Chaos Theory – A gentle introduction

Question: Do snails understand mathematics?

Ok here’s an idea.

Start with the value 0 and 1 and then apply the following rule. Take the last 2 numbers you have and then add them, to make a new number and repeat.

Doing that, we get the following sequence:-

0+1 = 1
1+1 = 2,
1+2 = 3,
2+3 = 5,
3+5 = 8,
5+8 = 13,
8+13 = 21
13+21 = 34
21+34 = 55
34+55 = 89
55+89 = 144
89+144 = 233
144+233 = 377

…which will go on forever. We could express this as “The next value” F(n+1) will equal “The current value” F(n) added to “The Previous value” F(n-1).

Or: F(n+1) = F(n) + F(n-1)

We repeat this process (mathematically: we iterate) to get each new value 1,2,3,5,8 etc.

Remember this iteration idea, because we’ll use it in future posts,
which you won’t read, or is that the comments?

An interesting chap, originally from Pisa, called Fibonacci first wrote about this sequence of numbers in his book Liber Abaci in 1202, although as a sequence of numbers, it had been known about since around 200 BC. It is known as a Fibonacci Sequence.

Why did the ancients find this interesting?

Well one thing that is immediately curious is if you divide one of the numbers, by its previous number you’ll find that the value you get starts honing in on a fixed value as you go higher in the sequence and that value doesn’t change.

As an example, at the start we have 3/2 = 1.5. Around the mid point we have 21/13 which is 1.6154 just a little larger than 1.5 and further up, 144/89 is 1.618 – all getting closer to some value just above 1.6. It turns out that the higher you go in this sequence, the closer any division will come to a very specific and fixed value… which is approximately: 1.61803398875.

This number has been known about for millennia. It was christened the Golden Ratio by Luca Pacioli way back in 1509. The Greek letter Φ (Phi) is often used to represent it and even Leonardo Da Vinci liked what it brought to his paintings when he used it to proportion one connected thing to another.

The Golden Ratio is an irrational number like π, meaning that it can’t be represented by a neat fraction. The digits on this number never end.

Ok – why interesting?

  1. The peace lily plant has 3 petals
  2. Buttercups have 5
  3. Chicory have 21
  4. Daises have 34.
  5. And sunflower heads have an interesting construction of spirals. If you count from 1 near the top, observe there are 34 on my example. In fact this is a relatively young sunflower, generally these will fully mature with 55 spirals.

Spot anything interesting?

The links go far deeper.

All individuals are different, but if you average across a large number of people the following will be found.

  1. Measure accurately from the navel to the floor and the top of the head to the navel and you’ll find it will approximate the Golden Ratio. Animal bodies have the same quality (Starfish, Sea Urchins, Dolphins – fins, to dorsal to tail).
  2. On faces (human and non human) the mouth and nose will be positioned at a golden ratio of the distance between the eyes and the bottom of the chin. Tests of attractiveness across large samples concluded that the most beautiful smiles were those where the central incisors were 1.618 wider than the lateral incisors, which themselves would be 1.618 wider than canines. The golden ratio is often considered to be an indicator of reproductive fitness and health.
  3. The ratio between the length of your forearm and your upper arm will closely obey the golden ratio across large numbers of samples. Same applies to the ratio between the lower leg and thigh bone (unless horrible quacks in Turkey have broken your legs to make you taller – lolz).
  4. The total width of your two front teeth (assuming you still have them) in the upper jaw, over their height gives the golden ratio. The width of the first tooth from the centre to the second tooth also yeilds the golden ratio.
  5. Every skull has a point at the top, where fissures meet. Technically it is known as the Bregma. Likewise at the back underside of the skull there is a point known as the Inion – which is a small bump at the base of the skull. If you measure the distance from the nasal bone to the bregma, and from the bregma to the Inion – you’ll find the golden ratio.

This ratio has been spotted all over the natural world – including our heart and in our cells prompting the question is this somehow biologically special?

Fun fact… if you cut a tube in the surfing world, it means you’ve managed to balance on your board inside the crest of a rolling wave. If you’re good enough, you will have the joy of riding through the tube formed by the water, as the wave rolls.

It goes without saying that a wave is a dynamic system consisting of mostly saline water, gravity and kinetic energy. A wave has zero “understanding” of a sequence of numbers and yet if you plot the curve of the wave, you’ll find a Fibonacci spiral.

A combination of our gravity, the density and motion of the water manage to result in this shape for the curvature of the wave.

As it happens this same spiral is stamped throughout the whole of nature.

Snail shells, self construct through the building formula of DNA and form a spiral shape adhering to the Fibonacci sequence. This doesn’t occur because Snails or DNA understand Fibonacci sequencing: it occurs because over time, this specific construction provides the best chance of survival. So snails are smarter than you realise, not because they are brainy, but because over millennia they, as a group of living organisms, have searched through all possible permutations and found one offering a high chance of survival, when their house is built a certain way.

Fibonacci could be said to be built into the fabric of nature. Snails have simply managed to grasp that deep natural truth.

Chaos Theory, or more correctly complexity theory, is a discipline exploring the link between the natural world and the abstract world of mathematics and believe me, what is interesting is just how many links there are.

Featured image: Photo by Bonnie Kittle on Unsplash

© text & images unless otherwise indicated Verax Cincinnatus 2022