0, 1,1,2,3,5,8,13,21,34,55,89,144,233,377,610,987,1597,2584,4181,6765

and

1,61803398874989484820458683436563811772030917980576...

What are these numbers, and why are they so important and interconnected? Step by step…

What is the Fibonacci sequence?

Thanks in part to the film *The Da Vinci Code*, the Fibonacci sequence has become one of the most famous and recognizable numerical sequences in the world, shrouded in a unique allure that straddles the line between mathematical rigor and symbolic imagery.

It’s not out of the question that someone might still be using it as a password—and not just to open a safe deposit box at a Swiss bank…

Fibonacci introduced it in the *Liber Abbaci* through a famous problem—almost a brainteaser—designed to illustrate the growth of a rabbit population starting with a single pair. 😊

Over the centuries, this sequence has revealed surprising properties, demonstrating an extraordinary ability to describe and interpret numerous natural phenomena, as we have already noted in previous articles.

For now, we will limit ourselves to recalling its fundamental rule: it is constructed using natural numbers, and each subsequent term is given by the sum of the two preceding terms.

What is the Golden Section?

It is an irrational number denoted by the Greek letter φ (phi); it is obtained when a line segment is divided into two parts such that:
major part : minor part = whole : major part
In formula:

(a+b)/a = a/b = φ

The golden ratio is a specific proportion between two quantities.
Imagine a line segment divided into two parts:

long side: a
short side: b

A division is “golden” when the following relationship holds:

(a + b) / a = a / b

That is to say:

The ratio of the whole segment to the larger part
is equal to the ratio of the larger part to the smaller part

Image taken from the website infinitoteatrodelcosmo.it

Visually, let's take a rectangle as an example

Imagine a rectangle that is slightly longer than it is tall. If the long side is 1.618 times the short side, you have a golden rectangle.

From this rectangle, you can:

• remove a square
• ottenere un rettangolo più piccolo con la stessa proporzione
• repeat the process

If you join the corners with a curve, you get a spiral similar to the one often shown in drawings of the golden ratio.

Image taken from the website infinitoteatrodelcosmo.it

This is how the golden ratio is known to the general public; it is an irrational number, we will find out later why it is important to underline this,

Let's try some math…

Let's start from the definition of the golden section:

with a>b>0.

Let's call this common ratio x:

Then we can rewrite the first member:

but since

therefore:

Now let's multiply everything by x:

or:

This is a quadratic equation. Let's apply the solution formula:

so the two solutions are:

But the ratio of positive lengths must be positive, so we discard the second solution, which is negative.

Remains:

Numeric value

φ≈1,618033988…

Proviamo una rapida verifica, come se scrivessimo un compito di matematica…

If

then it also satisfies the fundamental property:

φ^2=φ+1

Infact:

e

so it matches… Exam passed 😊

The connection between the Fibonacci Sequence and the Golden Ratio

The strict connection formula is:

where

The golden ratio is the limit, as n tends to infinity, of the ratio between two consecutive terms of the Fibonacci sequence, in which the term n+1 appears in the numerator and the term n in the denominator. Precisely for this reason, this ratio tends to a number greater than one, that is, to the divine proportion.

But is this relationship only a mathematical relationship, or can it also be read in a symbolic sense?
And are the terms of the sequence just numbers, or can they allude to something further?

And is that limit that tends toward infinity only a mathematical procedure, or can it also symbolically evoke man's natural tension toward improvement, toward growth, toward a fulfillment that can never be fully possessed?

Our answers—and we deliberately say ours—exit here from the strictly mathematical realm to enter that of harmony: that harmony which, almost three centuries after Fibonacci, would inspire an author we will shortly explore.

From this perspective, one might perhaps see:

• in the term placed in the numerator an image of the Divinity, however each religion wishes to understand it;
• in the denominator, an image of man, always marked by a lack, by an incomplete measure with respect to the fullness of the divine.

The relationship would then become a figure of dialogue: that dialogue that many men maintain with the Divinity, through rites, functions, prayer or interiority.

Moving further back in time, we come to Fra' Luca Pacioli, the great Renaissance interpreter of the divine proportion, as well as the one who systematically developed double-entry bookkeeping, destined to become the foundation of modern accounting.

Luca Pacioli

“IF YOU DISCOURSE WELL IN ALL THE ARTS YOU WILL FIND THE PROPORTION OF ALL BEING MOTHER AND QUEEN AND WITHOUT HER NONE CAN BE EXERCISED”

Luca Pacioli, De Divina Proportione, 1509

A few centuries after Fibonacci, it was Fra’ Luca Pacioli who brought the intrinsic properties of the golden ratio to the attention of the educated public, enshrining it in his famous treatise De Divina Proportione.

In chapter V, Pacioli declares that he wants to expose four properties “a necessaria”, to which he then adds a fifth suitability: overall, therefore, five fundamental properties.

1. Unit

The divine proportion is one in its essence: it does not divide into different species, but remains identical to itself. For this reason, Pacioli associates it with the idea of ​​perfect unity.

2. Trinity

Although it is one, it manifests itself through three terms: the whole, the greater part, and the lesser part. In this ternary structure, Pacioli recognizes a symbolic reflection of the Trinity.

3. Irrationality

The divine proportion cannot be expressed precisely by a numerical fraction, since it is irrational. For Pacioli, this incommensurability evokes something that exceeds man's full rational comprehension.

4. Immutability

It remains constant and invariable in its relationships: it does not change with the changes in the figures in which it manifests itself. In this sense, Pacioli considers it a sign of a superior stability.

5. Generative and cosmic virtue

Finally, for Pacioli, divine proportion has a constructive function: it influences the order of regular bodies and the beauty of the geometric structure of the world. It is not only a measure, but also a principle of formation.

For Luca Pacioli, therefore, the golden ratio is "divine" not only for its mathematical elegance, but also for the richness of its symbolic correspondences. It is one, yet manifests itself in three terms; it is irrational, and therefore exempt from full numerical reduction; its internal law is immutable; and finally, it exerts a generative function in the order of geometric bodies. Thus, the golden ratio becomes, in Pacioli's thought, a meeting point between mathematics, metaphysics, and the harmony of creation.

Luca Pacioli

“IF YOU DISCOURSE WELL IN ALL THE ARTS YOU WILL FIND THE PROPORTION OF ALL BEING MOTHER AND QUEEN AND WITHOUT HER NONE CAN BE EXERCISED”

Luca Pacioli, De Divina Proportione, 1509

A few centuries after Fibonacci, it was Fra’ Luca Pacioli who brought the intrinsic properties of the golden ratio to the attention of the educated public, enshrining it in his famous treatise De Divina Proportione.

In chapter V, Pacioli declares that he wants to expose four properties “a necessaria”, to which he then adds a fifth suitability: overall, therefore, five fundamental properties.

1. Unit

The divine proportion is one in its essence: it does not divide into different species, but remains identical to itself. For this reason, Pacioli associates it with the idea of ​​perfect unity.

2. Trinity

Although it is one, it manifests itself through three terms: the whole, the greater part, and the lesser part. In this ternary structure, Pacioli recognizes a symbolic reflection of the Trinity.

3. Irrationality

The divine proportion cannot be expressed precisely by a numerical fraction, since it is irrational. For Pacioli, this incommensurability evokes something that exceeds man's full rational comprehension.

4. Immutability

It remains constant and invariable in its relationships: it does not change with the changes in the figures in which it manifests itself. In this sense, Pacioli considers it a sign of a superior stability.

5. Generative and cosmic virtue

Finally, for Pacioli, divine proportion has a constructive function: it influences the order of regular bodies and the beauty of the geometric structure of the world. It is not only a measure, but also a principle of formation.

For Luca Pacioli, therefore, the golden ratio is "divine" not only for its mathematical elegance, but also for the richness of its symbolic correspondences. It is one, yet manifests itself in three terms; it is irrational, and therefore exempt from full numerical reduction; its internal law is immutable; and finally, it exerts a generative function in the order of geometric bodies. Thus, the golden ratio becomes, in Pacioli's thought, a meeting point between mathematics, metaphysics, and the harmony of creation.

Conclusion

The Fibonacci sequence and the golden ratio are linked by a profound connection, such that the former can be regarded as the numerical unfolding of what the latter achieves as the perfect form of proportion. It is not, therefore, a mere juxtaposition of a series of numbers and a geometric formula, but the manifestation of the same law in two different forms: one arithmetic, the other proportional.

Since the ratio of consecutive terms in the sequence gradually converges to

𝜑

The arithmetic growth of the Fibonacci numbers finds its fulfillment in the balance of the divine proportion. In this sense, the sequence is not merely a progression built through addition, but the ordered path through which the number seems to move toward a higher measure, toward a form of completeness that no longer belongs solely to the realm of calculation, but to that of harmony.

Thus, what initially presents itself as a series of successive sums is transformed, at its limit, into a law of harmony: Fibonacci represents the evolution of the number, the golden ratio its proportional equilibrium. One shows movement, the other measure; one expresses growth, the other the perfect form toward which that growth tends. For this reason, their connection has spanned the centuries, fascinating not only mathematicians and geometers, but also philosophers, artists, and minds inclined to recognize in numbers a more secret fabric of reality.

We began with a pair of rabbits—that is, with a problem that seemed simple and almost playful—to explore the mathematics of numbers; from there we moved on to the geometry of form, and from there we can continue our journey toward the aesthetics and metaphysics of proportion. This is precisely one of the most surprising aspects of Fibonacci’s legacy: from a fundamental numerical rule, a vast horizon opens up, in which calculation becomes contemplation, and measurement can be read as a sign of a deeper order.

As a side note to this discussion, and almost as a natural extension of the topic, it is worth noting that Fibonacci is not merely a figure in the history of mathematics, but continues to this day to spark cultural, collector’s, and symbolic interest. That is precisely why we offer for sale papers and a facsimile manuscript dedicated to him: artifacts that allow us not only to study his work but also to physically preserve its charm and memory.

That is precisely why we offer for sale cards and a facsimile of Fibonacci’s manuscript, designed for those who wish not only to read about this tradition but also to preserve a tangible and precious memento of it.