Many of us have heard of the Fibonacci sequence. When I was younger, the Fibonacci sequence fascinated me. I memorized over twenty numbers in the sequence! I marvelled at how someone could discover this “cool math stuff.” However, although I knew all the numbers in the Fibonacci sequence, I didn’t understand the applications of this amazing string of numbers – the Golden Ratio.

## The Fibonacci Sequence

Before we get into talking about the Golden Ratio, let’s discuss the Fibonacci sequence. Italian mathematician Leonardo Pisano Bigollo discovered the Fibonacci sequence in 1202. He was known as Fibonacci, which means “son of Bonacci”. He developed the sequence to calculate population growth (specifically, a rabbit population) and applied it to finances and trades. These were both very important among merchants. So, what is the Fibonacci sequence? The first few digits of this sequence are 0, 1, 1, 2, 3, 5, 8, 13…. Can you spot the pattern?

Each number in the Fibonacci sequence is the sum of the two previous numbers. For example, 0+1 = 1, 1+1=2, 2+1=3, 2+3=5 and so on.

This is modelled by the expression: F_{n}=F_{n-2} + F_{n-1}, where F_{n} is the next number in the sequence.

## Applications

What I find most interesting about the Fibonacci sequence (and math in general!) is that we can actually see it in nature! For example, sunflowers, pinecones and many fruits such as pineapples and cauliflowers all exhibit the Fibonacci sequence. If you could count the number of seeded spirals in a sunflower, you would find they add to a Fibonacci number. If you divided them up based on which way they open, the number opening each way would still add to a Fibonacci number!

It also models the growth of a honeybee population. Drone honeybees all have only one parent, as they hatch from unfertilized eggs. That one parent will be a female bee, which has two parents – a queen bee and a drone bee. Therefore, the drone bee will have one parent, two grandparents, three great-grandparents, five great-great-grandparents and so on. This is exactly like the Fibonacci sequence!

Now we know more about the Fibonacci sequence. Let’s go a little further into its importance and take a look at the Golden Ratio.

## The Golden Ratio

The Golden Ratio is a unique mathematical equation or relationship between two numbers. We don’t know who first discovered the Golden Ratio, but the Ancient Egyptians used it in their architectural designs. However, Martin Ohm is supposedly the first person to use the phrase “Golden Ratio” to describe the relationship. So…what is the relationship?

The algebraic equation: (a+b)/a = a/b =ɸ, where a > b models the Golden Ratio. The Greek letter ɸ (phi), denotes the answer to this equation, which is about 1.618.

So, what connects the Fibonacci sequence and the Golden Ratio? Well, the Golden number (1.618) can be most easily approximated by using Fibonacci numbers in yet another equation. By dividing F_{n } by F_{n-1, }the answer will be approximately 1.618. The bigger F_{n} is, the closer to ɸ the answer will be. For example, 2/1 = 2 and 3/2 = 1/5 and 5/3 = 1.666… and so on until you get to numbers like 55/34=1.6176.

## More applications!

Interestingly, the Golden Ratio is also sometimes called “divine proportion,” because it is present in nature. As they are connected, all of the Fibonacci sequence’s applications also apply to the Golden Ratio. The Golden Ratio can be seen in marine creatures’ shells, flower petal arrangements, tidal wave patterns, the shape and construction of our ears and even space! Leonardo Da Vinci even used the Golden Ratio when painting the famous Mona Lisa. Her perplexing smile is attributed to the Golden Ratio!

So, hopefully, you can now see how interesting the Fibonacci sequence and Golden Ratio are. Hopefully, some of the amazing real-world applications in science and nature have inspired you! I’m glad that I now more deeply understand my childhood fascination with the Fibonacci sequence, and how mathematics and nature are so closely and intricately intertwined.

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