Antoni Gaudí was a Spanish architect who worked almost entirely in Barcelona. His unconventional

style was characterized by freedom of form, voluptuous color and texture, and organic unity (Collins). Much

of Gaudí’s inspiration came from the natural world. He also incorporated his love of geometry and into his

work. One of Gaudí’s most famous works, the Expiatory Church of La Sagrada Família, features

hyperboloids, hyperbolic paraboloids, helicoids, and conoids.

Antoni Gaudí was born in Reus, in the Catalonia region of Spain, on June 25, 1852. As a child he

attended the Escolapian School, where excelled in geometry and arithmetic (The Sagrada Família). In

1869, Gaudí moved to Barcelona to enroll in the Science Faculty at the University of Barcelona where he

studied various braches of mathematics as well as chemistry, physics and geography. He was emitted into

the Barcelona Province School of Architecture in 1874 and obtained his architect’s diploma four years later

in 1878 (Roe 12).

During his childhood, Gaudí suffered from rheumatic fever. The illness forced him to spend a great

deal of time resting (The Sagrada Família). To occupy his time, Gaudí liked to observe nature. From his

observations, he noticed that there are an infinite number of forms in nature, some which are highly suitable

for structures, while others are more appropriate for decorative purposes. He also noted that structure and

decoration frequently occur simultaneously in nature; that nature creates structural forms, based merely on

functionality, that are both statically perfect and extremely beautiful (Roe 7).

There have been many different styles of architecture employed throughout time. However

architecture, from the early Egyptians to the present day, has always been based on simple geometry.

Traditional architecture involves lines, two-dimensional figures and regular polyhedrons combined with

spheres, ellipses and circles (Roe 8). Gaudí’s observation of nature led him to see that these regular forms

either do not exist in nature or, if they do, only rarely. He used the forms of nature in his architecture just as

they are in reality, and not as artistic intellectualization. Gaudí believed that if an architect “looks for the

functional in his work, he will ultimately arrive at beauty. If he looks for beauty directly, he will only reach art

theory, aesthetics, or philosophy,” abstract ideas that he had no interest in (Bassegoda Nonell 8).

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Gaudí observed that in nature many structures are composed of fibrous materials, such as wood,

bone, muscle or tendon (Bassegoda Nonell 9). From a geometric perspective, fibers are ruled surfaces.

Ruled surfaces are surfaces that can be “generated by the movement of one straight line that follows a

particular route” (The Sagrada Família). There are four distinct ruled surfaces: helicoids, hyperboloids,

conoids and parabolic paraboloids. Gaudí saw these surfaces in nature and incorporated them into his

architectural design for one of his most famous works, the Expiatory Church of La Sagrada Família.

The Expiatory Church of La Sagrada Família was commissioned in 1874 by the Spiritual

Association of the Devotees of St. Joseph. The plot of land the church was to be built on was finally

purchased in 1881, after sufficient funds

had been donated and the foundation was

laid on March 19, 1882. The architect

Francisco de Paula del Villar designed the

church crypt in a neo-Gothic style. Shortly

after building began on the crypt, Villar

resigned from the project; his successor

was Antoni Gaudí. Gaudí would continue

to work on La Sagrada Família until his death in 1926. After his death, the project was continued under the

direction of his associates (The Sagrada Família).

Antoni Gaudí knew that he would not live long enough to see the completion of La Sagrada

Família. While he was still alive, Gaudí organized the entire project and its construction so that it could be

carried out following his ideas after his death (The Sagrada Família). He defined important parts of the

project in plaster models, so that subsequent architects could use his guidelines for the construction. To

help with the interpretation of his plans, Gaudí used only geometrical forms and established all the laws of

relation between them. In 1936, during the Spanish Civil War, Gaudí’s workshop burnt down and the

models he had left to guide the construction of La Sagrada Família were destroyed. Many pieces of the

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plaster models were recovered; published plans and photographs of the original models were also

conserved and Gaudí’s followers left books with all the information they had noted down after talking to

him. The models were able to be reconstructed using the recovered plaster pieces and the photographs.

The reconstructed models allowed the church to continued to be built following Gaudí’s original nature

based design, a design composed of the four distinct ruled surfaces. (The Sagrada Família).

The first ruled surface is the helicoid. The helicoid is “a ruled surface generated by a straight line

that revolves according to a spiral around a vertical

axis” (The Sagrada Família). In nature, helicoids

can be observed in trees; the helicoid is the form of

a tree trunk (Bassegoda Nonell 9). The helicoid,

according to Gaudí, represents the ascending

movement that relates earth and heaven. The

spiral staircases within La Sagrada Família are

helicoids. (The Sagrada Família). These staircases

also form fractals when look at form above. The

staircases leading up the central main towers of the

church create a fractal spiral design reminiscent of a

spiral seashell. Helicoids were also used to create the

shafts of the

double twisted

columns.

The

hyperboloid is “a surface generated by a hyperbola which revolves

around a circle or ellipse” (The Sagrada Família). According to

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Gaudí, hyperboloids were symbolic of the movement and properties of light. Consequently, he used

hyperboloids on all the points where light had to pass through to be diffused and transmitted from one part

of the church to another. Light from the attic has to pass into the nave through the vaults, so Gaudí pierced

the vaults with hyperboloids. Hyperboloids were also used on the windows in La Sagrada Família (The

Sagrada Família).

A conoid is “a surface formed by a straight line which is displaced above another straight line and

above a curve” (The Sagrada Família). To construct a conoid, “a sinusoidal guideline is traced on the

ground and a straight bar placed at a certain height as a second guideline. Resting on these two guidelines,

one a straight line and one a curve, are the straight

lines that generate the surface, which are the whole

group of tensed strings linked to the upper bar that

seek the line marked below to act as a guideline for

raising the undulating wall” (The Sagrada Família).

The conoid is a form frequently found in the leaves

of trees (Bassegoda Nonell 9). Gaudí used the

conoid form on the façades and the roofs of the

Provisional Schools of La Sagrada Família (The Sagrada Família). For the roof, wooden beams are the

generatrices that rest on the sinusoidal profile of the

façades on one side, and on the central interior main

beam on the other (Jamiel).

The fourth rule surface, the paraboloid, is “a

twisted surface of parabolic sections which is the result

of displacement of a straight line above two other lines

that cross in the space” (The Sagrada Família). The

webs formed by tendons between the fingers of the

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hand are a natural example of a hyperbolic paraboloid. The hyperbolic paraboloid is the ruled surface that

Gaudí used most frequently. Generally the shape is bound by four straight lines and can be generated from

a twisted quadrilateral, a quadrilateral with the four sides in different planes. Due to the fact that all

generatrices of the shape rest on two straight lines, Gaudí found this shape representative of the Holy

Trinity. He believed that “one of the straight lines represented the Father, and the opposite side the Son,

the Holy Spirit was the generatrix that supported the two and joined them permanently” (The Sagrada

Família). Gaudí used hyperbolic paraboloids in La Sagrada Família for the roof of the nave; the central

towers of the church, dedicated to Jesus Christ, Virgin Mary, and the four Evangelists, are formed by

vertically stretched hyperbolic paraboloids. (The Sagrada Família).

Antoni Gaudí was a visionary architect who incorporated his love of geometry and nature into his

work. One of Gaudí’s most famous works, the Expiatory Church of La Sagrada Família, features

hyperboloids, hyperbolic paraboloids, helicoids, and conoids. The complex geometries of La Sagrada

Família so coincide with its architectural structure that the church gives the appearance of being a natural

object in complete conformity with nature’s laws.

Works Cited

Bassegoda Nonell, Juan. Antonio Gaudi: Master Architect. New York: Abbeville Press Publishers, 2000.

Collins, George R. “Antoni Gaudí.” Encyclopedia Brtannica. Encyclopedia Britannica Inc., 2011. Web. 14

Nov. 2011. <http://www.britannica.com/EBchecked/topic/226989/Antoni-Gaudi>.

Jamiel, Abby, Dawn Ely, Mike Moran, and Ryan Bailey. Nature Breathes Mathematics: Mathematics as the

Language of Natural Architecture. 2010.

Roe, Jeremy. Antoni Gaudí. New York: Parkstone Press International, 2009.

“The Sagrada Família.” La Sagrada Família. Web. 31 Oct. 2011.

<http://www.sagradafamilia.cat/sf-eng/index.php>.

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