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  Last updated May 29, 2017 at 12:26 pm

I have to confess, I’m hopeless at maths. I wish I weren’t because the bits I do understand reveal a beauty in the natural world that can only be expressed, …well, mathematically. But I can appreciate that this elegance in maths that has lead to some excesses and what can only be described as mathematical quackery. And I abhor quackery in all its forms as intensely as I adore the order of the natural world.

One classic example from within the little bit of maths that I do understand and its attendant abuse in mystical and mythical thinking is the Fibonacci Sequence and the Golden Rule. All too often these mathematical expressions are both invoked in the real world where they don’t exist and underappreciated where they do have an expression in reality.

Let’s start with the Fibonacci Sequence.

The Fibonacci Sequence

Named after the Italian mathematician, the Fibonacci Sequence is where a number is the sum of the previous two numbers in the sequence. So 1, 1, 2, 3, 5, 8, 13, 21 and so on.

One of the interesting features of the Fibonacci Sequence is that the ratio of any two successive numbers in the sequence approximates the irrational number Phi.  As you use sequentially higher numbers in the sequence, you get better approximations of Phi and each one will alternate between being a little bit above, then a little bit below the value you’re after.  Thus 3 divided by 2 equals 1.5 (under), 5 by 3 is 1.66666… (over), 8 by 5 is 1.6 (under), 13 by 8 equals 1.625 (over) and so on.  Phi to the first dozen or so digits is 1.618033988749894… and, being irrational, just keeps going on forever like it’s more famous cousin Pi. If you keep dividing successive Fibonacci numbers, you will never get to Phi but you will keep getting closer and closer with the over and under pattern all the way.

(Image source)

See what I mean? That is elegant! A simple, yet complex pattern that arises out of some straight forward maths. And it gives rise to a whole swag of mathematical phenomena with golden titles, The Golden Ratio, Golden Rectangle, Golden Spiral and The Golden Rule.

Golden Maths

The concept of the Golden Ratio goes back to the ancient Greeks.  There is one solution[1] where you can divide a line into a and b so that the total length (a+b) divided by the long part (a) is the same to a divided by b:

And that unique solution, dividing a by b gives a ratio of 1.6180339887…, where have we seen that number before? It’s Phi! And here is where the quackery starts to creep in.

There have been claims that the Golden Ratio occurs throughout nature and human design. One proposed example is the joints on human fingers. The claim is that the length of successive joints on human fingers conform to the Golden Ratio. Well, I’ve just measured the four fingers of my right hand (I’m left handed) and none show the ratio 1.618… and even the average (1.584) is different.

Similarly, there are claims that segments through a dolphin, the ratio of the length of the human forearm to the hand, the proportions of the cochlear and many other measurements all conform to the Golden Ratio and give values of Phi when divided. In short, they don’t.

And don’t think that these fanciful ideas are some product of the past.  In 2012 a Belgian gynaecologist claimed that the proportions of human female reproductive organs conformed to the Golden Ratio, claims he went on to publish in the peer-reviewed journal Ultrasound Obstetrics and Gynaecology. He used ultrasounds to measure the reproductive organs of 5,000 women to back up these claims.

However, Verguts had to exclude older and younger women from the data set because reproductive organs change shape over time, and this should have sounded some alarm bells about data manipulation.  Verguts claim then became that, for women between the ages of 16 and 20, the average dimensions of their reproductive organs came out at 1.6 – which he thought was close enough to the Golden Ratio to hold some meaning.

Perhaps it’s the evolutionary biologist in me that asks “what meaning?”.  Let’s put aside the data manipulation and ask “what possible benefit could there be in the dimensions of a woman’s reproductive apparatus conforming to the Golden Ratio?”  It can’t be one of aesthetics – these are all internal organs excluded from view – and natural selection doesn’t select for aesthetics anyway.  Perhaps they are supposed to perform better because of these dimensions?  I fail to see why or how evolutionary processes would produce human female internal anatomy to conform to the Golden Ratio, there is no selective advantage.

This whole study seems to be nothing more than folly.  It has all the hallmarks of bad science done backwards: decide on a conclusion then go looking for supporting data and cherry-pick your way through the data you collect to get the support that you want for your original thesis.

Golden Rectangles

Let’s move along with the maths and have a look at the Golden Rectangle. If you take that line segment a and turn that into a square, you end up with a rectangle:

The dimensions of this rectangle are 5:3 (5 being a+b, 3 being a). Did you see that? 5 and 3 are successive numbers in the Fibonacci Sequence! But let’s leave that for a minute and have a look at where people have been seeing Golden Rectangles popping up in the real world.

In human aesthetics, the Golden Rectangle is supposed to be particularly pleasing to the eye and devotees have gone off looking for it in art, sculpture and architecture.  But, once again, it remains elusive.

While it has been claimed that the proportions of the Parthenon conform to the Golden Rectangle, it depends on where you define the various lengths (the base of the plinth or the base of the columns? The height to the top of the columns or to the top of the eaves? etc).  Yes, you can get numbers approximating 1.6 but no exact measurements of a ratio accurate to more than a couple of digits. The same with the UN building in New York and the Grand Mosque of Kairouan.

Golden Rectangles are claimed to have been found in the face of the Mona Lisa but again, it all comes down to the definition of the beginning and end of the lines and no consistent measurements have borne this out.

Contrary to the claims of some, the canvasses used by the great masters rarely if ever are Golden Rectangles.

Golden Spiral

Take a Golden Rectangle and produce a bigger one of dimensions a+a+b by a+b. And you can repeat this pattern ad infinitum:

Note that, if a = 5 and b = 3, then a+b = 8, a+b + a+a+b = 13 and so on, up the Fibonacci Sequence! This pattern only works if the next square’s dimensions are exactly 1.6180339887… that of its predecessor.  It’s lovely and elegant and about as complicated as my very limited mathematical abilities allow me to appreciate.

Now draw a spiral through this set of rectangles like this:

You generate a logarithmic spiral. When conforming to these dimensions, this particular spiral is called the Golden Spiral and, once again, some people have claimed that it occurs in nature, such as the shell of the cephalopod Nautilus.  And, once again, they are wrong.  The spiral shape of Nautilus is logarithmic but it is not a Golden Spiral.

Rotational Growth

We’ve seen plenty of examples of where the golden maths and the Fibonacci Sequence do not occur in nature and art but there are genuine expressions in nature of this beautiful maths if you know where to look. Mostly, they are involved in the phenomenon of rotational growth.

Suppose you are a plant trying to bunch as many seeds into the smallest possible area as possible. How would you do it? Plants start by growing one seed then turning on their axis to start growing the next, turn and grow, turn and grow. The secret for packing the seeds as tightly as possible is to turn by exactly the right amount each time.

It turns out that the optimal turn is 0.6180339887… of a full rotation between each successive seed. That value is surprisingly sensitive.  Have a look at this page  where there is a pattern generator where you can decide the degree of turn for each successive seed.  If you are out by one in the third decimal place (0.617 or 0.619) the pattern rapidly breaks down.

Does 0.6180339887… look familiar? It’s Phi again! Ignoring the 1 in front of the decimal place (which would only mean a complete rotation), the optimal turn to bunch seeds as closely together as possible is Phi. And, when you do that, out pop some more magical numbers.

If you look at the head of Sunflower you will see two sets of spirals picked out by the seeds, one set running clock-wise, the other counter clockwise. Because the Sunflower has used the Phi-turn rule in growing its seedhead, the number of clockwise spirals and anti-clockwise spirals will always be two successive numbers in the Fibonacci Sequence!

Pattern produced by close-packing rotational growth. 21 clockwise spirals and 13 anti-clockwise: two successive numbers on the Fibonacci Sequence.

The same pattern is found wherever close packing and rotational growth are brought together. Think of pine cones or pineapples. I have a Monkey Puzzle Tree (Araucaria araucana) and the spikey little leaves are arranged along the branches in exactly the same way. And it’s all because of the same, simple rules of growth.

Measuring Nature

As an evolutionary biologist, when I look at claims of patterns in nature, I look for two things: some precision and ubiquity for the expression of that pattern and some adaptive benefit underlying why that pattern exists.  When it comes to finding the Golden Ratio, the Golden Rectangle or the Golden Spiral in nature they are difficult to accurately measure and they are not ubiquitous.  There is no apparent adaptive benefit for an organism to adhere to these beautiful but abstract mathematical principles.  Alarm bells go off in my head: here be fertile ground for nonsense.

On the other hand the relationship of the Fibonacci Sequence, Phi and rotational growth in nature is precise and ubiquitous: it’s found anywhere where there is rotational growth and optimal close packing.  And, what’s more, there is a definite selective advantage in being able to pack as many seeds within this growth form, there is a reason why plants will have evolved this very specific and neat trick.

Yes, there is magic and wonder in nature but we are often blinded as to where it can be found.

[1] There are in fact two possible solutions to this proposition, the other is known as the Golden Ratio Conjugate and I’d like to thank my mathematical mate Simon Pampena for pointing this out.

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About the Author

Paul Willis
Paul is a respected leader in the science community with an impressive career in science. He has a background in vertebrate palaeontology, studying the fossils of crocodiles and other reptiles. He also has a long history as a science communicator, with a career spanning as Director of The Royal Institution of Australia, presenter and host for Australia’s Science Channel, working for the ABC on TV programs such as Catalyst and Quantum as well as radio and online. He’s written books and articles on dinosaurs, fossils and rocks and is finding new ways to engage the people of Australia with the science that underpins their world. Follow him on Twitter @fossilcrox.


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