When Le Corbusier visited Sagrada Familia in Barcelona, he was not impressed with the naturalistic design of the towers or, of course, with the Gothic Revival plan. What attracted mostly his attention was the warped surface in the roof of a small building, scarcely larger than a shed, housing a provisional school.

Roof, Assembly Hall, Chandigarh. Le Corbusier, 1951. Photograph: Eduardo Guiot. CC BY 2.0.
Such interest is not surprising: in contrast with his disdain for the architectural instruction of the École de Beaux-Arts, Le Corbusier had a high regard of the pedagogical model of the École Polytechnique, a Parisian school endeavouring to provide basic scientific instruction to engineers of all disciplines. Later on, the students coming out of this school completed their studies in a number of écoles d'aplication, that is, schools that allowed them to apply the scientific knowledge learnt at the École Polytechnique to practical, technical issues. The founder of the École Polytechnique, Gaspard Monge, was also the father of Descriptive Geometry.[1] It comes as no surprise, then, that the concepts, figures and methods of this science appear frequently in Le Corbusier’s oeuvre, in particular in the later periods, when he was trying to escape the narrow bounds of the minimalist vocabulary of the rationalism of the Twenties: hyperbolic paraboloids in the Philips pavilion in Brussels, one-sheeted hyperboloid in the Assembly Hall in Chandigarh [ 1 ], a double curvature surface in the roof of Ronchamp, or shadow theory in the Tower of Shadows, also in Chandigarh.
This issue is not as simple as it may seem at first sight. We usually take for granted that descriptive geometry deals with surfaces in a neutral, scientific, aseptic way. However, the construction of the notion of surface as presented by descriptive geometry has undergone a long historical process, walking on the line between artisanal practices and learned science. First, Euclid defines the concept of surface in his Geometry[2] as a face of a solid and uses it to describe the notions of solid angle and diameter of a sphere, but this is all. Also, there is nothing in classical geometry about orthogonal projection and only some theorems about central or conic projection in Euclid’s Optics. This fact is essential; until the advent of Computer Science, the most efficient, visually intuitive, and historically relevant way to control the properties of ruled, developable, warped, or double curvature surfaces was orthogonal projection, as we will see.
Admittedly, Archimedes and other Classical geometers dealt with the area and the volume enclosed by specific surfaces, such as the sphere, the cone, and the cylinder. In the long run, these abstract problems metamorphosed in a typically Medieval science, practical geometry, which should not be confused with the abstract geometry of Euclid or the ruler-and-compass geometry of medieval artisans, in particular stonecutters. This practical geometry, studied by clerics such as Hugh of Saint Victor and Gundissalinus,[3] had three branches. Planimetry dealt with the measure of planar areas; Cosmimetry taught how to measure volumes and surface areas of solids; and finally, Altimetry solved the problem of the computation of the height of inaccessible objects.[4] Of course, Cosmimetry brought back Archimedean problems and surfaces, although it dealt again with simple bodies: the sphere, the cone, or the cylinder. However, it is worthwhile to mention that the methods of Altimetry, dealing with similar triangles, have some points of contact with those of Late Medieval and Renaissance masons, who also used triangulation.[5]
That brings back the issue of orthogonal projection. Before dealing in depth with this subject, we must define the term. Etymologically, “to project”, from the Latin “proiectāre” means “to cast forward”; there is no projection when a drawing depicts objects placed on the same plane. Although this may be striking, the examples of Antique architectural or technical drawings in orthogonal projection, in this restricted sense, are virtually non-existent. Such a remarkable piece as the Forma Urbis Romae, a huge marble plan of Imperial Rome, preserved as a large series of fragments, depicts the town as a foundation plan, leaving aside the hills and valleys of the city.[6] As far as we know, the artifact that comes closer to an orthogonal projection in all Antiquity is a papyrus in the Petrie collection, depicting a small Egyptian shrine in the shape of a pyramid frustum in front and side views. Although the slope of the edges of the frustum is slightly different, both views attain to the same height.[7] All this suggests the idea of an orthogonal projection, although the condition of the papyrus, reduced to a series of elongated fragments, and its uniqueness, does not allow to reach firm conclusions.

Richard of Saint Victor, “Commentary to Ezekiel”, c. 1175.
Our present conception of orthogonal projection seems to have taken shape in the 12th and 13th centuries, between the clerical and the artisanal media. Some miniatures in the “Commentary to Ezequiel” by Richard of Saint Victor depict arcades drawn frontally, with no foreshortening [ 2 ], clearly passing in front of battlements and walls, also depicted frontally.[8] It is worthwhile to remark that, according to John of Toulouse, Richard was a student of Hugh of Saint Victor, although this fact is contested by some scholars. In any case, it is generally accepted that Hugh had lived in the Parisian abbey of Saint Victor from 1115 to his death in 1141, while Richard was the prior of the same abbey from 1162 until his death in 1173, so both Richard and his illuminators were probably aware of the geometrical work of Hugh.

Villard de Honnecourt, “Sketchbook”, c. 1230. Longitudinal section and side elevation of Reims cathedral.
The miniatures in the “Commentary on Ezekiel” do not need to represent exactly arcades and walls; in contrast, architectural plans and elevations, even the simplest ones, require the precise placement of pillars, ribs, triforia, socles and windows. Several plans in the Villard de Honnecourt sketchbook, such as the one for the cathedral of Meaux and the church designed with Pierre de Corbie “inter se disputando”, provide clear examples of horizontal projection, depicting both the pillars and the plan layout of the vault ribs. At the same time, the sketchbook includes an internal longitudinal section and an external side elevation of the nave of Reims cathedral [ 3 ]. Both drawings depict objects in clearly distinct planes, thus furnishing a neat example of vertical projection. The elevation includes both the aisle external walls and the clerestory, which are separated by the width of the aisles. In the same way, the section shows the socle and the aisle windows, as well as the nave pillars, the triforium and the clerestory.[9]

Reims cathedral, nave, c. 1220. Quadripartite vault. Photograph: José Calvo.
From this moment on, orthogonal projection took the place of the representation tool of choice in architectural drawing. As in the Reims drawings, both orthogonal elevations and plans are used to control the complexities of Gothic spatial geometry, as the remarkable collections of Gothic drawings in Strasbourg, Vienna, Siena, and Segovia show.[10] It is worthwhile to remark that in this large corpus of drawings, orthographic projection was used in strictly architectural drawings both for plans and elevations, as in the Villard sketchbook. However, when preparing construction diagrams, such as the German Grundrisse or the Spanish ones included in the manuscripts of Hernán Ruiz, Rodrigo Gil de Hontañón or Alonso de Vandelvira,[11] Gothic masons used orthographic projections only for the plans. In contrast, in elevations they used a disarticulated scheme that showed all ribs, including diagonals and tiercerons, in true shape. If they had shown these ribs in true orthogonal projection, the result would have been elliptical arches, since these ribs are oblique to the projection plane. At this moment, nobody in Europe knew how to draw an elliptical arch representing the projection of an oblique circular arc. Further, such a representation would have been useless for masons, who were interested in the true shape and curvature of the ribs.
19th century elevations and 21st century scans of rib vaults show that ribs of quatripartite vaults do not overlap with transverse arches when seen in vertical projection. This showcases a crucial change in the geometrical vocabulary in medieval architecture. Generally speaking, Romanesque architecture uses simple surfaces, or, at most, combinations of them: half cylinders for barrel vaults, portions of cylinders for pointed barrel vaults, half or quarter spheres for domes and semidomes, intersections of cylinders for groin vaults, lunette vaults and windows in round walls. Intrados joints between courses are, generally speaking, parallel, except in domes; this layout generates ruled surfaces.
Anyhow, in quadripartite Gothic vaults it is quite difficult, or indeed impossible, to lay out the intrados joints as parallel or convergent lines [ 4 ], since diagonal and transverse ribs do not overlap in elevation. This led to the use of warped surfaces in the severies of quadripartite vaults, a first form of resistance against the simple, canonical surfaces in Romanesque architecture. Further, Gothic masons did not attempt to depict the intrados joints of the severies; they simply laid them out using cerces or light struts connecting two ribs.[12] This explains why, while mastering the horizontal layout of very complex sets of ribs in Late Gothic, masons did not use orthogonal elevations in construction drawings, by contrast with architectural drawings of the period.

Ginés Martínez de Aranda, “Cerramientos y trazas de montea”, c. 1600. Diagram showing warped surfaces.
However, Renaissance construction in ashlar required precise elevations, coordinated with horizontal projections, in particular to control the layout of warped surfaces. Nineteen-century geometricians will later point out that generatrixes that are neither parallel nor convergent lead to warped, non-developable surfaces. Much earlier, this notion was identified by Renaissance stonecutters on a purely empirical basis. Spanish texts, such as Alonso de Vandelvira or Ginés Martínez de Aranda, allude to these surfaces as “engauchidas”, from the French “gauche”, left-handed.[13] Of course, the term carries a strong connotation of sloppiness and irregularity.
Martínez de Aranda showed in a remarkable didactical drawing the notion of gauche or warped surface [ 5 ].[14] First, we must remember that while two straight lines in a plane may be either parallel or convergent, in a spatial geometry there is a third possibility: both lines may be skew lines, that is, lines that are not parallel but do not intersect. In space, two parallel or convergent lines determine a plane; that is, there is one and only one plane that passes through both lines. In contrast, skew lines do not determine planes; in fact, no plane can pass simultaneously through two skew lines. Thus, skew lines may not be represented in a single projection. In order to show the skew condition of a pair of lines, we must provide at least two different projections, as Martínez de Aranda does. The result resembles a pair of helicopter blades. Other stonecutting writers, such as Philibert de l’Orme, also present these apparently intersecting lines, although they are not as explicit as Martínez de Aranda about the concept.[15] These surfaces generated by skew lines are gauche surfaces; according to Descriptive Geometry, no template, either rigid or flexible, can be applied exactly over a gauche or warped surface.

Jean-Baptiste de la Rue, Traité de la coupe des pierres, 1728. Biais passé.
All this does not mean that Martínez de Aranda was thinking in the same mental frame that 19th century geometers. Quite to the contrary, he defined masons, including himself implicitly, in the introduction of this manuscript as “men who stick to physical matter”, and his understanding of these issues seems to be purely empirical.[16] In particular, he constructs and applies templates to warped surfaces, in contrast with other writers, in particular Alonso de Vandelvira.[17] Although this author constructs templates for rere-arches, a particular constructive element resting on a lintel and an arch, he states clearly that these should be used only as an auxiliary device for the computation of angles between the edges of the voussoirs and that the voussoirs for rere-arches should be dressed using the tiresome and time-consuming squaring method. By contrast, Martínez de Aranda’s templates are meant to be applied directly on warped surfaces. In other words, Martínez de Aranda put together another form of resistance against the canonical status of non-warped surfaces.
Most stonecutting writers of the 17th and early 18th centuries, such as Mathurin Jousse, François Derand and Jean-Baptiste de la Rue marched in the opposite direction. Admittedly, they dealt with pieces involving warped surfaces, such as the corne de vache and the biais passé, two kinds of skew arches with one or both oblique springers [ 6 ]. However, they did not use templates on these surfaces; rather, they resorted to the time-and-material-consuming squaring method.[18] In contrast, taking their cue from De l’Orme, they used approximate cylindrical and conical developments for non-warped surfaces, which is quite reasonable considering that the generatrixes of a cylinder are parallel, while those in a cone are convergent.[19]
All this led Amedée-François Frézier to differentiate between the aristocracy of non-warped surfaces, which he labelled as “regular” and the plebeian gauche surfaces.[20] As in any class stratification, there was also a middle class, those surfaces he called régulierement irrégulieres. At the same time, the class segregation launched by Frézier went much further than the stance of Derand.[21] This author eschewed templates for warped surfaces; however, when the squaring method involves planes orthogonal to the face of an arch, as in the biais passé and the corne de vache, the loss of material and effort brought about by the squaring method is limited, and in this case, he placed no objection against warped surfaces dressed by the squaring method.

Amedée-François Frézier, La théorie et la pratique de la coupe des pierres et des bois … 1737-1739. Biais passé with a cylindrical intrados and elliptical intrados joints.
Thus, while Derand’s mistrust of warped surfaces stemmed from practical considerations, Frézier’s rejection was based on conceptual reasons. The biais passé had been built by centuries and Frézier, as a result of his Encyclopaedic approach, explained the traditional solution to the piece. However, he tried to put forward a réguliere version of the piece, while keeping the dressing advantages of the orthogonal bed joints [ 7 ]. This led him to design an improved variant of the biais passé with a cylindrical intrados cut by planes orthogonal to the faces of the arch, but oblique to the springers.[22] The intrados joints of the piece are oblique sections of an elliptical cylinder, resulting in elliptical curves. Paradoxically, all this leads to a much more complex tracing and dressing process. As far as we know, no built example of Frézier’s enhanced biais passé has been found.

Amedée-François Frézier, La théorie et la pratique de la coupe des pierres et des bois … 1737-1739. Using the straightedge while dressing stones.
It is worthwhile to remark that the concepts of “ruled” or “developable” surfaces do not play an important role in Frézier’s treatise, other than his insistence on the use of the straightedge [ 8 ] in the dressing process.[23] Of course, he seems to be thinking of the sphere, the most perfect surface for Renaissance theorists, as a “regular” surface and in fact he associates it usually with the cone and the cylinder. This state of events was to change in 1772, when Leonhard Euler published "De Solidis Quorum Superficiem in Planum Explicare Licet”, that is, “About solids whose surfaces may be developed into a plane”, putting forward the equations that such surfaces must fulfil in order to be classified as developable.[24] This issue has a host of applications, as time will show, but it seems that Euler was mainly interested in cartography.

Charles Leroy, Traité de Stéréotomie, 1844. Ellipsoidal vault solved using lines of curvature.
Anyhow, Gaspard Monge, the founder of Descriptive Geometry, was to take back the subject to the field of stonecutting. He was Professor of the Theory of Stonecutting in the Military Engineering School at Mézierès. Remarkably, his extensive scientific production about a host of different subjects includes little more than a single paper on stereotomy, the science of the division of solids, which includes stonecutting as a practical application.[25] Trying to illustrate a new concept, lines of curvature, Monge used as an example a rather far-fetched problem in stone construction, that of an ellipsoidal vault with three different axes or scalene ellipsoid. He imposed two constraints to the problem: bed joints should be generated by orthogonals, or more strictly speaking, normals to the intrados surface. Also, these joints should be developable surfaces. Both constraints had some practical sense. The generation of bed joints by normals to the intrados surface avoids acute angles between the voussoir faces, which may suffer dents during the transportation, hoisting and placement processes. At the same time, the developable nature of these surfaces allowed the application of flexible templates to the bed joints. However, oval vaults had been built for centuries without the use of such sophisticated control methods; to consider them as mandatory seems rather excessive from a practical standpoint. In order to generate a developable surface, normals to the intrados surface must follow lines of curvature. In contrast with traditional oval vaults, lines of curvature drawn in the surface of a scalene ellipsoid do not lay on horizontal planes, but rather go up and down [ 9 ]. Monge was so proud of his findings that he proposed to build the Assembly Hall of the French Republic, under discussion at this moment, in the shape of a scalene ellipsoid, with ribs following lines of curvature, and even that the speaker should be placed under a node of lines of curvature called the umbilical point.[26]

Jean-Nicolas Hachette, Traité de Géometrie Descriptive, 1822. Arrière-voussure de Marseille.
This suggests that Monge was enforcing here the same concept of orthodoxy than Frézier, excluding all forms of resistance to “regular” surfaces, not only for practical reasons, but rather on conceptual grounds. Anyhow, the reader may ask whether an ellipsoid with three different axes, a non-developable surface, may qualify as “regular” surface. It seems that the acceptance of the scalene ellipsoid in the canonical realm of accepted surfaces stems from analytical, not strictly geometrical, reasons. The ellipsoid, whether the scalene variant or the well-known ellipsoid of revolution, with two axes of the same length, is a cuadric or second-degree surface, just as the sphere, the cylinder and the cone. Such simple mathematical representation must have been quite pleasing to the eyes of the scientists of the Enlightenment and the French Revolution.
Trying to “push further the boundaries of this happily conquered realm”, in Gino Loria’s words, one of Monge’s students, Jean-Nicolas Hachette, went further.[27] As Monge with the lines of curvature, he had proved a theorem that he needed to illustrate. Hachette’s theorem is highly abstract: it states that if two ruled surfaces have a common generatrix and they share the same tangent plane in three points, then they share the tangent plane all along the common generatrix.[28] Monge had insisted in the practical applications of Descriptive Geometry in the introduction to his text, so Hachette needed a practical application of his theorem. He found it in the Arrière-voussure de Marseille, a classical stonecutting piece designed to solve a door or window opening with a wooden frame crowned by a semicircle [ 10 ]. Using a segmental arch for the inner face of this piece prevents the window pane to collide with the surface of the vault covering the opening.[29] However, all three surfaces in the intrados of this piece are warped. Once again, as Monge with the lines of curvature and the scalene ellipsoid, Arrière-voussures de Marseille had been built for two centuries without the need of Hachette’s sophisticated procedure, and in fact, it is impossible to tell traditional Arrière-voussures built using the traditional solution from the ones dressed by means of Hachette’s sophisticated technique.[30] Thus, we may surmise that Hachette’s focus did not lie on the practical application of the piece, but rather in his use as an illustration of his theorem.

Antoni Gaudí, Palacio Episcopal de Astorga, rere-arch over the main door, 1899-1893. Photograph: José Calvo.
From this moment on, Descriptive Geometry textbooks, taught as a foundation subject in the host of Polytechnic Schools that spread through Continental Europe, included not only second-degree warped surfaces as hyperboloids and paraboloids, but also other warped surfaces as the Arrière-voussure de Marseille, the corne de vache and the biais passé, that could not present the same analytic credentials, trying again to push forward the frontiers of this new science.
The focus of these issues went back to Spain as a result of a strange turn of events. While most architectural schools in Continental Europe adopted the pedagogical model of the Paris École de Beaux-Arts, at the start of the 19th century, the architectural instruction of the Academia de Bellas Artes de San Fernando in Madrid followed the model of the École Polytechnique, with a strong emphasis on Descriptive Geometry, construction and, in particular, Stereotomy. In the wake of the Catalan Renaixença movement, a new architectural school was opened in Barcelona. It lacked any kind of support from the Spanish central government, up to the extent that it had to be financed by the Barcelona municipality and provincial administration. This school had enough difficulties to open a new front about didactic choices and, basically, adopted the pedagogical model of the Madrid school, which had gained its independence from the San Fernando academy in 1844.
As a consequence, Antoni Gaudí, trained in the Barcelona school, had a solid background in Descriptive Geometry and Stereotomy, which led him to an innovative use of warped surfaces and other figures in the catalogue of 19th century Descriptive Geometry. In addition to double-curvature surfaces in Casa Milá and many other locations, a clear example of his approach to warped surfaces can be seen in a remarkable rere-arch, leaning in a round and a pointed arch, in the Episcopal Palace in Astorga, an example of his early Gothic Revival style [ 11 ]. Rear-arches span the area between a lintel and an arch, or two arches, as in the Arrière-voussure de Marseille, but these arches are usually round or segmental. There is not a single example in Gaudí’s most probable sources, Leroy’s treatises on Descriptive Geometry and Stereotomy of a rear-arch resting on a pointed arch.[31] This suggests that Gaudí did not limit himself to a mechanical application of Leroy’s models; rather, he assimilated the methods explained in 19th century stereotomy treatises in order to innovate in the field of warped surfaces, putting forward again a new form of resistance against the canonical use of developable and “regular” surfaces.

Félix Candela, Los Manantiales Restaurant, Xochimilco, 1956. Photograph: Dge. CC BY-SA 3.0
Monge’s text was disruptive in another sense. Up to this moments, warped surfaces had been connected almost exclusively with stonecutting. As Sakarovitch stressed, the carpenter works with lines, and the coppersmith with developable surfaces; in the preindustrial world, warped surfaces can only be materialised in stone, brick or earth.[32] Frézier had tried to extend the field of application of his new science to woodwork, to justify the title of his book: La theorie et la pratique de la coupe des pierres et des bois … ou traité de stéréotomie. However, the section on woodworking in his book is surprisingly short and, in any case, deals with thin veneers that can only be materialized as developable surfaces.[33] Monge was much more ambitious: he tried to extend the field of application of Descriptive Geometry to all branches of the nascent industrial technology.[34] In the context of the Industrial Revolution, the difference between developable and warped surface was essential, since warped surfaces cannot be materialised in sheet metal without cutting the sheet; of course, such process brings about material waste and execution difficulties.
Le Corbusier used to quote Auguste Perret saying that “We are building in concrete, but we still think about stone”. In addition to his own oeuvre, a breakthrough in these fields was driven by two Spaniards. Eduardo Torroja y Caballé was a full professor of Descriptive geometry at the Universidad Central de Madrid, who had published a Teoría Geométrica de las líneas alabeadas y las superficies desarrollables (Geometric Theory of Warped Lines and Developable Surfaces).[35] His son Eduardo Torroja Miret, an outstanding civil engineer, used single-sheeted hyperboloids on the roof of the Hipódromo de la Zarzuela Madrid (1941), and in the Cuba de Fedala in Morocco (1956).
While Torroja Miret stayed in Spain after the Civil War, Félix Candela went to exile in Mexico, after his involvement with the Republican Army as Engineers Captain. He had followed the architectural courses at the Madrid school, but he got his degree in 1935, in the eve of the Spanish Civil War. In Mexico, he carried out some projects as an architect, but he worked more frequently as a builder, offering to architects a sensible and inexpensive construction method based in thin concrete shells in the shape of a hyperbolic paraboloid; he was familiar with this warped surface as a result from his instruction in Descriptive Geometry in the Madrid school [ 12 ].[36] In this way, warped surfaces leapt the increasingly narrow frontiers of ashlar construction to the expanding realm of one of the 20th century typical materials, concrete.

Le Corbusier, Philips Pavilion at the Brussels Universal Exhibition, 1958. Photograph: Wouter Hagens. CC BY-SA 3.0

Le Corbusier, Roof, Assembly Hall, Chandigarh. 1951. Photograph: Eduardo Guiot. CC BY 2.0.
Candela’s most fruitful years, the Fifties and the Sixties, overlap with Le Corbusier’s late period, where the vocabulary and methods of Descriptive Geometry furnished him a way to break the limits of the “regular” surfaces of the Twenties, such as cylinders and cuboids, to delve into a free and complex language including the hyperbolic paraboloids in the Brussels Expo Philips Pavilion [ 13 ], the one-sheeted hyperboloid in Chandigarh’s Assembly Hall, [ 1 ] [ 14 ] or even shadow theory in the Tower of Shadows in the same complex. Thus, this “form of resistance” involving warped surfaces, in contrast with the “regular” forms in his early production, can be seen as a chain in a long process that has its roots in Medieval architecture.