exotic fresh pineapple with green leaves
Just For Fun

The Fibonacci Sequence and the Golden Ratio

We have previously touched upon Fibonacci sequences when discussing different types of sequences for GCSE maths so you may already be familiar with them. This blog will go into more detail and show you some more uses of the Fibonacci sequence that you probably wouldn’t expect!

What is the Fibonacci sequence?

The Fibonacci sequence is a sequence of integers (whole numbers) starting with 0 and 1. The next term in the sequence is the sum of the previous two terms. So this would make the next term 1 because 0 + 1 = 1. And the next term 2 because 1 + 1 = 2. And so on. Here are the first few terms of the sequence (calculations are given below and each term is colour-coded so you can clearly see how they are calculated).

Fibonacci sequence

This is the most commonly used Fibonacci sequence, although really it is just one of a larger family of “Fibonacci-style” sequences that have the same term-by-term rule just with two different starting numbers. For example, if I started a Fibonacci sequence with the numbers 2 and 4, I would get this sequence:

Fibonacci-style sequence

Although the sequence is believed to have been used in India long before, it was first introduced to Europe in the 13th Century by an Italian mathematician by the name of – yes, you guessed it, Fibonacci. Although his birth name was in fact Leonardo Bonacci, with the nickname of Fibonacci deriving from a shortened version on “Filius Bonacci” (the Latin for son of Bonacci). In memory of the man who made popular the most famous sequence in mathematics, a statue was erected of him in his hometown of Pisa in the 19th century, some 600 years after his death.

A pretty cool sequence, right? Anything that earns you a statue 600 years after your death must be something special. Well, Fibonacci didn’t know it at the time, but his legendary sequence has plenty of interesting properties…

The Golden Ratio

Let’s focus in on the original Fibonacci-style sequence: 0, 1, 1, 2, 3, 5, …

What I’m going to do is for each term from the 3rd term onwards, I’m going to divide this term by the previous term to get a ratio.

So, for the third term (1), I will divide it by the second term (also 1), giving me 1 ÷ 1 = 1.
Then, for the fourth term (2), I will divide this by the third term (1), giving me 2 ÷ 1 = 2.
And so on…

This may seem completely pointless and contrived, but if I keep doing this I get these results in the below table:

TermNumberPrevious NumberRatio of Number / Previous Number
3rd111
4th212
5th321.5
6th531.666666667
7th851.6
8th1381.625
9th21131.61538461538462
10th34211.61904761904762
11th55341.61764705882353
12th89551.61818181818182
13th144891.61797752808989
14th2331441.61805555555556
15th3772331.61802575107296
16th6103771.61803713527851
17th9876101.61803278688525
18th15979871.61803444782168
19th258415971.61803381340013
20th418125841.61803405572755

This may still seem unremarkable to you, but maybe if I go even further to the 50th term and plot them on a graph it will look a bit more interesting.

Graph showing ratios of consecutive terms in Fibonacci sequence

What is happening is that the ratio we get each time is getting closer and closer to a number. The more times we do it, the closer we get to this magical number.

What is this magical number I hear you ask?

It’s 1.61803398874989.

This number can also be written in surd form as:

Golden ratio in surd form

This number is known as the Golden Ratio and is commonly known as the Greek letter phi (Φ).

The Golden Ratio is a number which has many different applications in nature and design.

For example, researchers believe that many traits that humans perceive as beautiful can be derived from the Golden Ratio.

An example of this is that the ideal ratio between the height of a person’s head and the width is 1.61803398874989 : 1. That is to say that if you divide the height of a person’s head by the golden ratio, that will be the ideal width of their head!

There are even more! You can even use this beauty calculator to work out how beautiful you are based on how close your ratios are to the Golden Ratio!

The Fibonacci sequence and Golden Ratio also appear in nature spookily often. For example, the number of spirals on pinecones, pineapples, flowers, even boring cauliflowers! For a visual explanation of this, I can’t show you any better than these excellent videos by Vihart: Doodling in Math: Spirals, Fibonacci, and Being a Plant (Part 1, Part 2 and Part 3). Hopefully you enjoy these and you don’t get too freaked out by the weirdness!

So, who would have thought that that seemingly trivial sequence that you briefly seen in school would turn out to be so fascinating and appear so much in nature?

In future blogs, I will start sharing some other cool integer sequences (although none of them match up to the Fibonacci sequence which has to be considered the daddy of all integer sequences).

This was just for fun, but if you need some extra help with maths, book in a free taster session.

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