**November 23 is Fibonacci Day!**

Who was Fibonacci and why is he relevant to biodiversity? Fibonacci was a mathematician from Pisa who lived around the year 1200 and popularised our modern number system instead of clumsy Roman numerals, having seen their advantages for accountancy. He also came up with a famous sequence of numbers which start 1,1,2,3, hence we celebrate him on 11/23.

Local Journalist Daphné Dossios recently put a series of questions about the Fibonacci sequence, and its relevance to the natural world, to the Director of the Beaty Biodiversity Museum, Dr. Quentin Cronk, for an article she was writing about the Fibonacci sequence. His verbatim answers are below.

### Q&A on the Fibonacci Sequence

#### 1) What is the Fibonacci sequence?

“The Fibonacci sequence is one of many sequences we use to describe growth. One type of sequence is the geometric sequence that we use if we wish to describe population growth: 2 rabbits make (say) 4 rabbits, which make 8, which make 16 etc. We also have a linear sequence where we add the same number again and again: 2, 4, 6, 8 etc. The Fibonacci sequence is a bit more complex. We get the next number by adding the previous two, so it goes 0, 1, 1, 2, 3, 5, 8, 13, 21, 34 etc. This is the way certain specialized things in nature grow, so it is useful.”

#### 2) In your opinion, what is interesting about this sequence?

“It is useful and interesting because it describes fairly well how certain things in nature grow. So for instance the spiral of a snail shell, which starts small, in the centre of the snail, and grows wider towards the edge of the snail. This doesn’t grow according to a geometric sequence, or a linear one, but (roughly) a Fibonacci one (in terms of the increasing area that each successive part of the spiral sweeps through). However, I say roughly, because actually, not all snail species conform to the Fibonacci sequence. There is an equation for this sort of spiral ^{(Note 1)}, of which the Fibonacci spiral is a particular case, but some species of snail do, pretty much, hit the exact spiral that comes from the Fibonacci sequence.”

#### 3) Is it relevant to talk about the Fibonacci sequence in botany? Why?

“Yes, well there are spirals in botany too, one of the best known being the overlapping spirals that you can see in the sunflower head. This is because spirals that have Fibonacci patterning, or near to the Fibonacci, result in very efficient packing of seeds in the sunflower head ^{(Note 2)}. So nature selects for efficient packing and the result is Fibonacci numbers of spirals, or something close to them.”

#### 4) Would you say that the Fibonacci sequence is omnipresent in nature?

“Well, spirals are omnipresent in nature. The precise sort of spiral found is the one that gives the organism the greatest chance of survival. Sometimes this happens to be a Fibonacci spiral (or near it). If we want to pack things tight and minimize overlapping (as in seeds in the sunflower head) one of the most efficient ways of doing this generates Fibonacci numbers of spirals. However, this is only under certain conditions. If the sunflower head is deformed or curved in any way, the numbers of spirals may drift away from the expected Fibonacci numbers.”

#### 5) What is the Golden Ratio or Divine Proportion? What is its relevance to botany?

“There is nothing particularly mysterious about this ratio. The fancy names ‘Golden Ratio’ and ‘Divine Proportion’ tend to mystify this simple number. All it is is a different way of looking at the Fibonacci sequence ^{(Note 3)}. If you divide a number in the Fibonacci sequence with its previous number, the answer is around 1.6. This is a helpful way of thinking about the Fibonacci sequence. It is useful in Botany because it gives us something called the ‘golden angle’ (137.5 degrees). We get this by dividing a circle into two segments [sectors], one being 1.6 times larger than the other. The smaller slice has an angle of 137.5 degrees. In many plants, successive leaves are produced with each 137.5 degrees displaced from the last. With this angle, no leaf is formed exactly on top of another, which is useful to avoid shading. However, this shouldn’t be taken too seriously, as the plant can’t produce leaves with that accuracy of spacing, so the actual angles tend to vary between 130 and 145 degrees, with an average of 137.5. Furthermore, there are variants of the Fibonacci sequence, such as the Lucas sequence, which gives a ‘Lucas angle’ of 99.5 degrees. This turns out to be about as good as the golden angle to avoid shading by successive leaves.”

#### 6) Is there something new to discover about the Fibonacci sequence in the field of botany?

“There is always something new to be discovered in Botany! And this certainly applies to Fibonacci patterns. A recent citizen science project examined 657 sunflowers. They found that most of the spirals came in Fibonacci numbers, a few were Fibonacci-related, but some were frankly weird, and we still don’t know what that means.”

### Notes

*Note 1.* This sort of spiral is called a *logarithmic spiral*. A logarithmic spiral is formed when something grows at an increasing rate (constant acceleration) but turns at a constant rate, i.e. grows one unit, then 2 units, then 3 units while it turns first 1 unit, then one unit, then 1 unit. A “golden spiral” is an example of this when the growth factor happens to be 1.618. If snail shells happen to match this, it is a coincidence.

*Note 2.* The left and right crossing arcs that are seen in a sunflower head are not actually Fibonacci spirals or even logarithmic spirals, although they look like it to the casual observer. However, they are patterned according to the Fibonacci sequence, as the number of left and right arcs visible in a sunflower head are usually two successive numbers in the Fibonacci sequence. This is because the golden angle (also called the Fibonacci angle) is used as the rotation between successive flowers originating in the sunflower head.

*Note 3.* A number theorist would find this answer a little bit disingenuous, as the ratio of successive terms in the Fibonacci sequence only __approximates__ the golden number (the higher the number in the sequence, the better the approximation). Actually, the golden number is 1.618033….. and it goes on without end, as it is an irrational number like pi (π). Like pi this number has great mathematical interest, but whereas pi has enormous practical applications too (not least throughout mechanical and electrical engineering), the golden number has virtually no practical applications, except as a totem in art and design.

### Further Reading

Douady, S., & Couder, Y. (1992). Phyllotaxis as a physical self-organized growth process. *Physical Review Letters*, 68(13), 2098.

Strauss, S., Lempe, J., Prusinkiewicz, P., Tsiantis, M., & Smith, R. S. (2020). Phyllotaxis: is the golden angle optimal for light capture? *New Phytologist*, 225(1), 499-510.

Swinton, J., Ochu, E., & MSI Turing’s Sunflower Consortium. (2016). Novel Fibonacci and non-Fibonacci structure in the sunflower: results of a citizen science experiment. *Royal Society Open Science*, 3(5), 160091.

### About the Author

Dr. Quentin Cronk is the Director of the Beaty Biodiversity Museum and a Professor at the Department of Botany in UBC’s Faculty of Science. Dr. Cronk has published over 140 papers on different aspects of Botany, including island biology, plant conservation, plant systematics and evolution, molecular systematics and, more recently, plant evolutionary molecular genetics. He has published several books including *The Endemic Flora of St Helena* (2000) and a book on the evolution of plant development *The Molecular Organography of Plants* (2009).