What came first? was it Fibonacci’s number sequence or the Divine Proportion?
Fibonacci and Divine Proportion are forever intertwined.
We all love Fibonacci. We are drawn to the pattern and the divine proportions that the Fibonacci numbers represent. But what came first?
This article is an overview of all those that came before Fibonacci, and also those that came after.
To gain the most from what follows, it will help if you understand that the Fibonacci sequence represents a pattern of divine proportions that had been observed for many centuries prior to Fibonacci’s life. More on the understanding of this can be found at the What is the Fibonacci sequence? article.
We will look at life before Fibonacci, to understand where his thinking stood in the history of divine proportion.
How did the Fibonacci and Divine Proportion become intertwined?
Throughout history and before, humans have been drawn to the complexities of the universe and sought to understand them. Way back before maths was even maths, scholars have always sought knowledge. That is what being a scholar was. To ‘know’ about life’s mysteries. In the evidence of these studies and their understandings we see many expressions of what was considered harmony in the natural world : expressions of the Divine Proportion.
A time before numbers
So we have to go way back. Back to before we played with these abstract numbers and bent them to the will of our civilisation.
There has always been an awe and wonder at the mysteries of the universe. The exploration of these mysteries has been expressed in a myriad of ways throughout history.
Each culture and era has developed ways to express their wisdom of these truths. From sacred customs, art, music, architecture. There have been eras when it was considered a great honour and responsibility as a human individual to be bestowed with this knowledge. Was the heart and mind even capable of such knowledge? such was the reverence for these existential truths. And there have been eras when it was forgotten, or eras when it just exists as a meme on Instagram.
Expression through architecture
There is evidence that the Egyptians used divine proportions as a mathematical proportion, measured with rope, in the construction of the pyramids. The right angled triangle we see in the image above would later be labelled the Pythagorus triangle and later still Kepler triangle. Whether they had a conscious understanding of it or not, the use of this indicates the presence of the divine proportions.
Scroll down to the section on Renaissance art and divine proportion and there is a link to a site which host some fantastic videos detailing the golden ratio and √5 rectangle in their art which is also handy to understand with reference to the Parthenon.
Phidias, a Greek sculptor (approx 500 BCE) is said to have used divine proportions in the construction of the large frieze and many of the large sculptural works that were originally inside The Parthenon.
The design of The Parthenon in Athens is said to be attributed to architect Ictinos. From the remains of the Parthenon can be seen the use of Golden Rectangles in the design of the front elevation and also in the √5 rectangle speculated to have been used for the ground plan.
Expression through shape …..
Plato, the Ancient Greek philosopher,, was busy with his embryonic solid shapes which were seen to set the basis for the structure of the universe. Many of these shapes use the divine proportions in their construction. The Dodecahedron came to symbolise the universe itself as it could house all the other solids. Being made up of pentagonal faces (the relationship between pentagon and pentagram is one in the same) the divine proportion is inherently present in the dodecahedron through its use of the pentagram which has the golden ratio embedded into its structure.
Pythagorus was around then too. Although not directly associated to the study of divine proportions he did much in the developing concept of ‘number’ with the metaphysical properties for number.
For an in-depth explanation of the pentagram and Pentad’s link to Divine proportion go to Dodecahedron
Being that the dodecahedron is made of 12 pentagons, the divine proportion is inherently present in the Dodecahedron.
….. and an understanding of proportion
Later, a Roman architect and engineer, Vitruvius, wrote a treaty on architecture in an effort to retain the classical dimensions he saw used in the previous Greek era. Referred to as The Ten books of Architecture it included many other subjects too. Leonardo da Vinci would later create the now famous ‘Vitruvian Man’ image as a nod to the laws of proportion with the human body and Vitruvius’s work.
In his book, Vitruvius references the design of the proportions of the human body, going into remarkable detail of the ratio relationships between features of face and body. It is suggested that the discipline of architecture follow similar rules when creating temples and use ratio in the proportions to best express pleasing symmetry.
Ancient Greece was a time when the process of thought developed fully, study and the possesion of knowledge was a camodity.
Into this landscape Euclid arrived.
Expression through words.
It was Alexandria where our recorded understanding of history began. A scholarlory city at the historical tail end of the Egyptians. Alexandria had what is now seen as the first library in existence in the west. So there was an impetus to make a written record of knowledge and get it into a book of some sort.
Alexandria was home to Euclid. Euclid made a big contribution to the mathematics (particularly geometry) with his Elements books which form much of our current understand of Euclidean geometry.
Euclid expressed this divine proportional relationship in words, words that were yet to become a mathematical language. Prior to this, the Divine Proportion, was still a concept, a knowledge, an inspiration of the wonders of the universe.
”Extreme to the mean”
Euclid of Alexandria
At the time, there weren’t any separate subjects of study as we think of today. If you were lucky enough to have had an education you were exposed to all the areas of knowledge. You were deemed a scholar. The great mysteries of life were studied and the scope of this subject took in the concept and thinking of number as well as the areas of arts (mostly sculpture at the time) and architecture, so one area blended into another, hense we get expressions of these big concepts about life in all forms.
At this time there was Hypatia, a female scholar and geometer teacher who did much to develop an understanding of conic sections and hyperbolic curves. (the idea of Hypatia has always fascinated me and I must look up more about her!)
We are still well before the birth of Christianity. And western history goes through the dark ages and so too does knowledge of divine proportion.
Expression through music
dimensions of instruments and lyre
many of these stringed instruments the origins of which came from the early understanding of harmonic resonances in times of Greeks. They explored the plucking of lengths of strings and the lengths that had the best harmonics between them. So the expression of Divine Proportion through music grew. What I find so fascinating about this is that maybe the intellectual knowledge of this proportion was lost in the years that followed but the experience of it stayed live in the making of music from these harmonies.
Fibonacci and Divine Proportion : Expression through numbers
It wasn’t until the 12th century BCE that Leonardo de Pisa – akaFibonacci- was around.
So just to recap….
That is hundreds of years since the Greeks and Romans, Christ has been born and the western religion of Christianity has taken hold. The dark ages have been lived though and Fibonacci is born into an Italy that finds itself just at the end of the dark ages and at the centre of trade routes through the Mediterranean sea.
Here he is! you see how there have been many before him involved with this Divine Proportion? and he wasn’t about to be the last either.
For more info on the child that became the man who brought use these numerals and with them his Fibonacci sequence read the article who was he and what was his true legacy?
As we have discussed in that article, Fibonacci was currently writing a book on the new hindi-arabic number system which he was introducing to the merchant traders of the mediterranean.
It was within this book that he used an analogy of rabbits breeding, known as the Rabbit Riddle to demonstrate how these numeral digits he was introducing could be added one to another. It is the numeral digits in this system that reflect the ratio of the divine proportion explained in the article What is the Fibonacci sequence? Up until then Roman numerals were being used and calculations with Roman numerals are possible but rather long winded. (don’t even think about multiplication!)
So you see that Fibonacci’s focus wasn’t on creating a number system that he would be famed for. That was just an interesting side note to him in the form of rabbits.
He brought us the numerals.
That is his legacy, it was hundreds of years later still that we came to call it the Fibonacci sequence.
But he fits into this timeline of history because it was an expression of the Divine Proportion using these new numerals. It was a clever way for him to show what these numerals were capable of. To demonstrate it to the merchant traders and persuade them to use this system, much easier and more independent than employing an accountant to use an abacus as they were used to at the time.
His focus wasn’t on the divine proportion. But in what he created, he played a crucial role in the development and expression of this knowledge that was to be passed down to us. He expressed the Divine proportion using numbers.
but there’s still more….
Expression through Art
Renaissance and Luca Paciolo
It is quite well known that the Renaissance artists used the golden ratio in their art. It is though, not exclusive to that time. Many artists in many eras have used it. This website, howartworks.com hosts a short and beautifully presented video showing how the √5rectangle is presented in three pieces from very different eras.
NOTE :30/1/23
there is plenty more info to come here, I am still gathering and processing the research and creating images so do return here in a few weeks!
to come: Leonardo di vinci , painters using golden rectangle. Expressions through Science. Kepler and his planets which takes us into the birth of modern science and expressions as maths and science notation
so what expression comes next?…..
