The Mathematical Secrets of Barcelona's Sagrada Familia

Authors: Sergi Muria Maldonado, Anton Aubanell Pou, and Jordi Font González (Professors of Mathematics Didactics, University of Barcelona). Date: June 30, 2026

As we approach the centenary of Antoni Gaudí's passing in 2026, the world continues to marvel at the Basilica of the Sagrada Familia. While many view its beauty as purely artistic calculated harmony, the true depth of the temple is revealed through the numerical patterns embedded in its stone. By analyzing the mathematical foundations, the building's balance and coherence transition from a visual mystery to a functional masterpiece.

The Quest for the Proportional System

The mathematician Claudi Alsina i Català dedicated significant effort to decoding the basilica's metrics. His research not only spanned his time at the University of Barcelona but also influenced the current lead architect, Jordi Faulí, whose doctoral thesis Alsina supervised.

Alsina recounted the "eureka" moment of his discovery in his memoirs:

"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... 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."

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The Significance of the Number 12

The number 12 is the heartbeat of the structure, serving both a spiritual and a geometric purpose.

1. Religious Symbolism

Gaudí blended architecture with biblical references:

• The 12 sons of Jacob.
• The 12 tribes of Israel.
• The 12 apostles.
• The crown of 12 stars (Book of Revelation).

2. Mathematical Utility

From a technical standpoint, 12 is an ideal number for proportions because it possesses a high number of divisors, allowing for flexible and harmonious ratios.

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The 7.5m Module in Practice

The entire temple is scaled using a base module of 7.5m. The heights of the various sections are simple multiples of this value, as shown in the following table:

Note: The Tower of Jesus Christ is the tallest point, nearly matching the height of Montjuïc hill. It is topped by a cross measuring $17\text{m}$ in height and $13.5\text{m}$ in width.

The Tower of the Virgin Mary

Standing at 138m, this is the second tallest tower. It is crowned by a 12-pointed star with a diameter of 7.5m.

• Geometry: The star is based on a regular dodecahedron.
• Detail: Pentagonal pyramid-shaped points rise from each face, creating a stunning effect under daylight and artificial lighting.

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Polyhedral Geometry and Structural Logic

Gaudí utilized various polyhedrons to crown the 12 towers of the basilica:

• Glory Façade: Dodecahedrons.
• Nativity Façade: Truncated irregular octahedrons.
• Passion Façade: Truncated cubes.
• Evangelist Spires: Regular icosahedrons (which house spotlights to illuminate the central cross).

The "Forest" of Columns

The interior is designed to mimic a natural forest. The structural efficiency is achieved through:

1. Catenary Arches: These are used to transfer loads to the ground without needing external buttresses.
2. Double-Helix Torsion Columns:
   • Base: Rounded, star-shaped polygons.
   • Form: The intersection of two opposing Solomonic columns.
   • Branching: Columns split into branches (like trees) to support the roof and towers.

# Conceptual logic of the height system
module = 7.5
heights = {
    "Apse": module * 10,
    "Transept": module * 8,
    "Nave": module * 6,
    "Side Aisle": module * 4,
    "Choir": module * 2,
    "Jesus Tower": module * 23
}

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Hidden Symbolism and Art

Beyond the primary modules, other numbers hold deep meaning:

• The Number 7: The canopy over the high altar is a regular heptagon (5m diameter), representing the seven gifts of the Holy Spirit.
• The Number 33: Found in the symbolic elements of the Passion façade.
• The Magic Square: The Passion façade features a magic square designed by Josep Maria Subirachs, echoing the numerical grids found in Albrecht Dürer's Melancholia I.