7.6 Gaudí’s Columns
Antoni Gaudí, one of the greatest architects of all times, led the Catalan Modernism with a very personal interpretation of the Art Nouveau movement, which combined Gothic elements, surrealism elements, and oriental influences. Gaudí used nature as his main source of inspiration: in his architecture, it is frequent to find references to natural elements, such as surfaces that resemble waves and support structures strongly inspired in the shape of trees.
In 1883, with just 31 years old, Gaudí started working on the Sagrada Família basilica, in Barcelona, where he explored fantastic combinations of shapes that make this temple, still unfinished, a masterpiece of architecture.
In this section, we will lean over a tiny part of this work, namely, the columns idealized by Gaudi, which can be seen in this figure.
Supporting columns of the Sagrada Família basilica in Barcelona. Photograph by Piqui Cuervo.
As seen in the figure, Gaudí imagined columns whose shape varies along the height. His goal was to mimic a stone forest and, for that, he modeled columns that branched from other columns (like tree branches), with the intersection between column brunches resembling tree knots. The variation of the columns’ shape is perfectly visible in some of these branches, in which the base is squared, but the top is nearly circular. In other cases, the column ends with a section in the shape of a star.
To model these columns, Gaudí spent two years elaborating a constructive scheme based on the intersection and union of helical shapes produced by the torsion of prisms [BarriosHernandez2006309]. In the simplest case, these prisms have a square section and suffer a torsion of sixteenths of a circle in both directions, i.e., \(2\pi/16=\pi/8\).
To generalize Gaudí’s approach, we will implement a function to deal with the general case of a prism with \(n\) sides, twisted at an arbitrary angle. For that, we will use the Khepri function surface_regular_polygon, which creates a regular polygon from the number of vertices \(n\), the polygon center \(P\), the distance \(r\) between the vertices and the point \(P\), and the angle \(\phi\) of the first vertex with the \(X\) axis.
To model the twisted prism, we will create a surface with the shape of the regular polygon and then we will extrude that surface to a height \(h\), while twisting it at an angle \(\Delta_\phi\) and applying a scale factor of \(e\):
twisted_prism(p, r, n, h, phi, dphi, f) =
sweep(line(p, p+vz(h)), surface_regular_polygon(n, u0(), r, phi), dphi, f)
To reproduce Gaudí’s approach, we intersect two of these prisms, both twisted at an angle of \(\pi/8\), but the first one in one direction and the second in the other direction. To be more realistic, we also apply a scale factor of 0.9, so that the column narrows as it goes up:
intersection(twisted_prism(xy(0, 0), 1, 4, 10, 0, pi/8, 0.9),
twisted_prism(xy(0, 0), 1, 4, 10, 0, pi/-8, 0.9))
The result is the leftmost column of this figure. The union of these prisms produces another of the shapes used by Gaudí, which is visible immediately to the right of the previous column:
union(twisted_prism(xy(5, 0), 1, 4, 10, 0, pi/8, 0.9),
twisted_prism(xy(5, 0), 1, 4, 10, 0, pi/-8, 0.9))
As done by Gaudí, we can complement the previous columns by doubling the number of prisms and halving the torsion angle:
intersection(twisted_prism(xy(10, 0), 1, 4, 10, 0, pi/16, 0.9),
twisted_prism(xy(10, 0), 1, 4, 10, 0, pi/-16, 0.9),
twisted_prism(xy(10, 0), 1, 4, 10, pi/4, pi/16, 0.9),
twisted_prism(xy(10, 0), 1, 4, 10, pi/4, pi/-16, 0.9))
union(twisted_prism(xy(15, 0), 1, 4, 10, 0, pi/16, 0.9),
twisted_prism(xy(15, 0), 1, 4, 10, 0, pi/-16, 0.9),
twisted_prism(xy(15, 0), 1, 4, 10, pi/4, pi/16, 0.9),
twisted_prism(xy(15, 0), 1, 4, 10, pi/4, pi/-16, 0.9))
The results are visible in the two columns on the right, in this figure.
Columns obtained by the intersection and union of twisted prisms with a square section.
