What is the Fibonacci sequence?

So we have looked at What are the Fibonacci numbers?. This article now looks at What is the Fibonacci sequence?

What is the Fibonacci sequence?

It is a name given to a series of numbers calculated by an Italian mathematician in approx 1100. This number sequence has come to be much revered.

If you just want to know what the numbers are and how to achieve them you can read What are the Fibonacci numbers to find all that out.

This following article explains exactly what the Fibonacci sequence is and why it is so revered.

So, what exactly is the Fibonacci sequence?

*This sequence is one expression of the divine proportion. There are many other ways to express it.

The sequence is an expression of the DIVINE PROPORTION

Now that the question of ‘What is the Fibonacci sequence?’ has been answered. We ask the next question (which is the more interesting question) …

What is Divine Proportion?

It’s a way to comprehend the patterns that manifest in nature and the universe.

It is a relationship between two parts of a whole. A specific relationship that is often reflected as a ratio or decimal which arise from using the numbers in the sequence in a certain way.

There we are, straight into maths language of decimals and ratios which puts people off. This is exactly what I didn’t want to do. This is how many people experience the Fibonacci Sequence and I believe it is so much more. We have to remember that using numbers is only one way to express the Divine Proportion.

Fibonacci’s sequence of numbers is one expression of divine proportion, it is an expression using numbers.

The Fibonacci sequence expresses the Divine Proportion using NUMBER DIGITS.

And the reason this is, or rather was, so exciting is because up until then these number digits didn’t exist in the west. Roman numerals were still being used….

At this point, it’s actually crucial that you gain two other important pieces of information before continuing to read. ….So… to make it as simple as possible I lay them out below for you.

But to fully get to know the information I suggest you follow the links in each paragraph. You see, its only when we really grasp the time that Fibonacci existed in with reference to cultural thought and arithmetic experience that we can fully appreciate what his sequence contributed. And it is my belief that much of this gets lost in today’s desire for memes and instant gratification etc.

1). Understand the Divine Proportion and its history

To fully understand the Divine Proportion it helps to look at the many other expressions of it in art, architecture etc. If I write all that in this article it gets really confusing, and Fibonacci can often be confusing enough.

So ‘What came first: Fibonacci or the Divine Proportion?‘ is the article for all of that information. That includes Euclid and many Greek artists, even some Egyptian achievements as well as what came since Fibonacci and how it even got that name.

2). Fibonacci’s true legacy

Spoiler alert: Fibonacci wasn’t even called Fibonacci when he wrote this number sequence. Also, what he REALLY contributed to at the time he lived was the introduction of the hindu arabic number system. This is the early version of the digits we use today.

He wrote a book called ‘Liber Abaci’ which laid out the use of new arithmetic numerals he was seeing used by traders who arrived from across the Mediterranean Sea. This number system became popular, and that is what we really have to thank Fibonacci for. ‘Who was Fibonacci and what was his true legacy‘ shines a light on who he was growing up and how he came to contribute these numerals that we all still use today.

So…. back to What is The Fibonacci sequence…. as I said before:

The Fibonacci sequence expresses the Divine Proportion using NUMBER DIGITS.

Each number digit in the Fibonacci sequence expresses a proportional relationship to its neighbouring number digit.

The ratio between each consecutive number turns out to be in this Divine Proportion.

This couldn’t have been done with Roman numerals. (at this time ratios were expressed as geometric constructions because number digits hadn’t yet been fully adopted , so there is speculation as to whether Fibonacci knew the importance of what he had discovered at the time)

If something is described as having ‘divine proportions’ it means that every part has a specific relationship to its neighbouring part, and to the whole . The whole is to the larger in exactly the same proportion as the larger is to the smaller.

To quote the other famous Leonardo da Vinci

“…each part is disposed to unite with the whole, that it may thereby escape its incompleteness”

Leonardo Da Vinci

But we have to back up and look through the recent centuries to fully understand the story that Fibonacci was to play his part in unfolding. link to other article.

He had put a formula on this elusive thing that humans had been striving to express. He was a pure mathematician that liked playing with numbers and discovered a unique characteristic of this particular sequence of numbers.

So let’s return to the numbers in the sequence again, these numbers go on and on. Keep adding previous two to get the next.

It’s difficult to imagine the ‘whole’ in logical thought. For Fibonacci it was a number experiment. He calculated the ratio between consecutive numbers in the sequence was always the same. **

You can read more detail about this ratio at What are the Fibonacci numbers? . Whereas this article is explaining what this sequence represents. What it actually is, not just what it does or how it behaves.

** to be precise this ratio becomes more exact the higher up the sequence

Fibonacci sequence and the new number system

I mentioned before about this book that Fibonacci wrote introducing this new number system. His sequence of numbers that displayed this divine proportional relationship appeared as a side note in this book of his as a short analogy about rabbits breeding.

It was almost a number game just to lighten the heavy reading material a little. Add one of these new number digits to the one before it and you get the next number in the sequence and so on.

He was a whizz kid with number remember, imagine him as one of the first coders of the internet boom. By playing with these number digits he formed a sequence that reflected the ratio of the divine proportion. He seemed to conjure magic with these new numbers.

This number sequence was just an interesting sideline to him. It was the adoption of the new number system he was most interested in, bringing it to the merchant traders of the time. This is the bigger legacy of Fibonacci and one that is pretty much overlooked.

Up until then these number digits didn’t exist in the west. Roman numerals were still being used. The image above shows a section of the text from this book, just to look at the inscriptions give you a suggestion of how different the world was back then. He was demonstrating the use of a new way of communicating number. We struggle these days to even imagine a world before the internet, and Fibonacci lived in a world even before numbers were what we know them as today.

Can you even imagine doing The Fibonacci Sequence with Roman numerals?!

A New Number System

This adoption of the new number system happened slowly though. The numeral digits that you and I know were becoming a really big deal. They began to replace the ‘mystery’ of number that existed before.

It is because we look from a contemporary viewpoint that we can’t imagine life without digit numbers. Fibonacci was at the cutting edge of the developing world in the 1100’s. He used his new number technology to express what was so revered in culture at the time. This understanding of the divine proportion. The awareness of the divine proportion and its use in art and architecture at the time of Fibonacci’s life made it a very popular topic.

The Fibonacci plot thickens..

At the time, this connection of the numbers to the divine proportion wasn’t really a big deal, an interesting sideline at most. It was the numeral digit system he introduced which changed the course of life and remains the origins of the digits we use today.

Fibonacci wasn’t known for The Fibonacci Sequence at the time. They didn’t even call it ‘The Fibonacci Sequence’ then. (how did that happen? read ‘What came first: Fibonacci or the Divine Proportion?’)

It was several hundred years later that Kepler associated this sequence of rabbit numbers with divine proportion and understood it to be involved with plant growth.

It was thanks to Fibonacci’s Rabbit Riddle calculations that gave rise to his sequence of numbers and the ratio between them.

And Bonnet in the 1700’s continued this thinking with the understanding of Phylotaxsis in leaf placement.

It wasn’t until the mid-late 1800’s that the numbers actually became referred to as The Fibonacci Sequence.

What is The Fibonacci Sequence?

I do hope this article has shed some light on what exactly The Fibonacci Sequence is. It is a tricky question to answer. But this article and the others I have linked to on this page and in further pages form a collection of articles which demystify who and what Fibonacci and the sequence is. it is the 101 resource for understanding where he sat in our history and what he pointed us towards.

The Fibonacci sequence is an expression using numbers of the Divine Proportional relationship we see reflected in the natural world

Remember that the sequence is just one expression of divine proportion. there are many other expressions (link) so now that we understand that we can now use these numbers as tools to visualise the patterns we see in the natural world.