The Tower of Jesus Christ, part of the Basílica de la Sagrada Família, was inaugurated last week. The basilica is the magnum opus of visionary architect Antoni Gaudí, considered the great master of Catalan modernism. Gaudí’s unique architecture is evident throughout Barcelona but his crowning glory is the Sagrada Família.Gaudí began his studies in 1852 when aged 16. He enjoyed geometry and wanted to use his knowledge of curves, surfaces and solids in a practical way. His mathematical studies deeply influenced his creation of unique architectural forms. While his grades were indifferent and he occasionally failed courses, Gaudí clearly stood apart, leading the school director awarding his degree to remark: “We have given this academic title either to a fool or a genius. Time will tell.” Any justification for such ambivalence has long since vanished.Many of the curves and surfaces found in Gaudí’s architecture reflect his study of forms found in nature. Arguing that straight lines and sharp angles are rare in the natural world, Gaudí felt they should not dominate buildings. His arches include parabolas and catenaries. A parabola is the curve followed by an object lobbed through the air, while a catenary traces the form of a hanging chain. The two curves are similar but the catenary is flatter near the vertex and steeper further away. It has the critical property that the net force acting on it is tangential to the curve, so that it is self-supporting, as every student of architecture learns.Ruled surfacesIf you hold a pencil vertically and rotate it around a horizontal circle, it will sweep out a cylinder. In a similar way, more exotic surfaces can be generated from straight lines. Gaudí used several such ruled surfaces in his designs, the most notable being the hyperboloid of one sheet and the hyperbolic paraboloid. We see the former in huge cooling towers and the latter in saddles and Pringles crisps. Architects are fond of ruled surfaces, which can be constructed from straight rods and beams. Some spectacular staircases in the basilica take the form of ruled surfaces called helicoids. To visualise one, rotate a pencil in a horizontal plane about its tip while simultaneously raising the plane vertically; the pencil will sweep out a helicoid. Structures in the form of spheres, ellipsoids and paraboloids can also be found in the basilica. Gaudí kept models of the five Platonic solids (cube, tetrahedron, octahedron, dodecahedron and icosahedron) in his workshop. Topping the Tower of Mary is an extraordinary form called a stellated dodecahedron, formed by placing pentagonal pyramids on the 12 faces of a dodecahedron.Nearer to heavenThe basilica became the world’s tallest church when the huge three-dimensional cross topping the Tower of Jesus Christ was installed last February, raising the height to more than 170m. The cross is an octa-cube, comprising eight cubes in a three-dimensional form. It is also a projection of a four-dimensional hypercube. For more information about the construction of the cross, see the official site blog.sagradafamilia.org. The blessing and inauguration of the tower by Pope Leo XIV marked the 100th anniversary of Gaudí’s death. His vision is nearing its full realisation – completion of the basilica, on which work began in 1882, is expected within 10 years.[ The pope in Spain: Leo XIV challenges EU over migration and rearmament policiesOpens in new window ]If you are heading for Barcelona, bring your old geometry book or – better yet – a copy of The Genius of Gaudí by Claudi Alsina and Roger Nelsen. This book explores the mathematical elegance underlying Gaudí’s extraordinary creations and includes hundreds of photographs, illustrating his use of geometry, with particular focus on his masterpiece, the Basílica de la Sagrada Família.Peter Lynch is emeritus professor at the School of Mathematics and Statistics, University College Dublin. He blogs at thatsmaths.com