Mathematical Patterns in Plants: Fibonacci and The Golden Ratio

By Ellie Archer

Spiral patterns abound in nature, manifesting themselves in plant petals, sunflower heads and pine cones. Remarkably, the maths behind these patterns all stems from one simple number: the golden ratio.

What is the golden ratio? Mathematically, it is the number ϕ such that ϕ = 1 + ϕ-1, approximately equal to 1.61803. Visually, if we draw a rectangle with sides in the ratio 1:ϕ, we obtain what is deemed to be the “perfect” rectangle (indeed Da Vinci used these rectangles to paint the perfect proportions for the Mona Lisa’s face). We can then construct a sequence of golden rectangles, as in the picture, inside of which we may inscribe the Fibonacci spiral, and it is precisely this spiral that we see exhibited so frequently in the natural world.

 

fibonacci

The Fibonacci Spiral (Image credit: people.rit.edu)

Perhaps the most classic example in nature is that of seed arrangement in a sunflower head. The seeds spiral out from the centre, and the problem is how to arrange the seeds to make the most efficient use of the space available. If you spiral too tightly, the seeds get squashed too close together and do not have sufficient space to mature. If you spiral too slowly, a lot of the space available isn’t utilised. It transpires that the most efficient path to take is a Fibonacci spiral.

Why is this? Imagine we are tracing a spiral by forming a seed, then rotating through a certain angle, then forming a new seed, and repeating. Suppose we make a quarter turn between forming each seed. We just end up with four straight lines of seeds, branching out from the centre in a cross shape. A similar thing occurs if we take any simple fraction (such as 5/7 or 1/31, giving seven or 31 branches respectively). Perhaps if we take an irrational number, say π, things may be better. But in actual fact π is very close to 22/7, so again we get seven branches, this time with very slight spiral behaviour.

ϕ is special because it is pretty much as far from a simple fraction as it could be.  Repeatedly applying the relation ϕ = 1 + ϕ-1, we can write it as the following infinite fraction:

This means that rotating through a fraction ϕ of a full turn avoids building up branches with empty space in between. Instead we get a pattern as shown in the sunflower, which is in fact a lot of Fibonacci spirals.

sunflower

Fibonacci spirals in sunflowers (Image credit: brilliantbotany.com)

The golden ratio is deeply connected to the Fibonacci sequence (0, 1, 1, 2, 3, 5, 8, 13, 21, …, where each term is formed by adding the previous two). Though the Fibonacci numbers were initially studied in relation to mating patterns in rabbits, it turns out that the ratio between two successive Fibonacci numbers is very close to ϕ, and indeed gets arbitrarily close as we take larger Fibonacci numbers. This means that the total number of spirals we see is usually a Fibonacci number, and if we count spirals going in opposite directions, we get two consecutive Fibonacci numbers, as shown in the pine cone below.

Pine

Fibonacci spirals in a pine cone. We count 13 and 8 spirals in each direction, both Fibonacci numbers. (Image credit: http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibnat.html)

There are countless examples of this in the plant world. Not only does this explain why three-leafed clovers are far more abundant than their four-leafed counterparts, but next time you see a daisy, take a look: chances are it’ll have 21 petals!

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2 thoughts on “Mathematical Patterns in Plants: Fibonacci and The Golden Ratio

  1. Pingback: Fibonacci | StrawberryCouture Hats

  2. I recent had a spiritual awakening, followed by a dream were the phi arch was shown to me, I had never even heard of this till this dream

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