Origins of the Fibonacci Sequence
In 1202, an Italian mathematician named Leonardo Pisano, more commonly known in history as Fibonacci, published a work that would have a profound impact on European mathematical thought: Liber Abaci, or The Book of Calculation. This treatise wasn’t intended for academic mathematicians but for merchants, craftsmen, and professionals of the time. Its purpose? To demonstrate the effectiveness of the Indo-Arabic numeral system, which was then little known in Europe, compared to the cumbersome Roman numeral system.
The Indo-Arabic numeral system, with its use of decimal numbers and the concept of zero, came from Indian mathematical traditions, which were much more advanced than the European ones of that period. During his travels in North Africa, Fibonacci encountered this knowledge and brought it back to Europe, where it became an essential tool for calculation.
In Liber Abaci, Fibonacci explained how to perform calculations with decimal numbers, how to apply percentages, interest rates, proportions, and division among goods. It was essentially a practical manual for mathematics applied to everyday life and commerce.
Among the many practical problems Fibonacci proposed, one became famous: the growth of a population of rabbits. The question was simple: how many pairs of rabbits will there be in one year, starting with just one pair, if every month each pair produces a new pair, which in turn becomes fertile from the second month onward?
The solution to this problem generates a numerical sequence in which each number is the sum of the two preceding ones:
1, 1, 2, 3, 5, 8, 13, 21…
This is what we now know as the Fibonacci Sequence.
In this example, conceived for purely didactic purposes, Fibonacci uncovered a simple yet surprisingly recurring mathematical pattern found in nature. And although Leonardo could not have imagined the future implications of his sequence, through that rabbit problem, he had unknowingly opened a window to one of the most fascinating mysteries of the natural universe.
What is the Fibonacci Sequence?
The rule is simple: each number in the sequence is the sum of the two preceding ones.
It starts like this:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89…
And so on, infinitely.
Despite its simplicity, this sequence hides a deep elegance. It is one of the ways in which nature seems to speak a mathematical language, repeating the same patterns in living forms that are very different from one another.
The Fibonacci Sequence and the Golden Ratio
The most fascinating connection of the Fibonacci sequence is its relationship with the so-called golden ratio, or the golden number, denoted by the Greek letter φ (phi).
This number has an approximate value of 1.6180339887… and is an irrational number, meaning it has an infinite number of non-repeating decimal places.
But what makes it so special?
Imagine you need to create an object (a window, a painting, a playing card, a logo, a vase, a work of art). How do you decide the right proportions to make it appear harmonious to the eye?
You could, of course, rely on your instinct… or you could use a “trick” that the human eye finds surprisingly natural: the golden ratio.
Now, imagine a segment. If you want to divide it in a “perfect” way, you could use a mathematical rule that says:
The ratio between the longer part and the shorter part must be equal to the ratio between the whole segment and the longer part.
This division produces a constant ratio: φ (approximately 1.618).
This golden proportion is the basis of many forms considered elegant or balanced, not because of magic, but because it works.
And the incredible thing is that this proportion spontaneously appears in nature as well, in the way plants grow, seeds are distributed, shells spiral, or even in the human body.
So, this number is not just useful in a school exercise: it is an ancient yet very relevant tool used by designers, artists, architects, and even engineers to create beauty, harmony, and efficiency.
Mathematically:
(a + b) / a = a / b = φ ≈ 1.618…
It is a proportion that, for reasons still under study and wonder, humans perceive as naturally harmonious and pleasant.
Now, back to the Fibonacci Sequence…
If you take two consecutive numbers from the sequence (for example, 21 and 13) and divide them by each other:
21 ÷ 13 ≈ 1.615
34 ÷ 21 ≈ 1.619
55 ÷ 34 ≈ 1.617…
The result gets closer and closer to φ, the golden number.
Magic? No, mathematics. But mathematics that seems to contain a form of universal aesthetics.
That’s why the golden number, and thus indirectly the Fibonacci sequence, appears in many diverse areas:
- In art (the proportions of the Parthenon or the Mona Lisa)
- In architecture (temples, Renaissance palaces, even credit cards)
- In music
- In photography
- In nature
It is not just a matter of beauty: often, these proportions also correspond to efficient solutions, optimal distributions, and forms that work.
And perhaps that is the true allure of the sequence: a simple numerical rule that hides a mysterious balance between efficiency and beauty.
Why Does It Appear in Nature?
Many of the natural phenomena that follow the Fibonacci sequence can be explained through a simple principle: optimization of space and resources.
Plants, for example, arrange their leaves along the stem in a way that prevents them from shading one another. The ideal angle for this distribution is approximately 137.5°, known as the golden angle.
This value, mathematically, is linked to the golden ratio and, consequently, to the Fibonacci sequence.
Nature tends to “choose” what works best. And often, what works best… is also surprisingly mathematically elegant.
Where Is the Fibonacci Sequence Found in Nature?
The Fibonacci sequence appears in many natural forms:
- Flower petals: many flowers have a number of petals that belong to the sequence (3, 5, 8, 13, 21…).
- Sunflower seeds: arranged in spirals that often correspond to pairs of Fibonacci numbers, such as 34 and 55.
- Pinecones and pineapples: the spirals of their scales follow Fibonacci numbers.
- Leaf distribution (phyllotaxis): leaves arrange themselves along the stem to maximize exposure to light.
- Shells (such as the nautilus): they grow according to a logarithmic spiral linked to the golden ratio.
These are not decorative details, but evolutionary strategies: ways to grow harmoniously and efficiently, saving energy and space.
Where Else Is It Found?
In addition to nature, the Fibonacci sequence also hides in:
- Classical and Renaissance art, through the “divine proportion”,
- Music, where some melodic structures follow similar proportions,
- Photography, with the rule of thirds and the golden spiral,
- Architecture of temples, cathedrals, and even modern works,
- Computer algorithms and finance, where Fibonacci-based predictive models are used to analyze trends and optimize economic decisions.
The Fibonacci sequence is also used in financial markets, where Fibonacci “retracements” are used to identify potential support and resistance levels on price charts. These patterns help analysts and traders predict potential trend reversals, based on the repetition of patterns that surprisingly reflect human behavior.
Even though these connections may sometimes be more poetic than scientific, the fact remains that the Fibonacci sequence has inspired creative minds throughout history, becoming a fundamental tool that spans not only pure mathematics but also art, nature, and the practical choices of our daily lives.
Reflections…
The Fibonacci sequence is also found in the human body: in the arrangement of bones, in the spiral of the ear, and in the number of segments in fingers. While not all of these connections have a rigorous scientific basis, they evoke a universal charm and a shared intuition: life seems to speak the language of numbers, and natural forms seem to obey a mathematical logic, invisible yet perfect.
So, the next time we observe a daisy, a pinecone, or the spiral of a snail shell, we can pause to reflect. In those small fragments of nature, we might recognize a deep order, hidden within the folds of chaos. An order that, perhaps, is not just a mathematical coincidence, but a proof that the beauty of the universe follows laws that transcend mere chance. And in this, we too, as observers, become part of a grand design, a fascinating intertwining of mathematics and life.