The year 2026 marks the centenary of the death of Antoni Gaudí (1852-1926), the architect of the Sagrada Família basilica in Barcelona. If the beauty of the building is already exceptional in itself, it gains even more depth when we discover the mathematical forms which underlie its unique architecture. By exploring the mathematical principles that structure the whole, the visual harmony of the monument takes on a new dimension, revealing an architecture where functionality, balance and coherence mutually reinforce each other.
Without a doubt, the person who most deeply studied the mathematics of the Sagrada Família is Claudi Alsina Català. Trained in mathematics at the University of Barcelona, he also supervised the doctoral thesis of the current chief architect of the site, Jordi Faulí.
In his memoirs, Claudi Alsina Català writes: “Everyone wondered if the design of the Sagrada Família was based on a module and a system of proportions guiding all of the metric relationships of the building. (…) One Saturday afternoon, sitting at my desk at home, with all the documents and all the data relating to this mysterious system of proportions – if it really existed… –, I discovered it. The module of 7.5 meters and the ratios between the divisors of 12 (1:4, 1:3, 1:2, 3:4, 2:3, 1) seemed to explain a multitude of things.
The number 12, a central element
It is not surprising that the number 12 occupies a central place in the structure of the building. Antoni Gaudí designed the Sagrada Família as a synthesis between architecture and religious symbolism, and the 12 is omnipresent in the Bible: the twelve sons of Jacob, the twelve tribes of Israel, the twelve apostles or even the crown of twelve stars from the Book of Revelation are just a few examples.
The dimensions of the basilica are closely linked to the number 12 and a module of 7.5 meters.
But its interest is not limited to its symbolic significance. From a mathematical point of view, 12 is a particularly good number for establishing proportions, because it has many divisors. It is precisely the relationships between these dividers which, according to Claudi Alsina Català, explain a large part of the system of proportions of the basilica. Considering these foundations, both symbolic and mathematical, the link between the structural elements and the number 12 is therefore not surprising.
The 7.5 meter module
Drawing on the work of Claudi Alsina Català, we offer you a brief visit to the Sagrada Família from a mathematical perspective. The dimensions of the expiatory temple are closely linked to the number 12 and a module of 7.5 meters. The building thus measures 90 meters long (7.5×12), 60 meters wide (7.5×8), while the main nave reaches 45 meters wide (7.5×6).
The heights follow the same principle: the highest vault is that of the apse, with 75 meters (7.5×10), followed by the vault of the transept, 60 meters high (7.5×8). Then come the vault of the nave, 45 meters (7.5×6), that of the side aisles, 30 meters (7.5×4), and finally the choir, whose height is 15 meters (7.5×2).
In harmony with the Montjuïc hill
The Tower of Jesus is the central and tallest tower of the Sagrada Família. With its 172.5 meters (7.5×23), its height echoes that of Montjuïc hill. It is topped by a four-branched cross, 17 meters high and 13.5 meters wide. Around it rise the four towers of the Evangelists, which rise to 135 meters (7.5×18).
With its 138 meters, Mary’s Tower is the second tallest in the basilica. It is topped with a twelve-pointed star resting on three supporting arms. With a diameter of 7.5 meters, this star is made up of a regular dodecahedron, each of whose faces is extended by a pentagonal point in the shape of a pyramid. The reflections of daylight and its nighttime interior lighting give it a unique beauty.
Polyhedra at the top of the towers
Polyhedra are also omnipresent in the towers of the Sagrada Família, as explained in this study by Claudi Alsina Català. The four towers of the Glory facade are topped with dodecahedrons, those of the Nativity facade with truncated irregular octahedrons and those of the Passion facade with truncated cubes.
At the top of each of the twelve towers rises a pinnacle above the polyhedrons. The towers dedicated to the evangelists are crowned with regular icosahedrons (solids made up of twenty faces) containing projectors which illuminate the large cross dominating the tower of Jesus. Just above each icosahedron is a sculpture symbolically representing the corresponding evangelist. The basilica also has numerous star polyhedra, particularly on the Nativity façade.
They’re not columns, they’re a forest!
The chain arches constitute one of the main structural elements of the basilica. This shape is particularly effective in transmitting loads to the ground without requiring other support elements. They are found in the system of inclined columns which supports the vaults of the interior naves, in the vaults and ceilings themselves, as well as on the Nativity facade.
Other numbers hidden in the Sagrada Família have a strong symbolic significance.
Inside the Sagrada Família, there are four types of columns. All are double helical twist columns. Their polygonal base takes the shape of a star with rounded contours and results from the intersection of two opposite Solomonic columns. Each is extended by a knot from which several ramifications emerge, similar to the branches of a tree, which support the towers and the roof of the basilica with remarkable efficiency.
The roof windows are also single-layer hyperboloids. As they are made up of straight lines, their construction is simpler while optimizing the capture and diffusion of light.
The symbolism of two numbers: 7 and 33
Other numbers hidden in the Sagrada Família have a strong symbolic significance. This is the case, for example, of the baldachin located above the main altar, a regular heptagon 5 meters in diameter, whose seven sides represent the seven gifts of the Holy Spirit.
On the facade of the Passion there is also a magic square with all the lines, columns and diagonals totaling 33, a number with obvious religious connotations. It seems to be inspired by the magic square depicted in the engraving Melencolia Icreated by the German artist Albrecht Dürer in 1514. Uncovering the mathematics behind the Sagrada Família only increases the beauty of the building and the admiration aroused by the genius of Antoni Gaudí.
