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Rob Eastaway - Joined Up Maths: Real and Contrived


This page has been archived. The content was correct at the time of original publication, but is no longer updated.
Created on 03 October 2008 by ncetm_administrator
Updated on 24 October 2008 by ncetm_administrator

In the first article, I described a truly awful and contrived puzzle that I contributed early on in my career with New Scientist. However, I wasn’t the first person to come up with a deeply contrived puzzle. One of the most notorious first came to light around 800 years ago, when Leonardo of Pisa (later to be known as Fibonacci) included a question about rabbits in his seminal work Liber Abaci.

As most mathematicians are aware, Fibonacci included a problem in which a pair of rabbits give birth to a mixed pair of rabbits after two months, and then again every month after that. The offspring follow the same pattern, each pair producing exactly one pair per month starting two months after they are born. The number of pairs of rabbits each month emerges as the famous sequence 1, 1, 2, 3, 5, 8…, and Fibonacci asked how many pairs of rabbits there would be after twelve months.

The model that Fibonacci used bears only the vaguest resemblance to the real behaviour of rabbits. Not only are litters generally much larger than two, but males are kicked out before they breed to avoid the chance of in-breeding. Rabbits do also have a tendency to die. This does make one wonder why Fibonacci came up with this puzzle in the first place. Did he come up with this artificial breeding pattern by chance, only for later generations to discover the remarkable properties of the ensuing number sequence? Or did Fibonacci already realise that his sequence was particularly significant, and he invented the rabbit problem as a means of revealing it? If the latter, then why did he not go on to reveal the significance of the numbers?

It is a historical mystery, but such mysteries simply add to the fascination of mathematics. One way in which maths “joins up” is through its rich history. The story of how mathematical ideas developed is like a long and intricate detective story. I find it disappointing that so little of this history is taught in school, since it could do so much to bring to life some the mathematics that we use today.

The story of the Fibonacci sequence is a particularly interesting one, because it also taps into the world of art, beauty and what I call ‘pseudo-beauty’, much of which centres on the so-called golden ratio.

I expect most reading this article will be very familiar with the relationship between successive terms in the Fibonacci sequence.

1   1   2   3   5   8   13
                         
  1   2   1.5   1.66   1.6   1.625...  

As one moves along the sequence, the ratio of successive terms converges to the number (1 + sqr(5)) / 2, roughly 1.618. This ratio, these days referred to by the letter phi, was known to the Ancient Greeks because of its appearance in some familiar basic geometry, including the pentagon. Yet remarkably, it was not until 400 years after Fibonacci’s death that Johannes Kepler first explicitly pointed out the profound connection between phi and the Fibonacci sequence.  (There are some who later nicknamed the sequence the Phi-bonacci sequence as a result.)

The discovery of this connection reinforced a belief, popularised by Leonardo da Vinci, that the golden ratio was the basis of the most beautiful forms in nature and in architecture. If you believe the popular myth, then if you measure the distances between various parts of your anatomy, you should find the ratio 1.62 cropping up all over the place. The truth, however, is that only very selective measurement produces the number 1.62. (And it’s enough to make a guy feel very insecure.) Many other so-called connections between the golden ratio and beautiful objects, such as its appearance in The Acropolis, also turn out to be spurious. It’s a case of fitting nature to the pattern rather than the other way around: pseudo beauty.

The real beauty of Fibonacci and the golden ratio is in its mathematical properties, and there is nothing fake about these. There are societies devoted to unearthing these exotic properties, but for me the most beautiful of all is one of the simplest. It is the connection between the Fibonacci sequence and the so-called Lucas sequence, which is generated using the same rules as Fibonacci, but starting with the terms 1, 3 instead of 1, 1.

Fibonacci 1 1 2 3 5 8 13 21 34...
                   
Lucas 1 3 4 7 11 18 29 47 76...

Three particularly intriguing properties connect these sequences. The first is that, apart from the numbers that appear in their first two terms, 1 and 3, the two sequences have no other numbers in common.

The second is that the ratio between successive terms in the Lucas sequence – and indeed all sequences of this form – also converges on the golden ratio.

And the third is that the ratio between each Lucas term and its corresponding Fibonacci term converges on the square root of 5.

These relationships have a genuine connection to many numbers that appear in plants and other natural forms. This, to me, is real beauty. And the discovery of these properties, and the story of how they were found, give a true sense of how maths can join up.

We live in an era where another form of joining up has also become important. Increasingly there are pressures to make maths more ‘relevant’, by demonstrating its connections with the real world. As an applier of mathematics, this should be right up my street. And it is true, I do enjoy finding connections between mathematical ideas and everyday life.

For example, Keele [where the joint conference took place] is close to the world centre of darts, Stoke on Trent, home of several champion players including Phil ‘The Power’ Taylor. Darts scoring is an obvious application of numeracy, but there’s some good maths there too, particularly around the question of where you should aim on a dartboard.

Most men, fancying themselves as potential world champions, copy the professionals and aim for treble 20 each time. What we fail to recognise is that the trajectory of our darts is prone to variability. The neighbours of the 20 segment are 1 and 5, not good scores. So if one’s darts are prone to slight variation in direction, it can be more effective to aim at treble 19 than treble 20, because of the scores in the neighbouring segments. An even shakier player is better placed aiming at treble 14.  Of course, if your dart playing is completely hopeless, you should simply aim at the bullseye. That way, you do at least maximise your chance of hitting the board.

Working out the optimal place to aim a dart requires mathematical modelling. That modelling combines geometry and probability – an unlikely pairing of two disciplines that rarely coincide in the curriculum. Indeed, there are many pupils who leave school with an A level in maths who would be unable to model the maths of a dartboard because of a lack of knowledge of one of the two strands. It’s an example of how maths in the real world (and in the pure world too) has a habit of joining up across topics, and why the subject needs to be taught in a joined up way.

On the other hand, there is a trend to attempt to join up maths to the real world at every possible opportunity. Just because maths CAN be used to model the world doesn’t mean that it always should be. At its most ludicrous, this manifests itself in PR stories that arise frequently in the press, in which an academic – typically sponsored by a company looking to promote a product – comes up with a ‘formula’ to improve performance in an everyday activity. My favourite formula was this one:
K = [F x (T + C) – l] / S

That, apparently, is the formula for a perfect kiss. All I can say is whatever you do, don’t leave out the brackets.

The risk of attempting to apply maths to everything is that it invites a backlash. Far from conveying a connection between maths and the real world, it confirms the view of some people that mathematicians are out of touch with reality.

There is increasing government support for initiatives under the umbrella ‘STEM’. This is a joining up of Science, Technology, Engineering and Mathematics. These subjects do indeed have many connections, and there is a case for integrating initiatives so that funding of these subjects is properly directed. I do, however, have two concerns. The first is that M comes last in STEM. It’s there because it helps to make a nice acronym, but perhaps inadvertently it does also send out the subliminal message that maths is the afterthought, the least important. The truth is, of course, that S, T & E depend far more on mathematics than Maths depends on them. I’d rather the acronym were METS.

My second concern is that if maths gets bundled together with the science subjects, its wider significance gets forgotten. Yes, maths is joined to science. But it is also joined to art, to music, to language, and it is also a pure discipline in its own right. STEM is the antithesis of what G.H.Hardy stood for, and a balance is needed.


To read Rob's first article, Maths that connects to everyone, and to read more about Rob and the background to these articles, click here.

To read Rob's third article, Joined Up Teachers, click here.


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