High-quality brick is used only at points of structurally critical performance, with a series of material downgrades to the level of clinkered waste from over baking at the brick kilns used as infill. Ruled surfaces can be considered as a principal component of a particular definition of the mathematics of space. Hyperboloids of revolution of one sheet

Hyperbolic paraboloids

Planar elements

Figure 6. Used well they point to an intellectual scope that re-engages curious minds with the compositional power of geometry. With their implementation as core to the description and production of the building we can cite them as evidence of the holism of Gaudi’s design approach. Though untested technically, it seems that the reverberation of the giant interior is positively enhanced by their presence on over 90 per cent of the surfaces. Every point on that surface will have at least one line passing through it that lies on the surface. Some ruled surfaces such as cones and cylinders are also described as developable surfaces; that is, surfaces that can be flattened out onto a plane without distortion or stretching of any kind, although cutting may be required. Despite the spatial complexity of the space, Gaudi employed an absolute minimum of materials both in quantity and selection. He used singly ruled surfaces such as helicoids and doubly ruled hyperbolic paraboloids for both structural walls and decorative elements that assert the muscularity of the rib vaulting supporting the main staircase landing above and so forming the crypt’s porch. And examining the stiffness of the cone made in this way relative to the flaccidity of the paper disc is a perfect lesson in imbuing materials with greater strength through the application of geometry – more with no more. Geometry of an amalgam of doubly ruled surfaces

Diagram indicating the geometry of an amalgam of all three doubly ruled surfaces: hyperboloids of revolution, hyperbolic paraboloids and planar fragments. Model of hyperboloid of revolution

A parametrically variable analogue model of a hyperboloid of revolution. Making a cone in this way is possibly the most haptic means of all to manifest the conceptual spatial leap from two to three dimensions. That it was executed entirely in Catalan vaulting (sandwich layers of structural tiles and special cement) held aloft by twisting helicoidal columns axially aligned off the vertical to correspond with the forces calculated from the hanging model all helps make this such a timeless building, despite missing the church it was designed to support. ‘Used

Figure 4. Doubly ruled surfaces, a subset of ruled surfaces, have at once a geometrical simplicity and a visual sophistication: they have two straight lines lying on the surface passing through any point on the surface, and the surfaces are doubly curved. The most recognisable hyperboloids of revolution are cooling towers of power stations. Model of hyperbolic paraboloid

A parametrically variable analogue model of a hyperbolic paraboloid. With the inauguration of the Sagrada Familia Church as a consecrated basilica on 7 November 2010, for the first time visitors could see the amalgam of hyperboloids of revolution, hyperbolic paraboloids and planar fragments that form the ceiling vaults and windows supported by helicoidal columns. Antoni Gaudf, Temple

Sagrada Familia, Barcelona, 1883-ongoing

The ceiling of Temple Sagrada Familia, fully completed in 2010. KHl

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Figure 5. As Gaudi wrote nothing of significance about his architecture during his career that spanned almost half a century, we do not know exactly why he devoted himself to doubly ruled surfaces to the virtual exclusion of all other geometry. He uses singly ruled surfaces such as helicoids and hyperbolic paraboloids for both structural wall elements and decorative elements that assert the muscularity of the rib vaulting supporting the main staircase landing

Guell Chapel porch undercroft, Santa Coloma de Cervello, Catalonia, Spain,

Figures 8 and 9. In terms of doubly ruled surfaces this building, at the middle stage of Gaudi’s career, was his launching point.