I was fortunate enough to spend the last two weeks on vacation in the south of Spain. Spain is a country composed of intricately layered history and traditions; influenced over thousands of years by its various inhabitants and conquerors: the Phoenicians, Greeks, Romans, Visigoths, Moors, and of course the Christians (the Reconquista ended with the surrender of Granada in 1492 to Ferdinand and Isabella, the same year Christopher Columbus made his famous journey). Its food, music, art, architecture, and customs display these intertwined influences in unique and sometimes surprising ways.

Helicoid and Hyperboloid, graphed using MapleI was particularly struck by the work of Antoni Gaudí, a brilliant Catalan architect, mathematician, and artist. Born in 1852, he studied architecture, with a strong grounding in mathematics, especially calculus and descriptive geometry. To sate my curiosity, I’ve done some further research into his work, particularly its mathematical content.

Spiral Staircase in Gaudí's Sagrada Familia CathedralDue to contracting rheumatism as a child, he spent a great deal of time in his hometown of Reus, walking and observing nature, which was to become the main source of inspiration for his later work. We can see the way in which he sought to imitate that which he observed in nature by his use of various ruled surfaces, which are surfaces created by sweeping a straight line through space. These included helicoids, hyperboloids, hypars, and hyparhedrons – I’ve created some 3-D plots in Maple to illustrate here. 

To explain a bit, the term “hypar” was introduced by the architect Heinrich Engel to describe a partial hyperbolic paraboloid, which is an infinite three-dimensional surface that has parabolic sections parallel to two coordinate planes, and hyperbolic sections parallel to the third coordinate plane.  Its equation when its axes of symmetry coincide with the coordinate axes is x^2/a^2+y^2/b^2=2*c*z. A hyparhedron is simply a complex surface created by joining two or more doubly curved surfaces along a straight line.

2 Hypars, or Partial Hyperbolic Paraboloids

Catenary curves for a=3, 10, 20Gaudí once said, “In the execution of surfaces, geometry does not complicate but simplifies the construction”. He was possibly the first to introduce geometrical properties such as these ruled surfaces into his designs; another example is his use of catenary arches. The catenary is a curve described by a heavy flexible cord hanging between two points; and symmetrical about the y-axis. Its equation is  y=a*cosh(x/a), where  a is the point of intersection with the y-axis.


Gaudí’s workshop was cluttered with small-scale models he used in order to develop his designs. Made of white plaster and hanging chain or wire, these models allowed him to study the geometrical shapes he used, and visualize how light and shadow would play upon them.

 Gaudí’s most famous work is the Temple Expiatori de la Sagrada Família in Barcelona. Begun in 1882, this astonishing cathedral has been worked on since the architect’s death in 1926, according to his original notes. We can see in its design many of the mathematical principles which Gaudí used, such as the arches built using catenary curves, as these were the easiest to build and could support their own weight. Today, computer tools are being used in the building process, with software programs helping to understand the geometry within Gaudí’s designs.

Much has been written about Gaudí’s work and his genius both in books and on the internet; I’ve only scratched the surface here and encourage anyone who has found this interesting to take a few minutes to learn more. Better yet, take a trip to Barcelona; feel free to tell them I sent you!

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