0 0, Zephyr, the number that did not exist. The Romans did not have the number "zero". It was the void they expressed with linguistic terms, for example non est, in accounting or legal documents, instead of zero they wrote nulla. Their system, additive and subtractive but not positional, did not require zero to function. However, precisely this absence limited the possibility of developing complex calculations, paving the way for the numerical revolution that was about to arrive

1 Who was Fibonacci? Leonardo Pisano called Fibonacci (from filius Bonacci, son of Bonaccio), was born in Pisa around 1170 and died shortly after 1240. Pisa, then a maritime republic, prospered thanks to trade. The dates are not precise because they are inferred from documents without having specific certificates or registers available. His father was a customs official in Bugia, present-day Algeria, which allowed him to come into contact with Arab culture and mathematics. He wrote several works: Liber Abbaci, Practica Geometriae, Liber Quadratorum. He is considered the greatest mathematician of the Middle Ages and this was recognized by Emperor Frederick II of Swabia and the Pisan Republic, which granted him a life pension for his scientific merits. This allows us to approximate a presumed date of death because the act of the Municipality of Pisa that assigns him the life pension is from 1240 and he died shortly after.

1 Liber Abbaci, the manuscript that changed Europe. The book of calculation is Fibonacci's most important and famous work. The first draft of 1202 has been lost, the second version, expanded, dates back to 1228 and is preserved at the National Central Library of Florence (BNCF). Written as a practical manual containing various demonstrations and riddles, the text changed mathematics in Europe by introducing Arabic numerals, the positional system and zero, making it possible to easily perform multiplications and divisions. It is the system we still use today, the position of the number determines its value: units, tens, hundreds, etc. from right to left. What we take for granted since elementary school, Roman numerals made practically impossible. Mathematics becomes a tool for commercial and mercantile activity.

3 Arabic numerals. Actually they were Indian... The decimal positional system, with base ten and the use of zero, that is the system we currently use, was born in India between the 6th and 7th centuries. It is interesting to note that, within the positional system, we also find the vigesimal system, which arose from the fact that using the fingers of hands and feet one could count up to twenty, or the base twelve system, like the months of the year.

From the 8th century onwards, Indian scientific and mathematical texts were translated by the Arabs, who imported and perfected the Hindu digits plus zero. Two versions of the digits developed – eastern and western – and it was precisely the latter that entered the European tradition. European Christians came into contact with them through the commercial and cultural routes of Islam. Here we open a curiosity. It is thought, but not confirmed, that Gerberto was the first who tried, unsuccessfully, to introduce Arabic numerals to Europe. He was born Christian, fell in love with Arab culture and science, hence the idea (historically unfounded) of his conversion to Islam. He was a very controversial and complex figure, with many legends that arose about him; these did not prevent him from becoming Pope with the name Sylvester II. A mathematician, like the current Leo XIV, became Pope.

The period in which Fibonacci lived was one of great tensions and interreligious wars, the Crusades were being fought and the military confrontation with Islam grew until it reached its climax with the battle of Lepanto. Despite this, numbers become a universal language that unites cultures and peoples. Starting from the Indians (Hindus), through the Arabs (Muslims) they reach the Europeans (Christians). This seems to suggest the existence of a universal Divinity. A millennium ahead of time it seems the embryo of the thought of Jung and Faggin, who starting from completely different positions, reach a very similar conclusion, the One.

5 The Fibonacci sequence. It appears, in chapter XII, as a practical solution to the rabbit problem. A pair of rabbits matures every month and generates a new pair. Assuming that the rabbits do not die (are we vegetarians?), how many pairs will there be after n months? The answer is a sequence of numbers in which each term is the sum of the two preceding ones: 1, 1, 2, 3, 5, 8, 13, 21, and so on. What was born as a practical solution to a demographic growth problem has become over the centuries one of the most studied and applied sequences, from nature to music, from art to finance.

Here is the original Latin text (Liber Abbaci, 1202/1228, Ch. XII):

"Cum posuerit unus homo par unum coniculorum in uno loco, qui par in uno mense generat alium par, et par ille qui natus fuerit in secundo mense generat alium par, similiter et par primus in omni mense generat alium par: quot paria coniculorum ex illo pario fiunt in fine unius anni?"

And here is the Italian translation:

"A man places a pair of rabbits in a place. This pair every month generates another pair, and the pair born begins to generate in turn starting from the second month; thus every pair, from the second month of its life onwards, generates another pair every month. How many pairs of rabbits will be formed in total at the end of a year?

This is the solution to the rabbit problem – Liber Abbaci (Ch. XII)

Thinking about the importance that the Sequence has had over the following centuries, the countless properties and connections we have identified, it is amusing to observe that Fibonacci had limited himself to solving a very practical problem, without any implications. He did not grasp the universal scope of his discovery. It is a bit like Christopher Columbus who discovered the American continent convinced he was in India, he had not understood. Amusing.

8 Mathematical games. Chapter XII of the Liber Abbaci contains not only the rabbit problem, but also other mathematical games and riddles of the Middle Ages, designed to amaze and convince readers to use Arabic numerals, at the time called the numbers "of the infidels". These games had a pedagogical and persuasive value: they showed how much simpler and more powerful it was to use Arabic numerals compared to Roman ones. Today we would say for communication & marketing. The dispute between the proponents of Arabic numerals and the proponents of Roman numerals went on for several centuries, we know the outcome...

The main problems are:

13 Abaci is not the plural of Abacus. As mentioned before, Roman numerals were not positional, lacked zero, and were unsuitable for complex calculations. Try performing simple arithmetic operations of addition, subtraction, and even multiplication and division with Roman numerals using paper and pen... We'll have fun doing this together on our social media channels; it will be amusing and entertaining. Unfortunately, emerging science, commerce, and economy needed to perform calculations quickly. To solve this problem, an instrument called the abacus was used, making arithmetic operations practical. There are two types, with the column type being the most common. It was essentially a rectangle where columns provided positional logic, identifying units, tens, hundreds, while each column had rings, buttons, or pebbles that identified numbers. It functioned very similarly to the English exchequer, from which the British finance minister takes the title of Chancellor of the Exchequer. It was suitable for quickly performing additions and subtractions, while multiplications and divisions were possible but slower as they were performed as repeated additions or subtractions. The introduction of Arabic numerals made calculation possible as we do it now, much more simply with just paper and pen. Afterward, various calculation aids were invented, such as Napier's Bones, Genaille's Rods, and the Pascaline, invented by Blaise Pascal, the forerunner of modern pocket calculators.

21 The Golden Ratio. φ ≈ 1.618033988...., this number is called phi, in honor of Phidias, the Greek sculptor. The Fibonacci Sequence relates to the Golden Ratio as bricks relate to a house, here's why:

Where the Golden Ratio emerges as the limit for n, number of elements, approaching infinity of the ratio between two Fibonacci Sequences, where the numerator has one more element (this is why it returns a number greater than one).

The ratio between the Sequence in the numerator and denominator can also be seen as the ratio between the perfect and imperfect, between the Divine and Human. Humans increasingly approach the divine without ever fully reaching it, in a continuous tension toward improvement, development, and growth, including inner growth. We tend toward perfection and almost touch it, yet never reach it. This continuous effort toward impossible perfection, which humans pursue through centuries, is reflected in many aspects of human nature. In economics, models cannot manage everything because an equation is always missing. In Freemasonry, the search for the Sacred word is constant, but it remains unknown, signifying the impossibility of reaching perfection, the divine. The Golden Ratio is not just a number, but a message, present everywhere in nature and where there is harmony. It is a universal symbol connecting mathematics, nature, art, and spirituality, as we shall see now.

34 The Fibonacci Sequence and Golden Ratio in nature. Many plants arrange leaves, petals, or seeds according to a pattern following the Fibonacci Sequence, often to optimize space and resources. In sunflowers, seeds are arranged in spirals, with two families of spirals often having consecutive Fibonacci numbers (34 and 55, or 55 and 89). In Romanesco broccoli, its fractal structure follows Fibonacci-related spirals. A brief note: fractal mathematics is quite recent; since 1975, it studies geometric figures, called fractals, characterized by self-similarity and fine structure. A fractal is an object whose shape or structure repeats infinitely at increasingly smaller scales, maintaining the same geometric characteristics at any level of detail. This principle leads to highly irregular and fragmented forms that cannot be described with traditional Euclidean geometry. Daisies, which everyone has plucked for the "loves me, loves me not" game, commonly have a number of petals belonging to the sequence - 21, 34, 55, or 89, so did whether our beloved returned our feelings depend on Fibonacci...?

Similarly, the spiral created with the golden ratio is present in nature in the Nautilus shell and in the shape of galaxies

Rabbits multiply according to Fibonacci's predictions; just spend a few hours in the beautiful Urban Park of Forlì to realize this...

55 The Fibonacci Sequence and Golden Ratio in architecture. Everything we see built following the rules of the Golden Ratio, we tend to perceive as beautiful. A first unintentional example - because the Greeks didn't know about it - is in the Parthenon of Athens, dating back to the 5th century BC, where the facade and columns are built according to golden proportions.

It was Luca Pacioli, the author of the book De Divina Proportione in 1509, and Leonardo da Vinci who studied the golden ratio and applied it to painting and architecture.

More recently, Le Corbusier, the famous architect, studied the proportion and developed the Modulor.

The Golden Ratio is thus a bridge between mathematics and aesthetics.

89 The Fibonacci Sequence and finance. The Fibonacci sequence is extensively used in technical analysis by traders in quoted financial markets, with the so-called Fibonacci Retracements.

These are tools that use horizontal lines to identify areas where a security's price might find support or resistance. It's a very technical topic that will be the subject of a specific dedicated article.

144 Alchemy and esotericism. As previously written, Fibonacci himself did not attribute any occult meaning to his numbers. Alchemical and esoteric interpretations arose centuries later, when the scientia numerorum was reinterpreted in light of hermeticism, cabala, and "natural philosophy." The strength of the sequence lies in its mathematical neutrality: precisely because it is objective, it becomes an ideal screen to project and accommodate different spiritual visions. However, some properties of the Golden Ratio have stimulated the sensibility of many scholars. The Franciscan friar and mathematician Luca Pacioli, in his treatise De Divina Proportione, describes the golden proportion (φ) as "divine" for its mathematical properties and harmonic perfection, attributing these mystical characteristics: it is one and triune because it reflects the Christian Trinity: one, infinite, and indivisible; it is universal harmony because it binds part and whole in perfect balance; it is irrational because although well-known, it cannot be captured, symbolizing divine infinity.

233 Let us conclude, for now. Fibonacci's Liber Abbaci is no longer just a manuscript, no longer just a manual of practical demonstrations. It is a bridge between past and future, between medieval manuscripts and contemporary scientific research. It is a work that has united different cultures, from India to Europe, passing through the Arab world, using the universal language of mathematics. It is a symbol of harmony, cooperation, tolerance, and dialogue. It is, even more, a sacred object, a relic, and its Sequence is a powerful symbol, a testimony of the divine, which we find wherever there is Harmony.

The Liber Abbaci: the masterpiece you can hold in your hands
Fibonacci's Liber Abbaci manuscript is not just a book; it is a symbol that has forever changed the history of knowledge and Europe.
Now you can own it, in a faithful and prestigious facsimile, available only on Antiquus.it
It's not a purchase: it's an investiture, a consecration.
It's making direct contact with the genius who introduced Arabic numerals, the Fibonacci sequence, and the Golden Ratio.
It's making direct contact with Harmony
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0 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181,...