Saturday, 30 August 2014

twenty miles of bad road

Whenever I’m in my home town in the UK, I have a choice of three cycling routes that I follow regularly, ranging in distance from 9.6 to 27.3 miles. Unfortunately, this summer a crucial section of road that is shared by all three routes has been closed to allow a new sewerage system to be installed, so I’ve had to find an alternative. This is the story of that new route, which at 20 miles is short enough that it can be repeated daily, weather permitting, without the need for intervening rest days.

The new route is shown on the map below as a series of red dots, which are followed in a clockwise direction. The last four miles are the same as the first four miles, except that they are ridden in the opposite direction.

Paula has been over in the UK for the last couple of weeks, and we did the route on each of the first three days, but then we had to find an alternative because the road between Skelton and Blencow was closed for repairs—the road surface on this section was so bad that it was often necessary to ride down the middle of the road to avoid the worst of the potholes.

Instead of turning right at Ellonby (the small hamlet between Lamonby and Skelton, which is not named on the map), we turned left, up yet another hill, through Lamonby, over another hill (see photo below), followed by a final climb through Greystoke Forest and a long, fast section to Johnby and Greystoke. We finally rejoined the original route at Blencow, having added 3.2 miles to the overall distance.

The first point of interest on the original route comes at the point marked X on the map: I feel as if I’m riding downhill in both directions (the first photo below was taken on the outward journey, while the second photo was taken on the return). Note that this road, like many other sections of the route, is just wide enough for a bike and an oncoming car to pass each other.

The first major hill starts immediately that the route turns right just before the village of Newbiggin, and the next photo was taken at the top. A deer crossed the road in front of me here earlier in the summer, but I’ve seen no further signs of cervine activity either here or elsewhere on the route.

The road then plunges down a steep hill, at the bottom of which a right turn leads onto another narrow road. The purplish mass on the right-hand side of this road (see photo) is a huge stand of rosebay willow herb. Stands of this opportunist weed are common on the sides of roads around here and are a spectacular sight in early summer.

The next uphill section begins in the picturesque village of Greystoke. The gradients are easy at first, but the last section to Johnby is quite arduous. The next photo shows Paula posing at the top of this final hill, which starts well before what appears to be the bottom in the picture. I should probably confess that we headed down the hill after I’d taken this photo, although we had passed this way in the uphill direction 30 minutes earlier.

The section from Johnby to Ellonby is relatively straightforward, except for the last half-mile, which provides a winding stretch of continuous but not unduly taxing uphill work. The shortest way back to Penrith from Ellonby is almost entirely downhill, but as noted above, a detour through Lamonby and Johnby adds a few more hills to the excursion. The next photo shows Paula approaching the first hill southwest of Lamonby.

We did the 20-mile circuit and the augmented version three times each in the first week, but then I thought that Paula needed a tougher challenge, so we followed the 23.2-mile circuit as far as the left turn towards Johnby, which we ignored. Instead, we continued towards Hutton Roof, turning left before we reached that village and heading towards Berrier.

However, this detour is not about adding extra distance, as the next photo illustrates. I’d stopped to take some photos of the fells to our right, meanwhile urging Paula to keep going (I would catch her up). Having taken the photos I wanted, I continued on my way but stopped again when I saw a splendid opportunity to capture the obstacle that lay in our path. This hill is by far the toughest on any of my regular routes (Paula can be seen as a tiny speck just starting the hill).

The second time we did this route, I decided that I wanted some photos of Paula tackling this hill, which is steepest near the top. Despite facing a 15–20mph headwind, she does look comfortable. The mountain behind is Carrock Fell, which is the only example of a gabbro intrusion in England (gabbro is chemically identical to basalt, but it cools deep below the surface rather than being extruded from a volcano).

Finally, I couldn’t resist including the following picture, taken at the top of a second hill, which follows the one I’ve just described but is much easier.

“Can’t we do something harder?” Paula seems to be saying.

Wednesday, 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.”

Wednesday, 6 August 2014


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:


   ‘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.

Friday, 1 August 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.

Pine cones also display the double spiral.

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.