**What maths defines natural beauty? Chyi Chung dives into the spirals of sunflowers to find out.**

Sunflowers – no strangers to being muses in art – also fascinate the minds of mathematicians. Behold, heads of tightly-packed seeds, each framed by a mane of bright yellow petals. Look again, look closer and descend into their spiralling beauty.

Alan Turing, better known for codebreaking during World War II and being subject to unjust homophobic prosecution, studied the frequent phenomenon that phyllotaxis – the geometric arrangement of seeds, leaves and petals of a plant – follows the Fibonacci series. He was not the only one. Previous attempts were made by Leonardo da Vinci and J C Schoute, a Dutch contemporary of Turing who counted the spirals of 319 sunflower heads. However, Turing’s premature death by suicide from cyanide poisoning prevented him from concluding his work.

The Fibonacci series defines each subsequent term as the sum of the two preceding it, with 0 and 1 taken as the first terms. Hence, it goes: 0, 1, 1, 2, 3, 5, 8, 13… As the series progresses, the fractions of consecutive Fibonacci numbers (2/1, 3/2, 5/3, 8/5, 13/8…) tend towards the Golden Ratio, a proportion favoured by aesthetics. Spiral phyllotaxis is easily observed in nature; a common example being sunflower heads (the looser arrangement of petals makes it more of a challenge), where seeds lie in the form of spirals, going either clockwise or anti-clockwise. The number of spirals headed in each direction is known as the parastichy number. Each sunflower head has two of these, which are often found to be adjacent Fibonacci numbers. This is where mathematics unmasks the science of the everyday.

An explanation of Fibonacci phyllotaxis stems from the biochemistry of the plant itself. In 1868, the study of meristems – plant ‘stem cells’ found at its growing tips – was conducted by German botanist Wilhelm Hofmeister, whose postulations have since been validated by electron microscopic images. Auxin, the growth hormone in plants, is quickly used up at the meristem. Hence, a primordium – newly specialised cell – that forms at the meristem tends to move outward, towards higher concentrations of auxin. Primordia also repel one another, with the greatest repulsion felt between consecutive primordia. This natural phenomenon was simulated by French physicists, Douady and Couder, in 1992. They modelled primordia as a ferrofluid dropped into a dish of silicon oil, magnetised at its perimeter. Each new drop was observed to move from its predecessor relative to the Golden Ratio, inevitably forming Fibonacci spirals. In other words, Fibonacci spirals are the most natural form of phyllotaxis.

80% of sunflower heads demonstrate Fibonacci phyllotaxis

In 2012, the centenary of Turing’s birth, a group of mathematicians and scientists at Manchester University launched the ‘Turing Sunflower Project’. In a bid to continue his work, it encouraged members of the public (dubbed ‘citizen scientists’) to grow sunflowers and submit sets of parastichy numbers. It took 4 years, and 3000 sunflowers, to compile the largest dataset of its kind to date: results from this summer showed that 80% of sunflower heads demonstrate Fibonacci phyllotaxis. Interestingly, some followed other geometric progressions, like the Lucas series (which has the same definition for subsequent terms but different initial terms), and some are yet to be classified. Professor Jonathan Swinton, one of the leading researchers, believes that future study lies in “creating models that take into account the full range of patterns, including the non-Fibonacci patterns”. Such models will undoubtedly shed more light on the everyday science of sunflowers..