Ěý
The golden ratio
The most beautiful number ever?
YOU have probably heard of the Fibonacci sequence, that list of numbers where the next digit is given by adding the previous two. It goes 1, 1, 2, 3, 5, 8, 13 and so on. But here’s something strange: work out the ratio of each number and its predecessor, and you start edging towards a specific number. Its first few digits are 1.618.
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This mysterious beast is the golden ratio, and it crops up a lot. Try drawing a diagonal line connecting two vertices of a regular pentagon. Divide the length of that line by the length of the pentagon’s sides and there it is. Something similar is possible with an equilateral triangle.
Read more: Fantastic figures – A tour of the most wonderful numbers
The world of numbers may be abstract, but it contains incredible sights. Come explore the many wonders in numberland
It turns out to be a quirk of maths. Imagine you have a number, A, and a larger one, B. If you set the numbers so that the ratio of B to A is the same as A + B to B, then that ratio is always the golden ratio.
That might have been the end of the matter, but the ratio has taken on a life of its own. Search for it online and you’ll be inundated with claims that ancient Greek architecture or the human face exhibit such proportions, and that people find it immensely aesthetically pleasing.
The truth is murkier. The human body has countless different proportions, and some of them seem to be close to the golden ratio, but not for everyone. The ancient Greek architects were aware of the golden ratio, so it is possible that they made use of it. To find out, just measure the ruins, you might say. But then there’s the question of which bits you measure – look hard enough and you will find the ratio if you want to.
A similar problem plagues studies that ask people to rate the aesthetics of artworks that incorporate the golden ratio and others that don’t. It’s not clear whether that judgement is really based on the ratio, or whether the association is learned or innate. Luckily, maths contains beauty enough without magic ratios.
Timothy Revell
Ěý
The Lyapunov exponent
The boundary of chaos
AND the weather on Tuesday will be exponential errors, followed by a loss in predictability. Because of Lyapunov exponents, it is impossible to accurately forecast the weather more than a few days ahead. Instead of predictability, there is chaos.
In the late 19th century, the Russian mathematician Aleksandr Lyapunov invented these numbers to describe how sensitive a system is to its starting point. Imagine, for example, throwing a ball across a field. Provided you know the angle and speed at launch, you can calculate where the ball will land to a good degree of accuracy without worrying about small effects like air resitance. If your measurements of the angle are a bit off, that doesn’t matter either. This situation would have a Lyapunov exponent of 0, or perhaps a negative value.
What does maths sound like?
Above that threshold of zero lies unpredictability. The weather is a case in point because tiny differences in starting conditions, such as in air pressure or temperature, grow exponentially over time to cause wildly different outcomes. If throwing a ball were like this, a launch angle of 30 degrees might arrive at catching height for your friend, while an angle of 30.00000001 degrees might land the ball on the moon. Mathematicians call this chaos.
Positive Lyapunov exponents make long-term weather forecasts impossible. As we can never measure wind speed, say, with total accuracy, an initial, barely noticeable error will grow so that in only a few days the forecast will be mostly error. In countries like the UK, where air currents are highly changeable, the Lyapunov exponent of the weather is much higher than in the tropics.
“We cannot predict the future. Any little uncertainty gets amplified exponentially by chaos,” says Francesco Ginelli at the University of Aberdeen, UK. Whether it is predicting the weather, the stock markets or the next president, Lyapunov exponents tell us our efforts are futile. But experience tells us we’re unlikely to stop trying.
Timothy Revell
Ěý
The Laplace limit
Why we don’t stray far from the sun
IN 1609 the great astronomer Johannes Kepler published a book called Astronomia Nova. This “new astronomy” delivered a bombshell: planets revolve around the sun in ellipses, not circles. But the equation at the heart of the revelations, Kepler’s equation, had astronomers’ heads spinning faster than the planets they were studying.
The formula describes the relationship between the coordinates of an object in orbit and the time elapsed from an arbitrary starting point. Actually solving it to find that location is fiendishly tricky. But eventually, astronomers saw it pointed to a particular number.
It took 150 years to find a mathematical way to solve it. The laborious process involved long strings of mathematics known as series expansions. Then the French polymath Pierre-Simon Laplace showed that this method would not work if the orbit was too elliptical.
“Kepler’s equation had astronomers’ heads spinning faster than the planets they were studying”
You can measure how far removed an ellipse is from a circle with a measure called eccentricity. A circle has an eccentricity of 0, and for values greater than that things become more skewed. Laplace found is that for orbits with an eccentricity of more than about 0.66 – now known as the Laplace limit – the method would not converge on a solution.
This means that “in general, orbits are less stable if the eccentricity is higher,” says Gongjie Li of Harvard University. Fortunately, Earth’s orbital eccentricity is about 0.02. Bodies farther out often have higher eccentricities. Pluto’s is 0.25.
This doesn’t mean that orbits with an eccentricity of more than 0.66 are impossible. Halley’s comet has an eccentricity of 0.9. But that’s best thought of as a fly-by rather than an orbit, really. The comet’s swinging loop brings it close to the sun, then catapults it into the coldest reaches of the solar system. Not a place we’d want to be.
Stuart Clark
This article appeared in print under the headline “Wonders ofĚýnumberland”
