day. Phwoar! 

You have to wash your hands thoroughly afterwards, and on first use prime the pump and dispose of what comes out. Quite a bit goes down the sink. I suppose somewhere there is a fish with a beard and a deep voice.

Snowing Like Buggery

Heavy snow was forecast today, with a weather warning for our part of the Pennines, but it was nothing compared to how things used to be.

The guinea pigs snowbound in their hutch at the end of the garden, 2013

Wednesday, 25th January, 1995. It started to snow at half past four in the afternoon, much earlier than forecast, and much more heavily. I was at work at the university, feet on desk, on the phone to someone in Newcastle. Big heavy flakes, like dinner plates some would say, reflected the office light back in through the window. I should have got out straight away. When I did, it was chaos in the car park. Any later and I would have had to spend the night in the sports hall. 

All over Yorkshire, people were trapped overnight at work, on trains, in churches and town halls, and in cars on the M62. Six people died trying to walk home. The audience at the West Yorkshire Playhouse in Leeds were forced to stay the night. They had been to see ‘The Winter Guest’, a play about a community cut-off during a blizzard. Realism at its utmost.

My wife, J, was six months pregnant, her last week at work before maternity leave. She worked seventeen miles away. How was she coping? No mobile phones in those days.

It took me two hours to do three miles. The traffic inched forward with longer and longer standstills. The road then steepened and the next few hundred yards took another hour. I turned round and went back to a pub car park, had a whisky and some crisps, and started to walk home.  

I crunched my way through the countryside under crystal-clear stars, along the middle of a now silent road. There were a few other walkers, all cheery and excitable, and a lot of abandoned cars. Where was J? Was she back already? There had been no answer when I phoned from the pub.

The house was in darkness. No messages. I waited up for a time. It seemed best to get some sleep, so I went to bed.

I was awoken by her scathing voice. Her car was outside in the middle of the street, deep tracks through pristine snow. It’s an interesting experience, shovelling two feet of snow off your drive at three in the morning.

It had taken her nine hours to get home. Apart from stopping for petrol and a bar of chocolate, she had simply kept going. She knew she couldn’t get out and walk. On getting stuck, she just went backwards and forwards wiggling the wheels until free. She was rather pleased with herself, the only car still on the road.

The snow melted quickly and we were able to rescue my car the following afternoon. 

Early the following week, it happened again. In no way was the university management going to be caught napping again.

This time I was helping to invigilate an exam for a hundred and fifty students. We were in a windowless, sports hall half a mile from the main campus. I had accompanied a student to the toilet (fortunately, they were still capable of removing and replacing their own pants in those days) when someone came to tell us we had an urgent phone call. Gary, one of the other invigilators, went to take it.

“Evidently it’s snowing like buggery* outside,” he whispered to us. “The examinations office say the university is about to be closed and we should all piss off home quick” (don’t ever think that university professors are urbane and well-spoken all the time).

I wasn’t too happy. It meant we would need to set another exam, and the students would have to prepare for it and retake at a later date. We whispered between ourselves and decided to tell the students how things stood. Gary made an eloquent announcement offering them the opportunity to leave early. None did.

Afterwards, we were supposed to be collected along with the examination scripts by university transport, but none came. We had to struggle back on foot. The university was locked up and abandoned. Even the security staff had gone home. We could not get into our offices. We walked to the car park and put the scripts in my car. I began the drive home.  

If anything, it was worse than the first time. The traffic was even slower and more halting, and I could see I was going to get stuck again. I had the bright idea of diverting by one of the back roads. Wrong decision!

Half way home there is a steep, down and up dip. I slid to the bottom and that was it. Car stuck in the middle of nowhere, boot full of students’ examination scripts. It was going to be hard enough to walk the rest of the way without having to carry the scripts as well. So, I abandoned them. I wasn’t going to spend all night in the car.

There it stayed for two days. What would have become of me had the car been broken into and scripts full of data flow diagrams and entity life histories scattered across the countryside? I suppose it would not have been the first time that examination papers have been lost. It must have happened at some time, somewhere. Or has it? 

Road near our house, 2013

 * a colourful expression I had not heard since childhood, meaning "a lot".

Hand Warmer

This solid-fuel pocket hand warmer has been at the back of a drawer, unused for forty years. I thought it was something from the past, but surprisingly you can still buy them. I bought mine around 1973. 

It consists of a small insulated metal case that fits in the palm of your hand and burns solid carbon fuel rods. They were mainly for climbers but I used it on wild camping trips in the Scottish Highlands when it was cold enough to freeze water inside the tent. It warns not to use it in bed, but I did. It was great for warming up your toes at the end of your sleeping bag. I could have died of carbon monoxide poisoning. 

I fired it up for one last time.

These days you can buy chemical hand warmers small enough to fit in a glove or sock. They look like tea bags. Apparently, they are not that warm and don't last very long. Mine gets quite warm and lasts all night. The only trouble is it makes you smell like you have been standing on a railway bridge above the funnel of a steam engine.

It's too nice to throw away. I'll put it back in the drawer for someone else to deal with. The kids won't use it. They are scared of matches. 

Upper Glen Nevis, 1975

Brendan and the Shared House

New Month Old Post: first posted 3rd February, 2019. (Not that old, but few current followers will have seen it).

I always assumed we would see each other again one day. We would go to the pub and get pissed and laugh about the people and the good times in the shared houses in Leeds. But it was not to be.

We would remember Ron, the guy who never stopped talking, notorious for ‘ronopolising’ the conversation with his mind-numbing ‘ronologues’ which always began “Did I tell you about the time I …”, and if you had ever been somewhere, done something or seen something, he had always been somewhere, done something or seen something better. He used to leave his towel draped over the hot water cylinder in the bathroom and it stank. He never washed it. You would think a hospital bacteriology technician would have been worried about bugs.

And Pete, who gassed the place out with the peculiar aromatic smell of Holland House pipe tobacco. He smoked even when it was his turn to cook, speckling the plates with ash. He once accidentally tipped the thing over my food and instead of being sorry just laughed and got on with his own unconcerned. Anyone would think he owned the place. Actually, he did. He was always asking “Can I trouble you gentlemen for some rent please?”

Then there was Nick, who could swear like only someone from the back streets of Manchester could, and Larry who made himself dainty little jellies and custards every Monday and lined them up uncovered on the kitchen table for several days (we had no fridge). And Roger, the Ph.D. student with his clever cryptic comebacks, and Paul with the outrageous ginger beard and silly Lancashire accent. And Gavin who was so well organised you had to make an appointment three weeks in advance just to ask him something. And Dave, the Geordie, who did an animated rendition of The Lampton Worm, and was on holiday when the electoral register form came, so we put his middle name down as Aloysius.

And who could forget ‘Pervy Pete’, the television rent collector, who came each month to empty the coin box, greeted us “hello mensies”, and lingered uninvited to take an unseemly interest in which bedrooms we slept? That television always ran out of money right in the middle of Monty Python or just before a punchline in Jokers Wild.

The others came and went, but Brendan and I stayed longest. We were from ordinary Yorkshire backgrounds, shared the same sense of humour and had under-achieved our ‘A’ Levels. Brendan was the liveliest among us, and the best looking. In his long Afghan coat, with his smooth young face and long centrally-parted hair, the kids in the street called him “that lad who looks like David Cassidy.” He made us laugh with his silly puns and deliberate misunderstandings. He could play guitar better than me and instantly put chords to almost any song at all. He could throw a lighted cigarette in the air and catch it the right way round in his mouth. He had an impossibly beautiful girl friend who was training to be a doctor.

We were both desperate to escape our mundane jobs, me from an accountants’ office and Brendan from a veterinary laboratory, and did so around the same time in 1977, me to university and Brendan on Voluntary Service Overseas (VSO). He dreamed of some idyllic tropical paradise where nubile young girls danced to the drum-beat naked in the twilight, and was dismayed to be sent to sub-Saharan Africa, to an isolated rural village in Northern Ghana called Pong-Tamale, around 400 miles from the coast. It was not even much of a change of job: he went to run a laboratory in a veterinary college.

Pong-Tamale in 2010 (click to play)

In those days, people still wrote letters, and I looked forward to his aerograms dropping through the letterbox with their exotic stamps and tales of distant Africa. Things were not easy. It was oppressively hot. He suffered tropical ailments and diseases. They were short of supplies and equipment. He asked to be sent books as there was little to read and no television, not that they always had electricity to run one.

Yet, after an initial term of eighteen months, he decided to stay. He found a salaried post for three years with the Overseas Development Ministry in the city of Kumasi, about two hundred and fifty miles to the south. Then, after a year back in England, he found a post at Mtwara in Tanzania, and then another at Morogoro. It sounded like a television wildlife documentary: horses, Land Rovers, lions, zebras, and trekking in the Ngorongoro highlands.

I saw him a couple of times over these years during his brief visits home. He was now married with children, and I was busy with my life too. Letters became less frequent. He suggested I visit them in East Africa but it was never the right time.

Then we lost touch. We both moved within a short space of time and I no longer had his address. Due to a downturn in the property market, we rented out my wife’s house where we had been living, and it was ten years before we finally sold it. In emptying it we came across various papers stuffed at the back of a cupboard by tenants, including a ten year old unopened letter from Brendan.

Replying after ten years seemed pointless. Perhaps I should have tried to find him, but didn’t. Did I fear the collision of past and present? We had surely both moved on.

But, it was already too late, as I distressingly discovered yet another decade later. Out of pure curiosity, I typed his distinctive name into a genealogy web site and was shaken to find a record of his death in 2001. It took more time to find what had happened. They had returned permanently to England in the nineteen-nineties, and Brendan had died suddenly of a massive heart attack at the age of 49. He had been living less than ten miles away. All that time ago, and I had no idea.

We’ll never have that drink now.