The mathematical secrets of Barcelona's Sagrada Familia

I visited this and several other Gaudi buildings about 15 years ago in Barcelona, and many of them are truly breathtaking, or at least dramatically original and unique. I went to the Gaudi museum as well and found it fascinating that the architect himself was not a professional mathematician - he did not use hyperbolic cosine to calculate the dimensions of the catenary curves, he traced the outline of hanging chains. Really interesting to hear about how he also heavily used ratios and symmetry. I love how artistic taste can be partially derived from math (but the math itself isn't sufficient to develop artistic taste)

Be sure to visit the basement museum to see the upside-down string + tiny sandbag weights model that was used to simulate the structure. Pure genius.

"Gothic cathedrals and Doric temples are mathematics in stone. Doubtless Pythagoras was the first in the Classical Culture to conceive number scientifically as the principle of a world-order of comprehensible things—as standard and as magnitude—but even before him it had found expression, as a noble arraying of sensuous-material units, in the strict canon of the statue and the Doric order of columns. The great arts are, one and all, modes of interpretation by means of limits based on number (consider, for example, the problem of space-representation in oil painting). A high mathematical endowment may, without any mathematical science whatsoever, come to fruition and full self-knowledge in technical spheres." ~ Spengler, Decline of the West

Gaudi stuff is cool. I used to live in Barcelona, I know all about it. But numerology is not the same as mathematics- nothing in the article comes close to sniffing at the math of classical architecture. And while many have said I have a well-developed architectural taste, and I certainly understand the artistic fascination and amusement with Gaudi’s works, I never understood the magnitude of architectural devotion that they have attracted.

Call me grumpy, but numerology does not make architecture more beautiful.

Vaguely related, there is also the unfinished Cathedral of Justo [1] near Madrid built by a solo developer since 1961 and essentially by hand and more or less from scrap and whatever people donated. Justo Gallego Martínez died in 2021 at age 96 and donated the building to some foundation for completion.

[1] https://en.wikipedia.org/wiki/Cathedral_of_Justo

"The temple’s dimensions are based on the number 12 and a module of 7.5m."

12 / 7.5 = 1.6 ~= Golden ratio

Such a spectacular building. I could spend all day watching those color gradients move across the walls and floor.

I went went once it's very beautiful inside! And the view from the top is also wow! Thanks for sharing this

Such a spectacular building. I could spend all day watching those color gradients move across the walls and floor.

I had to put this one through Claude, but it boils down to:

> A culture's felt sense of proportion, ratio, and spatial order manifest directly through the hands of masons and sculptors, without necessarily needing the mathematical formalism of proofs, axioms, and treatises.

Not sure how I feel about this, as the Familia was absolutely built in a context of formalised mathematical sciences.

I had to put this one through Claude, but it boils down to:

Not sure how I feel about this, as the Familia was absolutely built in a context of formalised mathematical sciences.

It seems somewhat important to me to know if something was done because it looked pretty, was random or because there was an intent to reflect maths, science, planetary alignment etc.

Whatever one might say about method, epistemically speaking, the aesthetic is prior to the mathematical. The mathematical is found in the analysis of the beautiful.

Mathematics and Art - https://en.wikipedia.org/wiki/Mathematics_and_art

Wikipedia on "Sagrada Família" - https://en.wikipedia.org/wiki/Sagrada_Fam%C3%ADlia (see "Geometric Details" section).

Gaudí used hyperboloid structures in later designs for Sagrada Família (more obviously after 1914). However, there are a few places on the nativity façade—a design not equated with Gaudí's ruled-surface design—where the hyperboloid appears.

Mathematics and Art - https://en.wikipedia.org/wiki/Mathematics_and_art

Wikipedia on "Sagrada Família" - https://en.wikipedia.org/wiki/Sagrada_Fam%C3%ADlia (see "Geometric Details" section).

Whatever one might say about method, epistemically speaking, the aesthetic is prior to the mathematical. The mathematical is found in the analysis of the beautiful.

Truth, beauty and goodness are known as the “transcendentals” for a reason. They are linked. They don’t come one after the other.

Be sure to visit the basement museum to see the upside-down string + tiny sandbag weights model that was used to simulate the structure. Pure genius.

[1] https://en.wikipedia.org/wiki/Cathedral_of_Justo

"The temple’s dimensions are based on the number 12 and a module of 7.5m."

12 / 7.5 = 1.6 ~= Golden ratio

I went went once it's very beautiful inside! And the view from the top is also wow! Thanks for sharing this

Call me grumpy, but numerology does not make architecture more beautiful.

That's not grumpy but ignorant. Aspect ratios are not numerology (per se)

Designing in an era where calculus exists, using chains and weights strikes me as gratuitous or onanistic.

The Ancient Greeks and Romans also used the same or similar empirical geometric methods to generate ellipses, parabolas, and hyperbolas in their architecture. The difference is, they were still 1000-2000 years away from having formalized calculus.

> I love how artistic taste can be partially derived from math [...] the math itself isn't sufficient to develop artistic taste

Strictly speaking, it isn't "math" as math is the science of quantity and structure, both of which are objective features of reality. We all perceive structure and quantity as it is instantiated in concrete things and ensembles of concrete things and so on. We all respond to and reason about quantifiable and structural properties of reality at varying depths all the time. All math does is pursue them intentionally and methodically. It isn't surprising, then, that a competent artist should intuit various mathematical truths. Indeed, quantity and structure as essential to art. The artist is therefore closer to a domain-specific application where such properties are understood in relation to the subject matter. This introduces a domain-specific aesthetic dimension that is not present in abstracted properties, though one can certainly make aesthetic judgements about abstracted properties.

> I love how artistic taste can be partially derived from math [...] the math itself isn't sufficient to develop artistic taste

That's not grumpy but ignorant. Aspect ratios are not numerology (per se)

Designing in an era where calculus exists, using chains and weights strikes me as gratuitous or onanistic.

Calculus exists, but analytic solutions generally don't. Gaudi's chains and weights serve as an incredibly elegant mechanical computer that were only surpassed in the last few decades by CAD. Designers used mechanical splines until the advent of CAD in the 70's/80's.

Authors: Sergi Muria Maldonado, Professor de Didàctica de les Matemàtiques, Universitat de Barcelona; Anton Aubanell Pou, Professor de l’Institut de Formació Continuada i professor jubilat de Didàctica de les Matemàtiques, Universitat de Barcelona, and Jordi Font González, Professor de Didàctica de les Matemàtiques, Universitat de Barcelona

Sagrada Familia

The basilica’s columns branch out, imitating the natrual structure of a tree.
David Herraez Calzada/Shutterstock

2026 marks 100 years since the death of Antoni Gaudí, the architect of the Basilica of the Sagrada Familia in Barcelona. While the temple’s beauty is extraordinary in its own right, it becomes even more profound when we explore the numerical patterns that lie behind its striking forms.

By contemplating the mathematical principles that underpin its structure, the visual harmony of the whole takes on a new dimension, endowing it with a renewed functionality, balance and coherence.

Mathematician Claudi Alsina i Català deeply studied the mathematics of the Sagrada Família. He undertook his initial studies in this field at the University of Barcelona, and supervised the doctoral thesis of Jordi Faulí, the architect currently in charge of the temple’s ongoing construction.

In his memoirs, Alsina stated:

Many had wondered whether the design of the Sagrada Família contained some module or system of proportions that guided the building’s metric relationships. (…) One Saturday afternoon, sitting at my desk at home, with all the data and documents on this mysterious proportional system – if indeed it existed – I discovered it. The 7.5-metre module and the ratios between the divisors of 12 (1:4, 1:3, 1:2, 3:4, 2:3, 1) seemed to explain a great deal.

12: the magic number

It is no surprise that the number 12 plays a prominent role in the structure of the church. Gaudí conceived the Sagrada Família as a synthesis of architecture and religious symbolism, and the number 12 features heavily in the Bible: Jacob’s 12 sons, the 12 tribes of Israel, the 12 apostles and the crown of 12 stars in the Book of Revelation are just a few examples.

But its significance is not merely symbolic. From a mathematical point of view, 12 is a number particularly well-suited to establishing proportions, as it has many divisors. According to Alsina, the relationships between these divisors account for much of the basilica’s proportional system.

The 7.5-metre module

Drawing on Alsina’s work, we invite you to take a brief mathematical tour of the Sagrada Família.

The temple’s dimensions are based on the number 12 and a module of 7.5m. It is 90m long (7.5 × 12) and 60m wide (7.5 × 8). The width of the main nave is 45m (7.5 × 6).

In terms of height, the highest vault is that of the apse, at 75m (7.5 × 10), followed by the vault of the transept at 60 metres (7.5 × 8). The nave vault is 45m high (7.5 × 6), the side aisle 30m (7.5 × 4), and the choir 15m (7.5 × 2).

The Tower of Jesus Christ is the central and tallest of the cathedral. It stands at 172.5 metres (7.5 x 23), close to the height of the landmark hill of Montjuïc. It is crowned by a four-armed cross, 17m high and 13.5m wide. Surrounding this are the four Evangelist Spires, which reach a height of 135m (7.5 x 18).

The Star of the Virgin Mary sits atop the tower of the same name.
Canaan, CC BY

The 138-metre high Tower of the Virgin Mary is the second tallest in the basilica. It is crowned by a 12-pointed star, which rests on three supporting arms. This star has a diameter of 7.5m, and is made up of a regular dodecahedron, with pyramid-shaped pentagonal points rising from each face. Its reflections of daylight and night-time illumination lend this star a unique beauty.

Polyhedral towers

Polyhedrons also feature prominently in the towers of the Sagrada Família. The four towers of the Glory façade are topped by dodecahedrons, the four towers of the Nativity façade by truncated irregular octahedrons, and the four towers of the Passion façade by truncated cubes.

On each of the 12 towers, a spire rises above these polyhedrons. Those dedicated to the evangelists are crowned with regular icosahedrons containing spotlights that illuminate the large cross that sits atop the Tower of Jesus Christ. Just above each icosahedron is a sculpture that symbolically depicts each evangelist. There are numerous star-shaped polyhedrons throughout the church, particularly on the Nativity façade.

The towers of the Nativity façade, crowned by octahedrons.
Yura Tarasovskyy/Shutterstock

A forest of towers

Catenary arches feature prominently as key structural elements of the church, as they are a highly effective way to transfer loads to the ground without the need for additional support. They can be seen in the system of sloping columns that support the vaults of the interior naves, the vaults and ceilings themselves, and the Nativity façade.

Inside the Sagrada Família, there are four different types of column. All are double-helix torsion columns – each has a rounded, star-shaped polygonal base, and is formed by the intersection of two opposing Solomonic columns. Above each one is a knot from which different branches emerge, similar to those of a tree, which very efficiently support the towers and the roof of the church.

The skylights in the roof are also one-sheet hyperboloids. Made up of straight lines, they are easy to construct, and optimise the capture and projection of light.

The symbolism of 7 and 33

The church has other deeply symbolic hidden features. Take, for instance, the canopy above the high altar, which forms a regular heptagon 5 metres in diameter, whose seven sides symbolise the seven gifts of the Holy Spirit.

On the left, Melancholia I by Albrecht Dürer, with a numerical grid visble in the top right-hand corner. On the right, the magic square designed by the sculptor Josep Maria Subirachs.
Jordi Domènech/Wikimedia Commons/, CC BY-SA

On the Passion façade there is a magic square in which the sum of all rows, columns and diagonals is 33. It appears to be inspired by the magic square featured in the engraving Melancholia I by Albrecht Dürer.

The mathematics that underpins the Sagrada Família makes it all the more beautiful. While the building itself is a sight to behold, a deeper understanding of the principles behind it inspires even greater admiration for the enduring genius of Antoni Gaudí.

This article is republished from The Conversation under a Creative Commons license. Original article.

Hacker Times