# The Geometry of Antoni Gaudi

** Under construction **

Antoni Gaudi i Cornet (1852-1926) was a well-known architect from Spain. He was born in 1852 as the son of a copper-smith. He studied architecture in Barcelona and combined an interest in history, mathematics and nature to create a rather unique style.

Fairly early in his career Gaudi made a display case for a glove maker from Barcelona, which was on display during the world fair in Paris in 1889. The wealthy entrepeneur Eusebi Güell i Bacigalupi noticed the display case, and upon his return to Barcelona he sought out Gaudi. This was the start of a lifelong friendship and collaboration. Güell had Gaudi design his main residence Palau Güell as well as Parc Güell in Barcelona. One of Gaudi's greatest works however became the Temple of the Holy Family (the Sagrada Familia) in Barcelona.

## Magic Square

Jesus and Judas with the Magic Square.

The passion facade of the Sagrada Familia was also worked on by the Catalan sculptor Josep Maria Subirachs (1927 -).

Close up of the magic square

Subirachs created a magic square where the rows and columns add up to 33, the age of Jesus at the time of his death. Two numbers are repeated: 10 and 14. If we add these numbers we get 48. The number 48 is the same number one obtains if the letter of INRI (Iesus Nazarenus Rex Iudeourum) are assigned a number according to their order in the Latin Alphabet.

 A B C D E F G H I K L M N O P Q R S T V X 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

So that 9 + 13 + 17 + 9 = 48

## Tesselations

Detail of pillar at the Parc Guell.
Wall at the Parc Guell.

Gaudi used mosaics in many of his works and he created several tiled floors and ceilings in the houses and parks he designed. The pillars in the Greek Theatre (Teatre Greque) show a random tessellation. The mosaic used to cover the surface of the pillar is laid out with no discernible pattern. The wall at the Parc Guell shows an overall pattern made up out of squares, but the colored mosaic squares all have their own (sometimes) symmetric pattern.

There are several true periodic tessellations. Many of them are based on the square, but there are also a couple of tessellations based on the hexagon and a wood inlay with a pattern consisting of triangles.

 A tessellation based on squares. Another tessellation based on squares, A hexagonal tessellation, but only 3-fold symmetry. Tessellation and optical illusion Triangular tessellation Another tiling

## Catenary Arches and Catenoids

A catenary arch is the shape one gets when we suspend a rope or chain from its endpoints. Gaudi used catenary arches in many of his projects. The advantage of the catenary arch is that it can be constructed from relatively light materials while still being able to support great weights.

In La Pedrera (also known as Casa Milà) a model of suspended chains is on view. A mirror below the model shows the reflected image of the structures. The reflected image clearly shows a collection of arched buildings formed by the catenoids, In the Sagrada Familia a similar model is on display, but the chains are now weighted with small bags. The latter model corresponds to the church that was planned at Santa Coloma de Cervello. The church was however never completed and at the present only the crypt is completed.

 Suspended chains form catenoids The reflection shows at outline of arched buildings Model with weights The design of the Church at Santa Coloma de Cervello.

## Hyperbolic Surfaces

### Hyperbolas

The hyperbola is a curve created when we slice a double cone vertically. This type of curve is used in the interior of the Sagrada Familia. The curves create fairly high vaults, and in the main church Gaudi used this to create pillars that resemble the structure of of tree complete with branched tree trunks.

 Cross section of a double cone. Sketch of a cross section showing hyperbolas Ceiling of the Sagrada Familia.

### Hyperboloids of One Sheet

A hyperboloid can be created if a column of strings is twisted about its central axis. Gaudi used this type of curved surface in the construction of some of the windows in the Sagrada Família in Barcelona.

The cloister walls have window created from 10 hyperboloid sheets which are arranged in a hexagonal honeycomb pattern.

 Twisting a cylinder gives a hyperboloid. Model of Hyperboloid Cloister wall, Sagrada Familia.