Sunflower seeds and square roots

Admirers of the Greek mathematician Pythagoras, who look for proof of his theories in constellations and even Tunisia's Great Mosque, have turned their attention to horticulture. Robert Matthews writes

Best known today for his theorem about right-angled triangles, the Greek mathematician Pythagoras was also the founder of a curious cult which believed that "all is number".

To those who see maths as no more than a way of totting up restaurant bills, this can sound pretty crazy. Yet its followers insisted numerological miracles could be found in all kinds of unexpected places.

Pythagoras is said to have found the mathematical underpinnings of music after noting that the harmonious sounds from a foundry were the result of the anvils having sizes in simple ratios to one another.

The Pythagoreans would doubtless have been delighted by last week's launch of a worldwide project to confirm the magical numbers lurking in, of all things, sunflowers.

Dreamt up by Jonathan Swinton, a biologist at Oxford University, the project aims to get to the bottom of claims that the heads of sunflowers contain a mathematical pattern.

It has long been known that the stems, petals and flowers of many plants exhibit a curious, spiral-shaped regularity. In the case of the sunflower, the seeds in the head are arranged in interlocking spirals swirling both clockwise and anticlockwise.

For years botanists have claimed that only certain "magical" numbers of these two types of spiral tend to occur - say, 21 spirals going clockwise and 34 anticlockwise, or 34 clockwise and 55 anticlockwise.

What makes these numbers so special is that they belong to a precise mathematical sequence - 0, 1, 1, 2, 3, 5, 8, 13, 21, 34 and so on - with each term being formed by adding together its two predecessors (0+1=1, 1+1=2, 1+2=3, etc).

This is the Fibonacci Series, named after the 13th-century Italian scholar who publicised the work of Indian mathematicians.

How plants could "know" about this esoteric sequence has long baffled scientists. Perhaps the most credible explanation is that explored in detail 20 years ago by two French mathematicians, Stephane Douady and Yves Couder, of the Laboratory of Statistical Physics in Paris.

Roughly speaking, they showed that if the number of spirals follows the Fibonacci sequence, then they can be packed around each other most efficiently, thus allowing the most seeds possible to be crammed on to the flower's head.

No one has been sure just how consistent this pattern is - and that is what Prof Swinton hopes to discover.

He and a team at Manchester University want people to grow sunflowers from seed and count the relative numbers of clockwise and anticlockwise swirls in the flower heads. Statistics in hand, they will put the various explanations to the test.

Chances are they will confirm the existence of at least some of the Fibonacci numbers. That alone would delight any latter-day Pythagorean as another manifestation of the mystical number Phi.

Better known as the Golden Mean and equal to about 1.61803, Phi has been found in all kinds of places, from the proportions of buildings such as the Great Mosque of Kairouan in Tunisia to the star-like pattern in a sliced apple.

And it's hiding in the Fibonacci sequence as well: as it continues, the ratio of any two successive terms in that series gets ever closer to the Golden Mean. So, for example, a flower with 21 spirals going clockwise and 34 anticlockwise gives a ratio of 1.61905. One with 34 clockwise and 55 anticlockwise spirals gives 1.6177.

Sometimes the explanation is relatively obvious. In the case of the sliced apple, the pattern is pentagonal, and the geometry of this five-sided polygon naturally leads to the appearance of the Golden Mean.

Yet sometimes the appearance of "magical" numbers in surprising places is more telling. Take the Great Pyramid of Giza, the relative dimensions of which have long been known to contain an amazingly accurate value for Pi, the ratio of the diameter of a circle to its circumference.

The pyramid has a perimeter of 920m, and a height of 146.5m. Dividing half of the former by the latter gives 3.14 - just 0.05 per cent away from the precise value of Pi.

This discovery spawned a rash of theories about the esoteric mathematical knowledge of its builders. A more down-to-earth suggestion is that it reflects the kind of tools used by ancient Egyptian surveyors.

It's plausible that they used a wheel-like device to mark out the perimeter before starting work. If the radius of the wheel was set equal to the unit of length used to measure heights, and the pyramid's perimeter was specified by a given number of turns of the wheel, a very accurate value of Pi would automatically be built into the final design.

All of this suggests the Pythagoreans were on to something, with the presence of magical numbers in natural phenomena pointing to some underlying law or regularity.

But that misses perhaps the most surprising thing about these numbers: that they emerge out of complete randomness.

For example, Georges-Louis Leclerc, an 18th-century French mathematician, showed that if a needle is dropped randomly on to a wooden floor, the chances that it will lie across the gaps between the floorboards depends on Pi.

Indeed, if you have the patience to do it a few thousand times, the orientation of the needle can be used to work out a fairly accurate value of Pi.

It has even been shown that the random scattering of the stars in the night sky can be used to generate a value of Pi that is just 0.06 per cent off its true value.

Whether sunflowers do contain the Golden Mean remains to be seen, but the Pythagorean view that all is number gives a whole new meaning to Pi in the sky.

Robert Matthews is visiting reader in science at Aston University, Birmingham, England