Eve is here. I’ve only been to Sagrada Familia once, but I was lucky enough to go on a Sunday morning when it wasn’t too crowded. If I lived in Barcelona, I would definitely visit from time to time. It’s a wonderful building.
Written by Sergi Muria Maldonado, Professor of Mathematics at the University of Barcelona. Mr. Anton Overel Pou, Professor at the Institute of Continuing Training and Retired Professor of Mathematics at the University of Barcelona. Jordi Font González, Professor of Mathematics at the University of Barcelona. Originally published on The Conversation
2026 marks 100 years since the death of Antoni Gaudi, the architect of Barcelona’s Sagrada Familia cathedral. The beauty of this temple is extraordinary in itself, but it deepens even more when you explore the numerical form behind its impressive form.
By contemplating the mathematical principles that underpin its construction, the visual harmony of the whole acquires a new dimension, giving it new functionality, balance and coherence.
Mathematician Claudi Alsina y Catala deeply studied the mathematics of Sagrada Familia. He conducted the first research in this field at the University of Barcelona, where he supervised the doctoral thesis of architect Jordi Fauli, who is responsible for the temple’s ongoing construction.
Alsina said in her memoirs:
Many wondered if the Sagrada Familia design included some kind of module or ratio system that guided the building’s dimensional relationships. (…) I discovered it one Saturday afternoon when I was sitting at my desk at home with all the data and documents about this mysterious proportional system. The ratio between the 7.5 meter module and the divisors of 12 (1:4, 1:3, 1:2, 3:4, 2:3, 1) seemed to explain a lot.
12: Magic number
It is not surprising that the number 12 plays an important role in the structure of the church. Gaudi envisioned Sagrada Familia as a synthesis of architecture and religious symbolism, and the number 12 features prominently in the Bible. The 12 sons of Jacob, the 12 tribes of Israel, the 12 apostles, and the 12 star crowns of Revelation are just a few examples.
But its importance is not just symbolic. From a mathematical perspective, 12 is a particularly good number for establishing ratios because it has many divisors. According to Alsina, the relationship between these divisors is a large part of the cathedral’s proportional system.
7.5m module
Take a quick math tour of the Sagrada Familia, inspired by Alsina’s work.
The dimensions of the temple are based on the number 12 and a module of 7.5 meters. It is 90m long (7.5×12) and 60m wide (7.5×8). The width of the main corridor is 45 m (7.5 x 6).
In terms of height, the tallest vault is the apse vault with a height of 75 m (7.5 × 10), followed by the transept vault with a height of 60 m (7.5 × 8). The height of the nave vault is 45 meters (7.5 × 6), the height of the side aisles is 30 meters (7.5 × 4), and the height of the choir seats is 15 meters (7.5 × 2).
The Tower of Jesus Christ is in the center of the cathedral and is the tallest. Its height is 172.5 meters (7.5 x 23), close to the height of the landmark Montjuic hill. At the top is a four-armed cross, 17 meters high and 13.5 meters wide. Surrounding it are four evangelistic spiers reaching a height of 135 meters (7.5 x 18).
The Star of the Virgin Mary sits atop the tower of the same name. Canaan, CC BY
At 138 meters high, the Tower of the Virgin Mary is the second tallest building in the basilica. The crown has a 12-pointed star and is supported by three supporting arms. The star is 7.5 meters in diameter and is made up of a regular dodecahedron, with each face having a pyramidal pentagonal apex. The reflections of daylight and nighttime lighting give this star a unique beauty.
polyhedral tower
Polyhedra are also prominently seen in the towers of Sagrada Familia. The four towers of the Glory facade are surmounted by dodecahedrons, the four towers of the Nativity facade have truncated irregular octahedrons, and the four towers of the Passion facade are truncated cubes.
Each of the 12 towers has a spire rising above these polyhedrons. The one dedicated to the Evangelist is crowned with an icosahedron with a spotlight illuminating the large cross at the top of the tower of Jesus Christ. Immediately above each icosahedron is a carving symbolically depicting each evangelist. There are many star-shaped polyhedrons throughout the church, especially on the Nativity facade.
tower forest
Catenary arches stand out as important structural elements in churches because they are a highly effective way of transferring loads to the ground without the need for additional support. They can be seen in the vaults inside the nave, in the vaults and ceiling itself, and in the system of sloping columns supporting the Nativity façade.
There are four types of columns inside Sagrada Familia. All are double-helix torsion columns, each with a rounded star-shaped polygonal base and formed by the intersection of two opposing Solomon columns. At the top of each is a knot, from which various branches, resembling tree branches, emerge, very efficiently supporting the roof of a tower or church.
The skylight on the roof is also a single hyperboloid. Constructed of straight lines, it is easy to construct and optimizes light capture and projection.
Symbolism of 7 and 33
The church also has other hidden features that are very symbolic. For example, consider the canopy over the main altar. This canopy forms a regular heptagon with a diameter of 5 meters, the seven sides of which symbolize the seven gifts of the Holy Spirit.
On the left is Albrecht Dürer’s Melancholia I, with a grid of numbers in the upper right corner. On the right is a magic square designed by sculptor Josep Maria Subirax. Jordi Domenech/Wikimedia Commons/, CC BY-SA
Passion’s facade has a magic square with all rows, columns, and diagonals totaling 33. This seems to be inspired by the magic square that appears in Albrecht Dürer’s print Melancholia I.
The mathematics behind Sagrada Familia is what makes it the most beautiful thing. The building itself is a highlight, but a deeper understanding of the principles behind it will only enhance your admiration for the timeless genius of Antoni Gaudí.
