20 August, 2014

competitive edge

I’m not by nature a competitive person, unlike my wife. In the 1980s, Paula fenced in the men’s competitions because there wasn’t enough challenge in the ladies’. And I can still remember being whacked by her several times when I inadvertently got in her way during a game of squash. For readers who’ve never played this game, the usual practice is to call a ‘let’ and replay the point.

And then there was the first time I took her rock climbing, when she complained that the climbs I was taking her on weren’t hard enough. As I quickly found out, she likes to push herself physically; she welcomes a challenge. And I’ve been doing my utmost to provide that challenge ever since.

About six years ago, Paula decided to change the gears on her bike while I was in the UK, and she kept trying to persuade me to do the same, but I resisted on the grounds that my existing gears were adequate for my purpose. I remember warning her that if I were to change, she would regret it. And that is how things worked out. My existing six-gear system started to malfunction, and I replaced it with a nine-gear system. But the three extra gears were all higher than top gear on my old system, and as a result I was able to go much faster on flat, open sections of road. Paula struggled to keep up.

However, she finally had an opportunity to retaliate earlier this year. Riding an ordinary bike over extremely rough ground has resulted in soreness around my elbows, so Paula offered to buy me a mountain bike. She’d seen one in a bike shop in Sheung Shui, so off we went to check it out.

It isn’t obvious in the photo below, but the tyres are about 5cm wide, even though Paula claims that because the surface that contacts the ground is slightly convex, the nominal width is a mere 2.2cm. However, I don’t buy this deception, which I regard as a blatant attempt to slow me down!

My new mountain bike. I’ve since changed to a more comfortable saddle.

Nevertheless, I’ve been able to do the journey to the west a couple of times with the new bike, although I didn’t get the gear sequence right on the climb over Saddle Pass the first time, which made it harder than it needed to be. We’ve even done the ‘grand tour’ once, which adds the long and winding road and the frontier road to the journey to the west, while I’ve been on the new bike. I’ve  succeeded on the hill a couple of times too! On the other hand, I’ve yet to do Liu Pok Hill, which is an optional add-on to the frontier road, on the new bike.

Perhaps I shouldn’t be making this admission publicly, but as we approached the top of the climb away from the frontier area, I let Paula choose whether to take the easy way or to turn right to the village of Liu Pok and the gruelling category 1 hill beyond, confident that she would be too tired to take the harder option. The fact that I would be too tired too isn’t relevant, apart from being able to shirk responsibility for the decision.

Anyway, Paula is in the UK for a couple of weeks, and we’ve been doing a 32km route through the countryside almost every day. This may seem rather a short distance, but being on the eastern fringe of the Lake District, the route does include several long hills. And it is short enough that it can be repeated day after day without any intervening rest days.

I’ll conclude this account with a photo of Paula that I took a few hours ago. It was taken on a fast (i.e., slightly downhill) section through Greystoke Forest. After cresting a short rise, a panorama of the northern Pennines opens up to the east of the road. Meanwhile, I’m saying nothing about what I have in mind for the coming winter in Hong Kong, but it will involve extending the grand tour to 120km, and I will get around to doing Liu Pok Hill, despite those 5cm tyres.

Paula: “The most scenic part of the route.”

06 August, 2014

hubris

Hubris is an ancient Greek word that deserves a more frequent airing. It refers to an overblown (and usually misplaced) self-confidence, but in its original meaning those who were guilty of hubris had usually infringed some taboo or other, and they would be punished by the gods for their presumption. Prometheus, who stole fire from the gods, is a typical example.

I’ve just come across some old exam papers from my time at Manchester University, and I was instantly reminded of the most gratifying example of hubris and its consequences in my personal experience. My main subject was geology, but at the beginning of my second year, I was faced with a decision. Which subsidiary subject should I choose to take? All the usual options were on the list—chemistry, physics, botany, zoology, metallurgy—but at the bottom of the page I spotted something unexpected. I could choose to attend a course in logic.

It turned out to be a lot harder than I had anticipated, but this wasn’t due to any intrinsic difficulty. Most of the others in the class were first-year students of mathematics, and much of the course focused on propositional calculus, which is logic reduced to a form of algebra.

You can imagine what happened. The lecturer started by introducing three axioms, from which everything else could be deduced. And the mathematicians in the front row of the lecture theatre were telling him what the next line should be before I’d finished writing down the previous statement. I resented this at the time, because I was writing down stuff that I simply didn’t understand, and I struggled. Fortunately, it did make sense when I was able to read the material at my own pace, and the payback would come in a three-hour examination at the end of the course.

I couldn’t suppress a little chuckle when I picked up the exam paper, which carried the following instructions:
Answer at least ONE question from each section [there were three sections] and ONE other question. At least TWO starred questions should be attempted by mathematicians.
The unstarred questions were easy then, although they make no sense to me now, almost half a century later. Here is an example:
 13. Explain the meaning of the constant terms occurring in Aristotle’s syllogistic.
And here is a question on the same subject for the mathematicians:
*14. Lukasiewicz has suggested the following axioms for Aristotle’s syllogistic:

   ΠaAaa
   ΠaIaa
   ΠaΠbΠcCKAbcAabAac
   ΠaΠbΠcCKAbcIbaIac

   ‘A’ and ‘I’ being primitive terms. Show that the axioms are independent of each other.
The second question seemed difficult then, but it would be impossible for me to answer now. And it is possible that the mathematicians at whom the question was aimed in the first place would have found it easy then. Still, I think that I was entitled to a little schadenfreude. It’s what should happen to people who are too cocky.

31 July, 2014

the mathematics of nature

In The Hitchhiker’s Guide to the Galaxy, Douglas Adams famously wrote that the answer to “the Ultimate Question of Life, the Universe, and Everything” is 42. He was wrong, of course. The correct answer is 1.618034 (to six decimal places), a number that is usually represented by the Greek letter Φ.

From whence did I dig up this number? The Greeks knew all about what they defined as the problem of dividing a line segment ‘in mean and extreme ratio’, a problem that is illustrated by the following diagram:
In this diagram, if point C is chosen so that the ratio AC/BC is equal to the ratio AB/AC, then both ratios are equal to Φ. This is the answer to that ancient Greek problem and is now most commonly referred to as the ‘golden ratio’. However, although Φ is usually referred to as a ratio, it is, like its cousin π, the ratio of the circumference of a circle to its diameter, irrational. This means that it is impossible to express as the ratio of two whole numbers, which in turn means that its decimal expansion continues to infinity without ever repeating.

It might seem that this an esoteric concept with little or no relevance to the real world, but if I bend the line AB through 90 degrees at C and add two extra lines to produce a rectangle, then things become more interesting. This so-called ‘golden rectangle’ has sides that are in the golden ratio, and it can be subdivided into a square and a smaller golden rectangle, and so on until the resulting square and rectangle are too small to measure. If I then inscribe a quarter-circle inside each square, the result is a very good approximation to a logarithmic spiral:


This is the same spiral that can be seen in the shells of snails and other gastropods:


The golden ratio is inextricably bound up with the geometry of the circle, as the following two diagrams demonstrate:


In the first diagram, an equilateral triangle has been constructed inside a circle. A and B are located at the mid-points of two of the sides, and BC is an extension of the line AB to the perimeter of the circle. The second diagram shows a regular pentagon and three of its possible five diagonals. In both cases, AB/BC = AC/AB = Φ. Note too that if the two missing diagonals are added to the second diagram, the result is a five-pointed star, or pentagram, which also includes the golden ratio in its properties.

In addition to producing geometric representations of the golden ratio, there are several ways to calculate its numerical value. Perhaps the most bizarre and unexpected involves taking two random numbers and constructing an arithmetic series according to the following rule: starting with the two random numbers, each new term is calculated by adding together the two previous numbers in the series. I’ve chosen the numbers 5 and 7 to begin with.
5, 7, 12, 19, 31, 50, 81, 131, 212, 343, 555, 898, 1453, 2351, 3804, 6155, 9959, …
The next step is to divide each number in the series by the previous number. I’ve split these calculations into two columns, for reasons that will quickly become obvious:


You will notice that this series of calculations starts with one value that is significantly lower than Φ and a second that is significantly higher, but in the next seven iterations both values gradually converge on the number that I quoted at the start of this account. However, this process can be continued ad infinitum, bearing in mind that only to the extent that the decimal places are the same in both columns can the results be regarded as accurate.

This process may seem no more than an intellectual curiosity, but there is one version of the arithmetic series constructed above that has wide significance in nature. That is the so-called Fibonacci series, named after Leonardo of Pisa (c. 1170–1250), which starts with the digits 0 and 1:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, …
The special significance of Fibonacci numbers can be glimpsed in the following photograph:


This is the central part of a sunflower. The seeds have arranged themselves in radiating spirals, with one set of spirals curving in one direction and a second set curving in the opposite direction. The surprising thing is that the number of spirals in each direction in any sunflower is always two consecutive Fibonacci numbers. Pine cones and pineapples are also constructed as two sets of opposing spirals, and it turns out that this happens because it produces the optimum packing arrangement in three-dimensional space.

Fibonacci numbers reappear in the petals of flowers. For example, lilies have three petals, buttercups five and daisies twenty-one. However, this is not an infallible rule: hydrangeas have four, six or ten petals, and other nonstandard examples include poppies and primroses.

The golden rectangle is often touted as offering the most aesthetically pleasing dimensions for a painting, and Renaissance artists such as Michelangelo, Leonardo da Vinci and Botticelli employed it extensively. The logo of Japanese carmaker Toyota can be enclosed within a golden rectangle.

The golden ratio has also been spotted in famous buildings such as the Taj Mahal, the Parthenon and the Great Pyramid of Khufu, although given that in the latter two cases significant parts of the original building are now missing, some wishful thinking may be involved. People who have identified the logarithmic spiral in the arms of spiral galaxies and the circulation of tropical cyclones may be guilty of the same offence.

That was my first impression. After all, attempts to see the golden ratio in human physiognomy seem to me to be a step too far. However, if the natural path of a seed in a flowerhead is outwards along a logarithmic spiral, and calcium carbonate molecules behave in the same way in the shell of a snail, then why shouldn’t atoms in a spiral galaxy follow the same rules?

And it is quite reassuring to think that everything is governed by a single number.

21 July, 2014

qualified success?

What is the purpose of a school? One should perhaps make a distinction between state-run and independent schools, but if considering the former you can award yourself a bonus point if you would have said that the principal function of a school is to facilitate the nationwide screening system that weeds out unsuitable candidates at strategic intervals. Education, if it takes place at all in a typical state school, is an incidental by-product.

Two years ago, then UK Education Secretary Michael Gove announced plans to abolish the current General Certificate of Secondary Education (GCSE), examinations for which are sat at the end of Year 11 (age 16) and which constitute the first of the sieves alluded to in the previous paragraph, and replace it with a ‘more rigorous’ General Certificate of Education (GCE) ordinary level (O level). It will not have escaped the notice of those who follow the English education system that O levels, which had been introduced in 1951, had been scrapped by a previous Conservative government, in the 1980s.

Part of Gove’s motivation was his desire to reintroduce a ‘world-class qualification’, which should stand as a reminder that politicians, who invariably think they know more about education than the professionals, can have a dangerously destabilizing influence on a country’s education system. Qualification? The BBC’s education correspondent used the same term, but passing an O-level exam is not any kind of qualification, although you will get a certificate that records your success.

Looking at Gove’s other initiatives, it is clear that he has no understanding of education as an organic process and is motivated almost entirely by ideology. His introduction of academies and free schools, which are beyond the control of local education authorities, have total control of their curricula and are allowed to employ unqualified teachers, is clear evidence of that, although these schools are subjected to scrutiny by the Office for Standards in Education (Ofsted).

Unfortunately, the schools landscape has changed considerably in England during my lifetime, invariably at the behest of politicians and rarely for the better. The Eleven Plus examination, which was used to decide whether a child went to a grammar school or to a secondary modern school, was introduced following the 1944 Education Act. However, both the exam and the grammar schools it fed were seen by the political Left as divisive and elitist, and the exam was abolished in most areas between 1964 and 1979, and most grammar schools were either closed or converted to comprehensive schools, which as the name suggests were intended to provide for the educational needs of all children, although I’ve always suspected that the architects of this policy were more interested in pegging back the brightest children by a notch or two rather than in raising the level that could be achieved by the least able.

Secondary modern schools were a new concept for the postwar period, the idea being to provide a technical education that purported to meet the needs of those children who were not academically inclined. The reality is that many became mere dumping grounds for those who didn’t make the grade, they were starved of funds, and they were regarded as providing a second-class education, especially by those who did make the grade.

I am not suggesting that there is no such thing as a ‘qualification’—the world would be a dangerous place if teachers, doctors, lawyers and accountants were able to practise their professions without a long and rigorous training period beforehand—but the notion that an O-level pass in English literature is any kind of ‘qualification’ is risible in the extreme, although a good mark in the A-level (advanced-level) equivalent will entitle you to apply to study the subject at university.

Another area where some form of certification is desirable is in skilled manual work: no one would employ a cowboy builder, plumber or electrician, you would think, although I regularly see examples of shoddy workmanship by the jacks of all trades (and masters of none) who build village houses in Hong Kong. The photograph below is of a wall that I watched being built a few years ago and is entirely typical. As you might guess from the undulations in the lines of bricks, no spirit level was used, and the structure consists of a single thickness of bricks with no interlocking, so its structural soundness is doubtful. A qualified bricklayer, in practice someone who has served a lengthy apprenticeship, could have been relied on to do a far better job, although his services would probably have been considered too expensive.


You will have noted that in the last example I suggested that experience is an adequate replacement for a piece of paper stating that you’re ‘qualified’. I can reinforce this point from my own experience: a few years ago, I learned about a course that one could enroll on to become a ‘qualified’ proofreader. I believe that it isn’t possible to teach someone to be a competent proofreader unless they already have some natural aptitude. But what’s so difficult about proofreading? Surely one only needs to be good at spotting mistakes.

However, the hardest part of proofreading lies not in spotting errors but in deciding whether or not the errors you have spotted should be flagged up. This will probably sound strange to anyone who has never tried proofreading, but if a mistake is the fault of the copy editor, then the publisher will have to pay for it to be corrected. Mistakes are rectified free of charge only if they were the typesetter’s fault, so the proofreader has to decide who to blame, and if it was the editor, whether the error is sufficiently trivial that it can be ignored.

But here’s the rub: although I’ve proofread hundreds of books, I’m totally unqualified for the task, if by ‘unqualified’ I mean that I don’t have a certificate of competence. On the other hand, given the way publishing is organized, a freelancer can get more work only if they can be relied upon to do a good job, and I was never short of work, so I take this as evidence that experience is a better indicator of ability than a piece of paper.

Since I started this essay, Michael Gove has been replaced as education secretary. Although it is too early to assess whether this change will have a significant impact on the education sector, it is reasonable to assume that we will hear more railing against that traditional right-wing bête noir, ‘trendy teaching methods’, using the now familiar buzzwords ‘rigour’ and ‘standards’, and a perpetuation of the myth that it is possible to learn more about a person’s intellectual capabilities from their answer to a question in a three-hour, sit-down exam than from their answer to the same question if given a week to write 5,000 words on the subject drawing on whatever documentary sources they deem necessary to construct their argument.

It is more important, and a more valuable skill, to be able to construct a rational argument than it is to memorize the information that supports that argument. However, helping pupils to develop such a skill really would be education, and both the present Conservative government and the last Labour administration have been more interested in a rubber-stamping process of training. And training is not education.