Ruled Surfaces

Elisavet Kiourtsoglou

In geometry, a surface is ruled if a straight line (skew line) passes through each one of its points. The cylinder, the cone, the helicoid, the hyperbolic paraboloid, the conoid and many other three-dimensional shapes follow the above mathematical definition. Xenakis (1958) first referred to the properties of ruled surfaces during his collaboration with Le Corbusier, regarding the design of the architectural shell of the Philips Pavilion, in the 1958 Brussels World’s Fair (Exposition Universelle et Internationale de Bruxelles). Philips, the Dutch company of lighting and broadcasting, chose to prize its technological innovations during this first world fair after the Second World War, on the theme of technical progress for a more humane world. Le Corbusier was chosen to provide the design of this temporary exhibition space, but the famous architect was far more interested in the audiovisual work Le Poème Electronique, with images projected inside the building during the exhibition’s time, accompanied by the music of Edgard Varèse. Le Corbusier handed his first, rudimentary sketches of the pavilion to Xenakis to transcribe them into accurate plans using geometry. In these sheets of paper there lied a plan of a stomach-like shape with two openings, bearing a scaffold as cover. Xenakis transformed these forms using segments of ruled surfaces. According to Xenakis himself (1958) the decision to use this type of forms was made due to their properties: the ruled surfaces enhanced the acoustics of the exhibition space; but most importantly, this type of surfaces are self-supporting, with no structure needed to bear their loads.

During the following months Xenakis tried to combine segments of hyperbolic paraboloids and conoids for the Philips project. Both were double ruled surfaces, which means that they contained two families of skew lines, each parallel to a common surface, but not to the other family’s lines. The visual outcome of using segments of double ruled surfaces was a three-dimensional shape that featured continuous development in two directions. The similarity of this architectural shape to the graphic score of Metastaseis of the same period (1953-54) is striking. In this work, which was to be his first and last attempt to embrace serialist practices, Xenakis created a series of pitches, from which he aimed to produce difference versions. During his numerous trials, Xenakis discovered that he could draw the stable evolution of a sound’s pitch. A straight line joining two points could represent a string instrument’s glissando (the continuous glide from one pitch to another). Based on this axiom, Xenakis built the first and last section of Metastaseis (bars 1-34 & 309-333), drawing two-dimensional forms that were generated by straight lines (Barthel-Calvet 2011). The evolution of glissandi in bars 309-314 of Metastaseis are drawn as a parabola generated by lines, that function as tangents (straight lines that touch a plane curve at a given point). Xenakis underlined this visual similarity between the graphic score of Metastaseis and the architectural shell of the Philips Pavilion (Xenakis 1958).

Due to a dispute with Le Corbusier over credits regarding the pavilion, Xenakis had to establish, in a lot of his writings, a connection between music and architecture, to explain the origin of this architectural form (Sterken 2004, 66-7). Truth be told, the Philips Pavilion does not resemble any previous work by Le Corbusier; its geometry flies away from anything else the French-Suisse architect had ever conceived. Nevertheless, the analogy between music and architecture is not straightforward. The horizontal section (plan) of the Philips Pavilion presents segments of parabolas quite like the forms created by the glissandi in the graphic score of Metastaseis. The three-dimensional pavilion is inspired by the two-dimensional music score.

Beyond the formal inspiration, we can also retrace a connection between the architectural space and the musical space that is symbolized as pitch versus time on the musical score. Digging back to his university records, Xenakis’s daughter Mâkhi discovered the only element preserved by her father: a coursework on graphostatics (Xenakis 2015). Common for a civil engineer’s education since the early 20th century, this graphic method allows for global control over a structure’s behavior and a visual apprehension of the forces applied on it. Xenakis, in this coursework, drew the influence lines on the critical points of a concrete bridge. The influence lines are tangents of a parabola, called the structure’s elastic line, that is useful to engineers when calculating the frame needed to resist to the loads applied on the future construction. In fact, this parabola shows the gradual fluctuation of the real load applied on the construction against which the engineer must build an appropriate steel frame. When, years later, Xenakis wanted to experiment with the entanglement of string glissandi, represented by straight lines, he used the same principle. In fact, the curves appearing in the Metastaseis score “envelop” the gradual evolution of the continuous change of frequencies (glissandi) of several instruments (Kiourtsoglou 2016, 204-6).

The architectural space and the musical space share the principle of continuous change in three or two directions respectively. In architecture, this describes a breakthrough: the Philips Pavilion was one of the first buildings whose surfaces were not based on a wall-ceiling principle; what was supposed to be a wall (the supporting frame perpendicular to the earth) was now designed to transform continuously, with no break, into a ceiling (the covering frame horizontal to the earth). Xenakis used projective geometry to create plans and three-dimensional views that would be read by engineers. Nevertheless, the design novelty of the Philips Pavilion’s ruled surfaces had to be aligned with an analogous construction innovation. This became obvious when the calculation and construction phase started. The Paris-based Eiffel construction firm proposed a steel frame structure, abolishing the self-supporting quality of the double ruled surfaces that were chosen by the Greek architect and composer (Xenakis 1958). The urgency and complexity of the construction were addressed by the civil engineer H.C. Duyster and his agency Strabed, known for its expertise in prestressed concrete constructions. The principal idea of Duyster consisted in dividing the hyperbolic paraboloids of the pavilion in numerous prefabricated concrete slabs of 1m2, that would follow the skew lines of every ruled surface. All the slabs would be fabricated in place and would be assembled by prepressed steel wires.

The pavilion was erected at the end of 1957 and remained in the exhibition area in Brussels until the end of 1958, when, regardless of Le Corbusier’s efforts to preserve the building as a permanent exhibition space, the Philips Pavilion was demolished. Its ruled surfaces had marked a new era in architectural design; a kind of design that would only be made standard in the late 90s when computers offered the possibility to draw and calculate the three dimensions of ruled surfaces.


Barthel-Calvet, Anne-Sylvie. 2011. “Xenakis et le Sérialisme: L’apport d’une analyse génétique de Metastasis.” Intersections: Canadian Journal of Music / Intersections: revue canadienne de musique 31 (2): 3-21.

Sterken, Sven. 2004. “Iannis Xenakis, ingénieur et architecte. Une analyse thématique de l’œuvre, suivie d’un inventaire critique de la collaboration avec Le Corbusier, des projets architecturaux et des installations réalisées dans le domaine du multimédia.” PhD diss. Université de Gent.

Kiourtsoglou, Elisavet. 2016. “Le travail de l’analogie dans l’œuvre de Iannis Xenakis.” PhD diss. Université Paris 8.

Xenakis, Iannis. 1958. “Genèse de l’architecture du pavillon,” Revue technique Philips 20 (1): 1-11.

Xenakis, Mâkhi. 2015. Un père bouleversant. Paris: Actes Sud.

How to cite

KIOURTSOGLOU, Elisavet. 2023. “Ruled Surfaces.” In A Xenakis Dictionary, edited by Dimitris Exarchos.