Let us ask ourselves: why do we learn mathematics? For calculations and applications. And for inspiration. The connection between mathematics and nature is inspiring and makes us think. Mathematics teaches us how to think. It is not just about solving for x. It is also about discovering how and why.
The Fibonacci sequence is one of the most famous patterns in mathematics. It has been part of human knowledge for centuries. It proves once again that we cannot draw a line separating one field of knowledge from another. If we do so, it is only for practical reasons, such as educational subjects and examinations. More often than not, we borrow knowledge and theories from different disciplines in order to gain a better understanding of our own.
The Fibonacci numbers show us that the whole universe is interconnected—the fabric of space itself, the galaxies, nature and human knowledge, all striving to understand the bigger picture. The Fibonacci sequence entered mathematics through observations of the natural world and inspired ancient mathematicians to find logical and practical explanations for natural phenomena that everyone could observe.
From the Fibonacci sequence, we can calculate the Golden Ratio, and from the Golden Ratio comes the spiral. The spiral can be seen throughout the natural world and across the universe. We encounter this pattern everywhere. The spiral is both a mathematical and a spiritual symbol of life, energy and rebirth. It is like the thread that holds everything together. It is like God's signature, reminding us that everything—from immense galaxies to the smallest fingerprints—has been created according to the same spiral design. It is the spiral that keeps everything in the universe connected and in order.
Galaxies share the same spiral shape as snail shells, sea shells, sunflowers, fossils, the structure of the human lungs, plants, cabbages, pineapples, strawberry seeds, animal horns, flower petals, pine cones, snakes, embryos, ears, storms, DNA, curly hair, hurricanes, tornadoes, tree branches and roots. Nature's design echoes the design of space itself. The spiral is a symbol of growth and evolution.
If space-time itself is shaped like a spiral, that may explain why the Creator chose this magnificent design for both galaxies and nature. It almost seems that we live in a universe built upon the divine Golden Ratio.
Who was Fibonacci? Fibonacci is one of the most famous names in mathematics. Leonardo Pisano is remembered for two remarkable achievements: first, for the famous sequence of numbers—0, 1, 1, 2, 3, 5, 8, 13...—and second, for introducing the Hindu-Arabic numeral system to the Latin-speaking world.
Leonardo Pisano was born in Pisa, Italy, towards the end of the twelfth century. He spent his childhood in North Africa, where his father worked as a customs official. He was educated by the Moors and travelled widely through Barbary (modern-day Algeria). Later, he travelled on business to Egypt, Syria, Greece, Sicily and Provence. In 1200, he returned to Pisa and used the knowledge he had gained during his travels to write Liber Abaci (published in 1202), in which he introduced the decimal number system to the Latin-speaking world.
The first chapter of Part One begins with these words:
"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 historians of mathematics point out that Indian mathematicians had described the same numerical sequence long before Fibonacci, and that he introduced their ideas to Europe rather than inventing the sequence himself.
One of the mathematical problems Fibonacci explored in Liber Abaci concerned the growth of a rabbit population under ideal conditions. Suppose that a newly born pair of rabbits—one male and one female—is placed in a field. Rabbits can reproduce from the age of one month, so at the end of its second month the female produces another pair of rabbits. Suppose that none of the rabbits ever dies and that every female produces one new pair every month from her second month onwards. Fibonacci asked a simple but fascinating question: How many pairs of rabbits will there be after one year?
At the end of the first month, the rabbits mate, but there is still only one pair. At the end of the second month, the female produces a new pair, so now there are two pairs of rabbits. At the end of the third month, the original female produces another pair, making three pairs altogether. At the end of the fourth month, the original female produces yet another pair, while the female born two months earlier produces her first pair, making a total of five pairs.
What is the Fibonacci sequence? A sequence is a list of items, usually numbers, arranged in a particular order. It is simply an ordered collection. When we say that the terms are "in order", we are free to define what that order is. They may go forwards or backwards, or follow any pattern we choose. A sequence usually has a rule that allows us to find each term or even the nth term.
The Fibonacci numbers are:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987...
The Fibonacci sequence is formed by adding the two previous numbers together. For example:
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 the Golden Ratio? The Golden Ratio (represented by the Greek letter φ, phi) is a special number approximately equal to 1.618. Its value begins:
1.61803398874989484820...
The digits continue forever without repeating in a pattern. In fact, the Golden Ratio is an irrational number, just like π (pi).
What is the idea behind the Golden Ratio? The Golden Ratio describes a relationship between two quantities. Two quantities are said to be in the Golden Ratio if the ratio between them is the same as the ratio between their sum and the larger quantity:
a/b = (a + b)/a
In visual arts such as painting and photography, as well as in architecture, the Golden Ratio is often used in composition because it is considered aesthetically pleasing. One famous example is the Parthenon in Greece.
There is a remarkable relationship between the Golden Ratio and the Fibonacci sequence:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34...
Here comes the surprise. If we divide any Fibonacci number by the one immediately before it, the result is very close to the Golden Ratio. In fact, the larger the numbers become, the closer the result approaches the Golden Ratio. The Golden Ratio is also known as the golden section, the golden mean, the golden number, the divine proportion and the golden proportion.
For example:
21 ÷ 13 = 1.615384615...
34 ÷ 21 = 1.619047619...
As we continue dividing successive Fibonacci numbers, the answers become increasingly close to approximately 1.618.
Where can we see the Fibonacci numbers? Bee populations are not the only place in nature where Fibonacci numbers appear. They can also be found in the beautiful shapes of shells. To understand this, let us begin with two small squares, each with sides of length 1, placed next to each other. Above them, draw a square with sides of length 2 (1 + 1). Then draw another square with sides of length 3 (2 + 1), followed by one with sides of length 5 (3 + 2), and continue in the same way.
Each new square has a side length equal to the sum of the side lengths of the previous two squares. The resulting collection of rectangles, whose sides are successive Fibonacci numbers and which are built from Fibonacci squares, is called the Fibonacci rectangles.
In nature, the Golden Ratio can often be observed in the way living things grow and develop. If we draw a quarter of a circle inside each square, we create a spiral. This is not a true mathematical spiral because it is made from circular arcs rather than a continuously changing curve. Nevertheless, it is an excellent approximation of the type of spiral that appears frequently in nature, such as in the shells of snails and other sea creatures.
The spiral is one of nature's most common patterns. In fact, it is difficult to think of all the things that have a spiral form. Snail shells, flower petals, pine cones, snakes, storms, DNA, curly hair and even galaxies are spirals—and that is still only a small part of the list.
Although we cannot be entirely certain why so many things grow in a spiral, it may simply be a matter of efficiency. A spiral is an excellent way of making the best use of space. If you look at the arrangement of seeds in a sunflower, for example, the spiral allows the greatest possible number of seeds to fit within the flower head. The more seeds a sunflower can produce, the greater the chance of creating future generations of sunflowers.
In many spiral-shaped plants, we can count the spirals and discover Fibonacci numbers appearing in almost the same order. Is this simply an extraordinary coincidence?
The visual design of the spiral is one of the oldest and most mysterious sacred symbols known to humanity. It is among the earliest examples of human creative expression, appearing in almost every ancient civilisation. The spiral has a universal appeal and a remarkable resonance with the human spirit. It is both simple and complex, mysterious and beautiful. As we have already seen, the spiral is deeply woven into nature—it is encoded in plants, animals, the human body, the Earth and the galaxies around us.
Mathematics can explain the algorithms, sequences and equations that create spiral patterns, but it cannot fully explain why the spiral has such a powerful appeal to the human imagination.
From a tiny baby to the vast universe, spirals surround us. They connect us to nature and to the greater universe itself. Perhaps that was part of the Creator's design—to create a symbol that unites humans, animals, plants, the Earth, the galaxies and everything beyond.
Here are two activities for you.
Activity 1: Make a list of everything around you that has a spiral shape. Your examples may come from nature, the human body or works of art. If an object is man-made, try to explain why it was designed in the shape of a spiral. What meaning does the spiral give to that object?
Activity 2: Try to create a drawing using a spiral. Make it interesting. What message are you trying to communicate through your picture?
In our lives, it is important to remain open to the deeper dimensions of everything that surrounds us. Whatever we do—in our professional lives or in our relationships with others—we should try to develop a sensitivity to the life around us. This sensitivity keeps us connected to everything, not only through its outward appearance but also through what lies deep within. It opens our eyes to the hidden beauty and wisdom that are sometimes right in front of us, yet remain unseen.
It fills our hearts with compassion and humanity for everything around us because everything is part of one great design, and we can draw strength from all that surrounds us. But in order to see and feel the beauty and wisdom around us, we must educate ourselves to recognise the signs that connect us all. Today, it is more important than ever to educate not only our minds but also our hearts. We will survive as a species only if we share educated hearts filled with mercy and compassion for one another. That is how we keep the great chain of energy unbroken—between ourselves and the universe.
Look at the sky. What do you see? No, we are not really looking at the sky. Behind that beautiful blue illusion lies the vastness of space—our home as well. We should remember that each one of us is made of stardust. Stardust exists in everything, both here on Earth and far beyond that blue illusion above us.
This should remind us that we are an important part of a much greater whole. We possess the gentle power to build and to create, not to destroy. That is who we are too—creators.
Remember that.
You are a creator!
(E. S. Lyubenova, LoveMaths Story for My Students)
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