THE MAGIC OF FIBONACCI NUMBERS

Let us ask ourselves - why do we learn mathematics? For calculations and  applications. And because of inspiration. The connection of Maths and nature is inspiring and makes us think. Maths teaches us to think. Mathematics  is not just solving for X. It is also finding out how and why. 

 The Fibonacci sequence is one of the most famous formulas in mathematics. It has been part of people's lives for centuries. It proves one more time that we cannot draw a line to separate one knowledge from another and if we do this this is just for some practical reasons like educational subjects and exams. More often we borrow knowledge and theories from different disciplines in order to find better understanding and explanation in our own discipline. 

The Fibonacci numbers are proving that the whole universe is one whole - the space itself, the galaxies, nature and the human knowledge which is trying to understand the bigger picture.  The Fibonacci sequences came to Maths from real life observations and provoked ancient mathematicians to find reasonable and practical explanations to some natural facts which everybody observes. From the Fibonacci sequence we can calculate the Golden Ration and after that the spiral. The spiral design could be seen in the natural world and in the universe. We can see the pattern of the spiral everywhere. The spiral is like a mathematical and spiritual symbol of life energy and of rebirth. It is like the connection that holds everything together. It is like God’s brand that is saying to us that everything - from the huge objects like galaxies to the smallest ones - like fingerprints - have been made following the same design of the spiral. The spiral that keeps everything in the universe connected and in order.  Galaxies have the same spiral shape as so many things in nature. Nature’s design copies the space one. The spiral is a symbol of growth and evolution.

If space-time is shaped as a spiral that explains why the Creator used this great design to form the galaxies and nature in that way. It looks like we do live in a golden ratio divine universe. The galaxies have the same shape as  snail shells, sea shells, sunflowers, fossils, the structure of human lungs, spirals in plants, cabbages, pineapples, strawberry seeds, animal horns, flower petals, pine cones, snakes, embryo, ears, storms, DNA, curly hair, hurricanes, turnados,  tree branches, roots, and even galaxies are spirals. 

Who was Fibonacci?  Fibonacci is one of the most famous names in mathematics.  Leonardo Pisano has been immortalised for two things: 1) For his famous sequence – 0, 1, 1, 2, 3, 5, 8, 13, ... ; 2) For popularising our modern number system in the Latin-speaking world. 

Leonardo Pisano was born late in the twelfth century in Pisa, Italy. Fibonacci spent his childhood in North Africa where his father was a customs officer. He was educated by the Moors and travelled widely in Barbary (Algeria), and was later sent on business trips to Egypt, Syria, Greece, Sicily and Provence. In 1200 he returned to Pisa and used the knowledge he had gained on his travels to write Liber Abaci (published in 1202) in which he introduced the Latin-speaking world to the decimal number system. The first chapter of Part 1 begins:

"These are the nine figures of the Indians: 9 8 7 6 5 4 3 2 1. With these nine figures, and with this sign 0 which in Arabic is called zephirum, any number can be written, as will be demonstrated." 

Many other mathematicians are pointing that Indian mathematicians were talking about the same number sequence long before Fibonacci and that he took their idea and introduced it in Europe. One of the mathematical problems Fibonacci investigated in Liber Abaci was about how fast rabbits could breed in ideal circumstances. Suppose a newly-born pair of rabbits, one male, one female, are put in a field. Rabbits are able to mate at the age of one month so that at the end of its second month a female can produce another pair of rabbits. Suppose that our rabbits never die and that the female always produces one new pair (one male, one female) every month from the second month on. The question that Fibonacci was interested in was how many pairs will there be in one year?

• At the end of the first month, they mate, but there is still only 1 pair.
• At the end of the second month the female produces a new pair, so now there are 2 pairs of rabbits.
• At the end of the third month, the original female produces a second pair, making 3 pairs in all.
• At the end of the fourth month, the original female has produced yet another new pair, the female born two months ago produced her first pair also, making 5 pairs.

What is a Fibonacci sequence? A Sequence is a list of things (usually numbers) that are in order. It's a collection. When we say the terms are "in order", we are free to define what order that is. They could go forwards, backwards, they could alternate into any type of order we want. A Sequence usually has a Rule, which is a way to find the value of each term, Nth term.

Fibonacci numbers are: 0,1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987… and so on.

The Fibonacci Sequence is found by adding the two numbers before it together. So; 0+1=1, 1+1=2, 1+2=3, 2+3=5, 3=5=8, 5+8=13,  8+13=21, 13+21=34 and so on.

What is a Golden Ratio? The Golden Ratio (symbol is the Greek letter "phi") is a special number approximately equal to 1.618. The actual value of the  Golden Ratio is equal to: 1.61803398874989484820... (etc.)

The digits just keep on going, with no pattern. In fact the Golden Ratio is known to be an Irrational number, just like number ‘pi’= 3.14159

What is the idea behind the Golden Ratio? The Golden Ratio refers to a simple equation that translates to mean, “two quantities are in the golden ratio if the ratio between them is the same as the ratio of their sum to the larger of the two quantities" or a/b = (a+b)/a.

In visual art like painting and photography,  and in architecture the Golden Ratio is used in composition because it is considered aesthetically pleasing. For example, the Panthenon in Greece.

There is a special relationship between the Golden Ratio and the Fibonacci Sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ...

And here is a surprise: when we take any two Fibonacci Numbers (one after the other), their ratio is very close to the Golden Ratio. In fact, the bigger the pair of Fibonacci Numbers, the closer the approximation. The Golden Ratio is also sometimes called the golden section, golden mean, golden number, divine proportion, divine section or golden proportion. For example: 21/13 = 1.615384615. 34/21 = 1.619047619 and if we keep dividing the other numbers in the sequence we will get approximately the same answer. 

 Where we can see the Fibonacci numbers? Bee populations aren't the only place in nature where Fibonacci numbers occur, they also appear in the beautiful shapes of shells. To see this, let's build up a picture starting with two small squares of size 1 next to each other. On top of both of these draw a square of size 2 (=1+1). We can now draw a new square – touching both one of the unit squares and the latest square of side 2 – so having sides 3 units long; and then another touching both the 2-square and the 3-square (which has sides of 5 units). We can continue adding squares around the picture, each new square having a side which is as long as the sum of the latest two square's sides. This set of rectangles whose sides are two successive Fibonacci numbers in length and which are composed of squares with sides which are Fibonacci numbers, are called the Fibonacci Rectangles.

In nature, the golden ratio can be observed in how things grow or form. If we now draw a quarter of a circle in each square, we can build up a sort of spiral. The spiral is not a true mathematical spiral (since it is made up of fragments which are parts of circles and does not go on getting smaller and smaller) but it is a good approximation to a kind of spiral that does appear often in nature. Such spirals  are seen in the shape of shells of snails and sea shells. 

The spiral is also one of nature’s most common configurations. In fact, it’s difficult to think of all the things that have a spiral pattern. Snail shells, flower petals, pine cones, snakes, storms, DNA, curly hair, even galaxies are spirals—and that’s not even most of them.

While we can’t be entirely sure why something grows in a spiral, it could be a matter of efficiency. A spiral is an excellent way to maximize space. If you consider the arrangement of seeds on a sunflower, the spiral is the best way to house the most seeds on the face of the flower. The more seeds you are able to house, the more future sunflowers are possible. In all these plants with the spiral shape we can count inside the spiral the numbers of the Fibonacci sequence, they appear in almost the same order. Is it just an incredible  coincidence? 

The visual design of the spiral is one of the oldest and most enigmatic sacred images known.  It is one of the earliest examples of human creative expression, appearing in nearly every society in the ancient world.  The spiral has universal appeal and has a mysterious resonance with the human spirit, it is complex yet simple, intriguing and beautiful.  The spiral pattern is found extensively in nature as we mentioned – encoded into plants, animals, humans, the earth and galaxies around us.  Mathematics can explain the complex algorithms, sequences and equations that make up a spiral pattern, but it can’t explain the appeal and fascination of the spiral to the human imagination.  

From a tiny baby to the vast  universe, spirals are all around us.  They link us all – to nature, and to the greater universe.   Maybe that was part of the divine idea – to design a symbol that joins humans, animals, plants, earth, galaxies and beyond.  

Here are two activities for you:

Activity 1: Can you make a list of everything around you that has a spiral shape. This could be from nature, the human body or art objects. If it is man made can you try to explain why they were made in that shape - the spiral? Is there any meaning that the spiral gives to the object?

Activity 2: Can you try to draw a picture using the spiral? Make it interesting. What is the message you are passing to the others by your picture?

In our life it is important that we are open to the deeper levels of existence of everything that surrounds us. It doesn’t matter what we are doing - in our professional life, or in our relationships with  others, we need to try to develop sensitivity about the life around us. This will keep us connected with everything not only in a way how it looks like but what is deep inside. It will open our eyes to the hidden beauty and wisdom which sometimes is just in front of us but we are not capable of seeing it. It will fill our heart with compassion and humanity for  everything around. Because everything is part of the grand design and we can charge ourselves from everything around. But to be able to see and to feel the energy of the beauty and wisdom around we need to educate ourselves for the signs that connect us. It is so important today to educate not only our brains but our hearts as well. Because we will survive as species only if we share our civilised educated hearts full of mercy and compassion for each other. That’s how we will keep the energy chain whole - between us and the universe. 

Look at the sky. What do you see? No, we are not seeing the sky, behind this blue illusion lies the vast space, our home as well. We need to remember that inside all of us there is a lot of stardust as building material. There is a stardust in everything here and there behind that beautiful blue illusion. This fact should remind us that we are a huge and important part of the whole. We are full of gentle power to build and to create, not to destroy. That’s who we are as well - creators. Remember this.

You are a creator!

(E. S. Lyubenova; LoveMathsStory for my students)