The Golden Ratio

Take a look at a branch. Notice that the leaves appear in a pattern, every leaf in a specific place on the branch at a certain angle. In fact this is a mathematical pattern leading to something called the golden ratio. Time for some math.

On mating rabbits

Imagine a pair of rabbits in a field, one male, one female. They’ll mate, of course – they’re rabbits. Newly born rabbits can reproduce after a month. Now, to reduce this to a math problem, let’s assume that the rabbits never die, and that females birth exactly one pair each month, starting at two months. How many pairs of rabbits do you have after a year?

This is a math problem that the mathematician Leonardo Fibonacci researched back in the twelfth century. It turns out that the number of pairs or rabbits follows an interesting sequence.

At the start, you have one pair. After one month there’s still only one pair, that can now mate. Another month down the line, there’s one new pair. After three months, there’s another new pair, leading to three pairs. Four months means two extra pairs, leading to five, and so on.

This is in fact a mathematical sequence, where each value n is equal to the previous value n-1 and the value before that, n-2:

1, 1, 2, 3, 5, 8, 13, …

Each value in the sequence (after the first two) above is the sum of the previous two. The next one in the sequence will be 8 + 13 = 21, the one after that is 13 + 21 = 34, and so on.

This sequence is called the Fibonacci sequence. You might be wondering: what the heck does that have to do with plants?

Fibonacci everywhere

Leaves growing on plants need to maximize their light input. That means evolution will ensure they are places optimally, and you know what that leads to? A Fibonacci sequence.

So Fibonacci is in plants and animal reproduction. Is it elsewhere?

Take a look at your finger. Start with your finger nail. We’ll assign the length of your finger nail the value 1. If you then look at the top part of the finger, it has a length of roughly 2. The next section has a rough length of 3, and then 5. Finally, If you’d take an x-ray, you’d find a bone in your hand attached to the finger that will have a relative length of 8. That’s a familiar sequence: 1, 2, 3, 5, and 8. It’s a Fibonacci sequence!

Okay, so I’ve been lying a bit. Although popularly accepted, science is not 100% on the side of Fibonacci fingers. It’s a little more complicated. Your little finger follows the pattern, but other fingers may not.

Suffice it to say, this sequence is everywhere.

The Golden Ratio

When you look at the ratio between subsequent values in the sequence, something interesting happens. From the first to the second (1 to 1) that’s 1, from the second to the third (1 to 2) it’s 2, and so on:

1, 2, 1.5, 1.67, 1.6, 1.63, 1.62

You’ll note that the values seem to converge to a specific value. Using some advanced math you can see that this converges to 1.6180339… I won’t reproduce that here for brevity’s sake, see here for details. And it’s also been two decades since I studied that kind of math in university.

This number — 1.618 — is the golden ratio. It has been studied by mathematicians, artists, and biologists since at least 500BC and maybe longer. Meaning the ratio has fascinated us for over 2500 years. Wow.

If you start digging, you’ll find that following our discovery of the ratio in nature, believers in its beauty have applied in things like paintings and architecture. Whether or not it is more aesthetically pleasing is still a matter of debate.

I’ve yet to find an application in writing, although there are some who say that the length of certain sections in stories and articles should follow the Fibonacci sequence. Hmm, yeah. Don’t know if I want to try that.

Conclusion

Once seen, the ratio cannot be unseen. Have fun.

Author: Martin Stellinga

I'm a science fiction and fantasy writer from the Netherlands

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