The Accidental Mathematician

La Sagrada Familia and the hyperbolic paraboloid


I’m travelling in Spain this month – mostly for mathematical reasons, but, well, it’s Spain. Last week I was fortunate to see La Sagrada Familia.

La Sagrada Familia is the opus magnum of the great Catalan architect and artist Antoni Gaudí. Gaudí was named to be in charge of the project in 1883, at the age of 31, and continued in that role for the rest of his life. From 1914 until his death in 1926 he worked exclusively on the iconic temple, abandoning all other projects and living in a workshop on site.

The construction is still in progress and expected to continue for at least another 20-30 years. The cranes and scaffolding enveloping the temple have almost become an integral part of it. That’s not exactly surprising, given the scale and complexity of the project together with the level of attention to detail that’s evident at every step. Almost every stone is carved separately according to different specifications. Here, for example, is the gorgeous Nativity portal. (Click on the photos for somewhat larger images.)

To call Gaudí’s work unconventional would be a major understatement. To call it novelty – don’t even think about it. His buildings are organic and coherent. Everything about them is thought out, reinvented and then put back together, from the overall plan to the layout of the interior, the design of each room, the furnishings, down to such details as the shape of the railings or the window shutters with little moving flaps to allow ventilation.

Gaudí’s inspiration came from many sources, including nature, philosophy, art and literature, and mathematics.

His nature-inspired designs, found everywhere throughout his work, are as much functional as they are decorative. Take a look at these columns at La Sagrada Familia. Gaudí didn’t just try to make them look like tree trunks with branches – he studied how exactly the branches of a tree support the weight of its crown, then applied the same principles to his columns:

“Do you want to know where I found my model? An upright tree; it bears its branches and these, in turn, their twigs, and these, in turn, the leaves. And every individual part has been growing harmoniously, magnificently, ever since God the artist created it.”

Not that there’s anything wrong with decorative.

That’s from Casa Batlló, another Gaudí building. With its curved lines and marine themes, it could be a rather opulent land version of Jules Verne’s Nautilus.

But you may be waiting to hear about the mathematics. As you might have noticed by now, Gaudí wasn’t particularly fond of straight lines or regular circles. So what did he use instead? Quadric surfaces and conic curves: parabolas, hyperboloids, ruled surfaces. For example, take a look at those tree-like columns again.

That’s a hyperbolic paraboloid: the surface formed by all straight lines that intersect three generically placed lines in three dimensions. It’s also an example of a ruled surface. There are many more of those in Gaudí buildings, for example as parts of roofs and staircases.

The surface you saw in this photo is a hyperboloid of one sheet, also a ruled surface. Gaudí’s circular windows often have the shape of the hyperboloid instead of the more conventional cylinder, to improve the lighting. Think about it: light rays travel in straight lines…

Here are a few arches from Casa Batlló: parabolas or hyperbolas? Notice that the floor in the last photo isn’t flat, either.

This is a part of a ceiling at La Sagrada Familia. Can you name these surfaces? I’m not sure I could.