# Gaudí’s geometric secrets: catenaries, hyperboloids and a deep symbology of the number 12

Geometry is at the center of Gaudí’s legacy. The Catalan architect did not stand out for his mathematical training, but throughout the Sagrada Familia, Park Güell and his various modernist works, the presence of mathematical forms and numerical relationships is constant, the result of personal interest and a **he studied thoroughly during his student days** of the use of curves in architecture.

Inspired by nature itself, curves, polygons and ruled surfaces can be found in every corner. These are some of the **mathematical figures that allow gaudí’s work to be appreciated with an additional dimension**. A unique architect who made the most of geometric elements.

## Catenary

Father, Son and Holy Spirit. The Holy Trinity has a high symbolic value for Gaudí and one of the many ways to represent it is through straight lines and curves. One of the elements most used by Gaudí is the parabolic or catenary curve.

The catenary was used mainly for the construction of suspension bridges, but **Gaudí took her to the Sagrada Familia and other of its buildings to provide great resistance**.

By definition, this curve is the **produced by a line when it is under the influence of the gravitational field**. That is, the shape that a rope would have if we hold it at both ends and let it fall.

The catenaries can be found in the columns of the Sagrada Familia, in the attics of “La Pedrera” or in the sloping passageways of Park Güell.

This shape can also be seen in the **funicular model of the crypt of the Güell colony**, a formation where a system of ropes and weights that hang on catenaries is related.

## Hyperboloids

Claudi Alsina, professor at the Polytechnic University of Catalonia, explains in his lecture ‘Gaudí’s geometric secrets’ some of the most relevant figures of the architect, among which the use of ruled surfaces stands out. Among them are hyperboloids.

Through a Alsina’s article for Gaussians we know the use of the hyperboloid of a leaf of revolution: formed by **lines that lie between two equal and parallel circles**. A surface generated by a hyperbola that rotates around a circle.

Gaudí discovered this geometric figure during his student days. After his passion for bells he ended up discovering that the lower part of them was a piece of hyperboloid. Years later, he incorporated the hyperboloid at the entrance to Park Güell, in the Calvet house and **in the vaulted windows of the Sagrada Familia** to let in the light.

These hyperboloid vaults have their center where the Gothic vaults had the key, with the difference that the hyperboloid allows to create a hole in that space and allow light to pass through.

In 1896, the Russian engineer Vladimir Shükhov built his hyperboloid tower. However, in 1888 Gaudí had already used this structure inside the vault of the Palau Güell. A surface of revolution that different architects would use during the 20th century.

## Hyperbolic paraboloid

In the Parc Güell Church we find numerous hyperbolic paraboloids. A surface generated by a straight line that is supported by two directive lines and always remains parallel to a plane called the director. **It is also known as a “saddle”**, due to its characteristic shape.

Gaudí felt a special interest in these paraboloids since **despite being a curved surface, they can be built using straight lines** based on varying the angle of inclination.

## The number 12

In an interview with Very interesting, Alsina explains that **the secret number of the Sagrada Familia temple is 12**.

The 12 refers to the months of the year, the tribes of Israel and the apostles. And it is also the **number of pillars supporting the basilica**.

Numerical relationships are very present and dimensions are not the result of chance. The number 12 can also be found in the proportions of the towers and different elements. All the proportions of the elements of the basilica involve dividers of 12.

## Helicoide

The **spiral stairs** They are another of Gaudí’s usual elements. For the architect, these are representations that relate the Earth to the sky and are based on helical shapes.

**Gaudí assimilated the helical shape to movement**, and the hyperboloidal to light. These helical forms are also found on the columns of the Sagrada Familia.

## Conoide

The Conoids define the **walls of the Provisional Schools of the Sagrada Familia**. It is one of Gaudí’s most original contributions to modern architecture. It is a surface that is determined in space by a line, a plane perpendicular to it, and a curve. An extension of the cones that we can find in nature, especially in leaves and flowers.

These Conoids can also be found in the **roof of Gaudí’s workshop**, specifically on the roof of the model store, next to the photographic studio. A **sinuous profile for distinctive roofs**.

## A work with its own “Gaudian geometry”

Gaudí’s work has been subject to multiple studies. This is how Alsina describes in the RSME Gazette the architect’s relationship with geometry:

“Gaudí despised the rest of his life the“ abstract ”mathematical training received. He always considered (according to his disciples of that time that mathematical models, with a high degree of abstraction and formalization, were, for his creative purposes, totally useless. However, Gaudí was able to abandon this academic mathematics and become, himself, a great synthetic geometrist “

In honor of Euclidean geometry, it is common to speak of “gaudian geometry“when it comes to encompassing and describing Gaudí’s geometric work and his predilection for ruled surfaces and curves. **unique vision of architecture where mathematical figures are essential** to fully understand the message.

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