José Calvo-López, Enrique Rabasa-Díaz / Warped Versus Regular Surfaces

Warped Ver­sus Reg­u­lar Surfaces

A Form of Resistance to Canonical Shapes, from Reims Cathedral to Le Corbusier

José Calvo-López, Enrique Rabasa-Díaz

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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
11

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
12

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
13

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.
14

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.

  1. 1

    Joël Sakarovitch, Épures d'architecture (Basel-Boston-Berlin: Birkhäuser, 1998), 185–283.

  2. 2

    Euclid, Geom­e­try, c. ‑300, book 11, def­i­n­i­tions 2, 11 and 17.

  3. 3

    Hugh of Saint-Vic­tor, Prac­ti­ca Geome­tri­ae, c. 1120; Domini­cus Gundissal­i­nus, Liber De Divi­sione Philosophi­ae in Partes Suas …., c. 1150.

  4. 4

    Roger Baron "Note sur les Vari­a­tions au XIIe Siè­cle de la Tri­ade Géométrique Altime­tria, Planime­tria, Cos­mime­tria," Isis 48, no. 1 (1957): 30–32; Stephen Vic­tor, Prac­ti­cal Geom­e­try in the High Mid­dle Ages. Artis Cuius­li­bet Con­sum­ma­tio, and the Pratike De Geome­trie (Philadel­phia: Amer­i­can Philo­soph­i­cal Soci­ety, 1979); Hugh of Saint-Vic­tor, Prac­ti­cal Geom­e­try: Prac­ti­ca Geome­tri­ae. Attrib­uted to Hugh of St. Vic­tor. Trans­lat­ed and with an Intro­duc­tion by Fred­er­ick A. Homann (Mar­quette: Mar­quette Uni­ver­si­ty Press, 1991).

  5. 5

    José Cal­vo-López, Stereoto­my: Stone Con­struc­tion and Geom­e­try in West­ern Europe 1200–1900 (Cham: Birkhäuser-Springer Nature, 2020), 610–612.

  6. 6

    Gian­fil­ip­po Caret­toni et al., La Pianta Mar­morea di Roma Anti­ca. For­ma Urbis Romae (Roma: Arti gra­fiche M. Dane­si, 1960); Emilio Rodríguez-Almei­da, For­ma Urbis Mar­morea. Aggior­na­men­to Gen­erale 1980 (Roma: Edi­zione Quasar, 1981); Jen­nifer Trim­ble et. al., Dig­i­tal For­ma Urbis Romae”, https:// exhibits.stanford.edu /fur; accessed August 16, 2022; Sakarovitch, Épures d'architecture, 27.

  7. 7

    Sakarovitch, Épures d'architecture, 23.

  8. 8

    Richard of Saint-Vic­tor, "Com­men­tary on Ezekiel", [ca. 1171–1190] MS lat. 14516, Bib­lio­thèque Nationale de France, Paris; Wal­ter Cahn, "Archi­tec­tur­al Drafts­man­ship in Twelfth-Cen­tu­ry Paris: The Illus­tra­tions of Richard of Saint-Victor's Com­men­tary on Ezekiel's Tem­ple Vision," Ges­ta 15, (1976): 247–254; Wal­ter Cahn, "Archi­tec­ture and Exe­ge­sis: Richard of St.-Victor’s Ezekiel Com­men­tary and Its Illus­tra­tions," The Art Bul­letin 76, (1994): 53–68; Karl Kin­sel­la, "Richard of Saint Victor’s Solu­tions to Prob­lems of Archi­tec­tur­al Rep­re­sen­ta­tion in the Twelfth Cen­tu­ry," Archi­tec­tur­al His­to­ry 49, (2016): 3–24.

  9. 9

    Vil­lard de Hon­necourt, et al., "Sketch­book" [ca. 1225]. MS fr. 19093, Bib­lio­thèque Nationale de France, Paris. Vil­lard de Hon­necourt and Carl F. Barnes, The Port­fo­lio of Vil­lard de Hon­necourt: A New Crit­i­cal Edi­tion and Col­or Fac­sim­i­le (Farn­ham: Ash­gate, 2009). There is an unre­solved dis­pute about whether this arti­fact is an album”, that is, a blank book whose sheets were drawn after­wards, or a port­fo­lio”, that is an assort­ment of orig­i­nal­ly inde­pen­dent sheets that was bound after­wards. Thus, we will use the neu­tral term sketch­book”.

  10. 10

    Roland Recht et al., La Cathé­drale de Stras­bourg, Dessins et Plans (Stras­bourg: Les édi­tions des Musées de Stras­bourg, 2015); Hans Koepf, Die Gotis­chen Plan­risse der Wiener Samm­lun­gen (Wien: Böh­lau, 1969); Johann Josef Bök­er, Architek­tur Der Gotik-Goth­ic Archi­tec­ture. Bestand­skat­a­log der Welt­grössten Samm­lung an Gotis­chen Bau­ris­sen … Im Kupfer­stichk­abi­nett Der Akademie Der Bilden­den Kün­ste Wien … (Salzburg-Munich: A. Pustet, 2005); Vale­rio Ascani, "Le Dessin d'architecture Médieval en Ital­ie," In Les Bâtis­seurs des Cathé­drales Goth­iques, ed. Roland Recht (Stras­bourg: Édi­tions les Musées de la Ville de Stras­bourg, 1989), 255–277; Anto­nio Ruiz Her­nan­do. Las Trazas de la Cat­e­dral de Segovia. (Segovia: Diputación de Segovia-Caja de Ahor­ros de Segovia, 2003).

  11. 11

    Hernán Ruiz el Joven, "Libro de Arqui­tec­tura", [ca. 1560], MS R‑39. Bib­liote­ca de la Escuela de Arqui­tec­tura de la Uni­ver­si­dad Politéc­ni­ca de Madrid, f. 46v; Rodri­go Gil de Hon­tañón, "Man­u­scrito" [ca. 1560], includ­ed in Simón Gar­cia, "Com­pen­dio de Arqui­tec­tura y Simetría de los Tem­p­los", 1681, MS 8884, Bib­liote­ca Nacional de España, Madrid, f. 25r; Alon­so de Van­delvi­ra, "Libro de Trazas de Cortes de Piedras", [ca. 1580], copy, MS R‑10, Bib­liote­ca de la Escuela de Arqui­tec­tura de la Uni­ver­si­dad Politéc­ni­ca de Madrid, f. 96v.

  12. 12

    John Fitchen, The Con­struc­tion of Goth­ic Cathe­drals (Chica­go: Uni­ver­si­ty of Chica­go Press, [1961] 1981), 117–122.

  13. 13

    Van­delvi­ra, "Libro de Trazas de Cortes de Piedras", ff. 58r, 58v, 60r; Ginés Martínez de Aran­da, Cer­ramien­tos y Trazas de Mon­tea", [ca. 1600], MS. 457, Bib­liote­ca Cen­tral Mil­i­tar, Madrid, 222–223.

  14. 14

    Martínez de Aran­da, "Cer­ramien­tos y Trazas de Mon­tea", 223.

  15. 15

    Philib­ert de l'Orme, Le Pre­mier Tome de L'architecture (Paris: Fréder­ic Morel, 1567), 126v.

  16. 16

    Por haber de estar los artí­fices con­tin­u­a­mente asi­dos a la mate­ria …” Martínez de Aran­da, "Cer­ramien­tos y Trazas de Mon­tea”. The sen­tence is includ­ed in the first page of the unnum­bered pro­logue to the manuscript.

  17. 17

    Van­delvi­ra, "Libro de Trazas de Cortes de Piedras", 46r.

  18. 18

    Math­urin Jousse, Le Secret d'architecture (La Flêche: George Griveau, 1642), 14–17; François Derand, L'architecture des Voûtes (Paris: Sebastien Cramoisy, 1643), 122–126; Jean-Bap­tiste De la Rue, Traité de la Coupe des Pier­res (Paris: Imprimerie Royale, 1728), 27–28, plate 17.

  19. 19

    Derand, L'architecture des Voûtes, 172–175.

  20. 20

    Amedée-François Frézi­er, La Théorie et la Pra­tique de la Coupe des Pier­res et des Bois … ou Traité de Stéréo­tomie (Stras­bourg: Jean Daniel Doulssek­er; Paris: Charles Antoine Jombert, 1737–1739), 1: 33–34, 2:35–39.

  21. 21

    Derand, L'architecture des Voûtes, 122–126.

  22. 22

    Frézi­er, La Théorie et la Pra­tique de la Coupe des Pier­res, 2:137–140, plate 37. See also Enrique Rabasa Díaz, "Los Arcos Oblic­u­os en la Traza de Can­tería," EGA Expre­sión Grá­fi­ca Arqui­tec­tóni­ca,, no. 2 (1994): 145–53.

  23. 23

    Frézi­er, La Théorie et la Pra­tique de la Coupe des Pier­res, 2: pl. 28.

  24. 24

    Leon­hard Euler, "De Solidis Quo­rum Super­fi­ciem in Planum Expli­care Licet," Novi Com­men­tarii Acad­e­mi­ae Sci­en­tiarum Pet­ro­pol­i­tanae, no. 16 (1772): 3–34.

  25. 25

    Gas­pard Mon­ge, "Des Lignes de Cour­bu­res de la Sur­face de l’Ellipsoïde," Jour­nal de I’Ecole Poly­tech­nique, no. 2 (1796): 145–65.

  26. 26

    José María Gen­til Baldrich and Enrique Rabasa Díaz, "Sobre la Geometría Descrip­ti­va y su Difusión en España," in Geometría Descrip­ti­va, Gas­pard Mon­ge (Madrid: Cole­gio de Inge­nieros de Caminos, Canales y Puer­tos, 1996), 55–93; Sakarovitch, Épures d'architecture, 309–313; Enrique Rabasa Díaz, For­ma y Con­struc­ción en Piedra. De la Can­tería Medieval a la Estereotomía del Siglo XIX (Akal: Madrid, 2000), 296–302.

  27. 27

    Gino Loria, Sto­ria del­la Geome­tria Descrit­ti­va, dalle Orig­i­ni sino ai Giorni Nos­tri (Milano: Ulri­co Hoepli, 1921).

  28. 28

    Jean–Nicolas–Pierre Hachette, Traité de Géométrie Descrip­tive…. (Paris: Cor­by, [1822] 1828), 96.

  29. 29

    Hachette, Traité de Géométrie Descrip­tive, 315–318. See also Rabasa, For­ma y Con­struc­ción en Piedra, 278–286.

  30. 30

    Enrique Rabasa Díaz, "Arcos Esvi­a­dos y Puentes Oblic­u­os. El Pre­tex­to de la Estereotomía en el Siglo XIX," OP, no. 38 (1996): 30–41; Sakarovitch, Épures d'architecture, 313–319.

  31. 31

    Charles–François–Antoine Leroy, Traité de Géométrie Descrip­tive … (Paris: Carilian–Goeury, 1834); Charles–François–Antoine Leroy, Traité de Stéréo­tomie … (Paris: Bache­li­er, Carilian–Goeury et Dal­mont, 1844).

  32. 32

    Sakarovitch, Épures d'architecture, 243–244.

  33. 33

    Frézi­er, La Théorie et la Pra­tique de la Coupe des Pier­res … 2: 484–486.

  34. 34

    Gas­pard Mon­ge, Géométrie Descrip­tive, Leçons Don­nées aux Écoles Nor­males, l’an 3 de la République … (Paris: Bau­douin, 1799), 1–4.

  35. 35

    Eduar­do Tor­ro­ja y Cabal­lé, Teoría Geométri­ca de las Líneas Alabeadas y de las Super­fi­cies Desar­rol­lables. (Madrid: For­t­anet, 1904).

  36. 36

    Col­in Faber, Can­dela the Shell Builder (New York: Rein­hold Pub­lish­ing Cor­po­ra­tion, 1963).

Fig­ures 2, 3, 5, 6, 8, 9 and 10 are in the pub­lic domain. Fig­ures 1, 12, 13, 14 are under a Cre­ative Com­mons license. Fig­ures 4 and 11 are authors’ photographs.

Bibliography

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