The Mathematics of the Ideal Villa

8/3/2019 The Mathematics of the Ideal Villa

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The M athem atics of theId ea l V illa

First pub lished in the Architectural Review,1947.


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2 The M athem atics of the Ideal V illa

There are two causes of beauty-natural and. customary. Natural is from geof!1etryconsisting in uniformity, that is equality and proport1o.n. Cust~mary beauty ISbegotten by the use, as familiarity breeds a love for things not In ~hemselves love-ly. Here lies the great occasio~ of errors, but always the true test !snatural, or _geometrical beauty. Geo~dncal figures are natur~lly more beautiful than Ir~egular ones: the square, the circle are the most beautiful, ~ext t,he parallelogr,am andthe oval. There are only two beautiful positions of straight I~nes, perpendicularand horizontal; th is is from Nature and conseq uently necessity, no other thanupright being firm, -Sir Christopher Wren, ParentaliaAs the ideal type of centralized building Palladia's Villa Capra-Rotonda (Plate1) has, perhaps more than any other house, imposed itself upon the imagination ..

Mathematical, abstract, four square, without apparent function and totally memo-rable, its derivatives have enjoyed universal distribution; and, when he writes of it,Palladio is lyrical.The site is as pleasant and delightful as can be found, because it is o n a small hillof very easy access, and is watered on one side by the Bacchiglione, a navigableriver; and on the other it is encompassed about with most pleasant risings whichlook like a very great theatre and are all cultivated about with most excellentfruits and most exquisite vines; and therefore as it enjoys from every part mostbeautiful views, some of which are limited, some more extended, and otherswhich terminate with the horizon, there are loggias made in all four fronts.'When the mind is prepared for the one by the other, a passage from Le Cor-

busier's Precisions may be unavoidably reminiscent of this. No less lyrical butrather more explosive, Le Corbusier is describing the site of his ' Savoye House atPoissy (Plate 2).le site: une vaste pelouse bornbee en dome aplati. La rnaison est une bolte enl'air ... a.u milieu des prairies dominant I.everger. Le plan est pur ... , II. a sajuste place dans l'agreste p~ysage de Poissy .. , , Les habitants, venus lci parce quecet~e campagne agreste etart belle avec sa vie de campagne, ils la contempleront,mal~ntenue intacte, du haut de leur jardin suspendu ou des quatre faces de leursfenetres en longueur. Leur vie domestique sera inseree dans un reve vlrglllen.!The Savoye House has been given a number of interpretations. It may indeed be

a machine for living in, an arrangement of interpenetrating volumes and spaces, anemanation of space-time; but the suggestive reference to the dreams of Virgil mayput one in mind of the passage in which Palladia describes the Rotonda .. Palladio'slands~ape is more agrarian and bucolic, he evokes less of the untamed pastoral, hisscale IS larger; but the effect of the two passages is somehow the same.Palladio, writing elsewhere, amplifies the ideal life of the villa. Its owner, from


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3 The Mathemat ic s of th e Ideal Villa

with in a fragm en t o f c re a ted o rde r, w ill w atch th e m atu rin g of h is p osse ss ion s an dsavo r th e p iquan cy o f con tras t b e tw een h is fie ld s an d h is gard en s ; re flec ting onm u tab ility, h e w ill con tem p la te th roughou t th e years th e an tiqu e virtu e s o f a s irn -p ie r race , an d th e harm on iou s o rd e rin g of h is life an d h is es ta te w ill b e an ana logyof p arad ise ..T h e anc ien t sage s comm on ly u sed to re tire to such p lace s , w he re b e ing o ften tim esvis ited by th e ir vir tu ou s frien d s an d re lation s , h avin g hou se s , gard en s , foun ta in san d such like p le asan t p laces , and above a ll th e ir vir tu e , th ey cou ld eas ily a tta in toas m uch h ap p in e ss as can b e a tta in ed h e re b e low . 3Perhap s th e se we re th e d ream s of V irg il; and , free ly in te rp re ted , th ey have

ga th e red aroun d th em se lve s in th e cou rse o f tim e all th o se id eas o f Rom an virtu e ,exce llence , Im pe rial sp len do r, and d ecay wh ich m ake up th e im agin a tive re con -s tru c tion o f th e anc ien t world . It wou ld h ave b een , p e rh ap s , in th e lan d scap es o fPou ss in -w ith th e ir p o rten tou s ap p arition s of th e an tiqu e -th at Pallad io wou ldh ave fe lt a t h om e ; an d it is p oss ib ly th e fun d am en ta ls of th is lan d scape , th e p o i-gnan cy o f con tras t b e tw een th e d isen gaged cu be an d its se tting in th e paysageaqreste, b e tw een geom e trica l vo l um e and th e app earance o f un im paired n a tu re ,wh ich lie b eh in d Le Corbu s ie r 's Rom an allu s ion . If a rch ite ctu re a t th e Roton d aform s th e se tting fo r th e good life , at Po issyit is ce rtain ly th e backgrou nd fo r th elyrica lly e ffic ien t on e ; and , if th e con tem po rary pas to ra l is no t ye t san c tion ed byconven tiona l u sage ! app aren tly th e V irgil ian n os talgia is s till p re sen t. F rom th ehygen ica lly equ ip p ed boudoirs , p au s in g wh ile ascen d in g th e ram p s , th e m em oryo f th e Georg ics n o doub t in te rpo se s itse lf; an d , p e rh ap s , th e h is to rica l re fe rencem ay even ad d a s tim u lu s as th e car p u lls ou t fo r Paris ..

How eve r, a m ore spec ific com parison wh ich p re sen ts itse lf is th at b e tw een Pal-lad io 's V illa F oscari, th e M alcon ten ta of c . 1550-60 (P la te s 3 , 4), and th e hou sewh ich in 1927 Le Corb u s ie r bu ilt fo r Mr. an d M rs . M ich ae l Ste in a t G arch e s(Pla te s 5 , 6).T h e se are tw o bu ild in gs wh ich , in th e ir fo rm s an d evocation s , a re su pe rfic ially

so en tire ly u n like th at to b rin g th em toge th e r wou ld seem to b e face tiou s ; bu t, ifth e ob sess ive p sycho logical and phys ica l gravity of th e M alcon ten ta re ce ives nop ara lle l in a h ou se wh ich som e tim es w ish e s to b e a sh ip , som etim es a gym nas ium ,th is d iffe ren ce o f m ood sh ou ld no t b e allowed to in h ib it sc ru tin y.

F o r! in th e firs t case , bo th Garch e s an d th e M alcon ten ta a re con ce ived o f ass ing le b locks (Pla tes 7 ,8); an d , a llow ing fo r varia tion s in roo f tre a tm en t, it m igh tb e n o ticed th a t b o th are b locks o f corresp ond in g vo lum e" each m easu rin g 8 un it


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The M athem atics o f th e Id eal V illa

n length , by 5% in b read th , by S in h e igh t. Then , fu rth e r to th is , th e re is a com -arab le bay stru ctu re to be ob se rved . Each house exh ib its (and conceals) an alter-a ting rhythm of doub le and single spatia l in te rvals ; and each house, read from

to back, d isp lays a comparab le tr ip artite d is tribu tion of lin es of support1 . ) .Bu t} at th is s tage, it m igh t b e be tte r to in troduce an a/most. Becau se , if th e d is-

ribu tion of bas ic horizon tal coord inate s is, in bo th cases , m uch th e sam e , th e rere s till. som e sligh t and sign ifican t d iffe rences re la ting to th e d istr ibu tion of thosein es of support wh ich paralle l th e facades; and thu s at Garches , read ing fromron t to back, th e fundam en tal spatia l in terval p roceed s in th e ratio of Y 2 : 1% :: l Y 2 : Y 2 , while at th e M alcon ten ta we are p resen ted w ith th e sequence2 : 2 :Y z . In o th er word s, by th e u se of a can tileve red half un it Le Corbu sie r ob ta in s aom press ion for h is cen tra l b ay and th e reby tran sfers in tere s t e lsewhere ; wh i Ieallad ia secu res a dom inance for h is cen tra l d ivis ion w ith a p rogress ion toward sis portico wh ich ab so lu te ly focu ses atten tion in th ese two areas . The one schem es, th e re fore , po ten tia lly d isp e rsed and poss ib ly equalitarian and th e o th e r is con -en tric and certain ly h ie rarch ical; bu t, w ith th is d iffe rence ob se rved , it might sirn-'ly b e added that, in bo th cases , a p ro jecting elem en t-extruded terrace or at-h ed p or tic o-o cc up ie s 1% un its in dep th .S tru ctu re s , of cou rse , are no t to be com pared ; and , to som e ex tent, both arch i-c ts look to s tru ctu re as a ju s tification for th e ir d isposition s . Thu s Pallad ia em -loys a so lid bearing wall; and of th is system he w rite s :t is to be observed, that those (rooms) on the right correspond with those on theft, that so the fabric may be the same in one place as in the other, and that thealls may equally bea~ the burden of the roof; because if the walls are made largeone part and small In the other, the latter will be mo~e firm to resist the weight,re~so~ of the nearne~s of the ~alls, and the former more weak, which will pro-

In time very great inconveniences and ruin the whole work."Pallad ia is conce rned w ith th e logical d isposition of m otifs dogm atically ac-p ted , bu t h e attem p ts to d iscover a s tru ctu ral reason for h is p lann ing symme-

wh ile Le Corbu sie r, who is p roving a case for s tru ctu re as a bas is for th erm al e lem en ts of des ign , con trasts th e new system with th e o ld and is a littlee c om p re he nsive .vous rap~elle ce "plan paralyse" de la maison de pierre et ceci a quoi nousmmes arrives avec la maison de fer ou de ciment arrneplan Jibre ..facade I ibre


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F igu re 1 M alcon ten ta and Garch es . Analyti-c al . d i ag rams .

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6 The Mathematics of the Ideal'Villa

ossature independante .fenetres en longueur ou pan de ve rrepllotistoit-jardin . " d '1 ' b t d s m eub le s 5et I'Interieur muni de "casiers" et debarrasse ..,e :.en corn rem en e . .Pallad io 's s tru ctu ra l sys tem m ake s it a lm ost n ecessary to repeat th e sam e p lan

on eve ry leve l of th e b u ild ing, wh ile p o in t sup po rt a llows Le Corb u .s ie r a flex ib learrangem en t; b u t bo th arch itec ts m ake a cla im wh ich is som ewhat In excess o f th ereason s th ey advan ce . Solid wall s tru ctu re s , Pallad io declare s , d em and ab so lu tesym metry; a fram e bu ild in g, Le Corb u s ie r announ ces , requ ire s a fre e arrangem en t:bu t th ese m ust b e , a t leas t p artly , th e pe rson al exigencie s o f h igh s tyle -fo r asym -m etr ic~1 bu ild in gs of trad ition al s tru ctu re rem ain s tand in g and even fram e bu ild -in gs of conven tional p lan con tinue to give satis faction .In both hou ses th e re is a piano nobile one floo r up , wh ich is linked to th e gar-

d en by a te rrace or portico and a fligh t (or fligh ts) of s te p s . A t th e M alcon ten tath is m ain floor sh ow s a crucifo rm hall w ith , symm etrica lly d isposed abou t it, twosu ite s of th ree room s each and two sta ircases ; bu t a t G arches th e re is no th in g 50read ily describ ab le . A t Garches th e re is a cen tra l h all an d th e re are two sta ircases ;b u t wh ile one of th e sta ircases occup ie s a s im ilar pos ition to those o f th e M alcon -ten ta , th e o th e r h as b een tu rn ed th rough an angle of n in e ty d egrees . F u rth e r , th een tran ce h all h as been revealed from th is leve l b y an asymm etrica l cu ttin g op en ofth e floor; an d th e te rrace (wh ich corre spond s to th e M alcon ten ta 's p ortico) hasb ecom e partly a re en tran t vo lum e ob lite ra ting a lin e of sup po rt, p laced in d is tin c t-ly le ss p e rcep tib le re la tion sh ip to th e p rin cipal room . Thus , a t G arches , th e cruci-form shape su rvives on ly ves tigia lly (pe rhap s it m ay b e though t to b e regis te red bythe ap se o f th e d in ing room ?); an d th e re fo re , in s tead of th e cen tra lity of Pal-lad le 's m ajor space , a Z-sh aped balance is ach ieved wh ich is ass is ted by th row ingthe sm all lib rary in to th e m ain apartm en t. F in ally I while a t th e M alcon ten ta th e reis a h igh ly evid en t cro ss axis , a t Garche s th is tran sve rse m ovem en t wh ich is in ti-m a ted by th e cen tra l vo id s of th e en d walls is on ly allowed to d eve lop im p lic itlyan d b y fragm en ts .T h e wall a t th e M alcon ten ta com p rise s th e trad itiona l so lid p ie rced by vertica l

open in gs w ith a cen tra l em phas is in th e po rtico an d sub s id iary accen ts in th eou te r w indows p laced toward th e extrem itie s of th e facad e . The doub le b ay in th ecen te r o f th e b u ild ing wh ich carr ie s th e uppe r ped im en ts of th e roof is exp ressedon th e on e fron t by a s in gle d oo r, on th e o th e r by a 'Rom an bath s ' m otif andhorizon ta ll~ , th e wall a lso fa lls in to th re e p rim ary d ivis ion s : base ; piano n'obil:,corre sp ond ing to th e Ion ic o rde r of th e portico ; an d sup e rim posed attic . T he b ase


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7 The M ath em atic s o f th e Id e al V illa

p lays th e p art of a p ro jec tin g , con s is te n tly su p porting san d upon wh ich th e hou sre s ts ; b u t, w h ile th e piano nobile and a ttic a re ru s tica ted , th e b ase is trea te d as ap la in su rface an d a fee lin g of even grea te r w e igh t ca rr ied h e re Is ach ieved by th ish igh ly em otive in ve rs ion o f th e u su al o rd e r.A gain th e s itu ation a t G arche s is m ore com p lex; an d th e re th e exp lo ita tion o f

th e s tru ctu ra l sys tem has led to a con cep tion o f th e w all a s a se rie s of h o rizon ta ls trip s-a s tra tegy wh ich p laces equa l in te re s t in b o th cen te r an d extrem ity o f th efacade and wh ich is th en m ain ta in ed by Le Corb u s ie r 's te n d en cy to su pp re ss th ew id e r span s o f th e doub le b ays . B y th e se m ean s any sys tem of cen tra l ve rtica laccen t and in fle c tion o f th e wall lead in g up to it is p ro found ly m od ified ; an d th eimm ed iate re su lt in th e gard en e levation o f Garch e s sh ow s itse lf in th e d isp lacin go f th e e lem en ts wh ich m ay b e con s id e red equ ival.en tto th e M alcon ten ta 's p o rticoan d su p e rim posed p ed im en t. T he se b ecom e sep ara te ; an d , tran sp osed as te rracean d roo f p avilion , th e on e occup ie s th e two (o r th ree ) b ays to th e le ft o f th e fa-cade , th e o th e r a cen tra l p os ition in th e so lid b u t an asymm e trica l on e in th ew h ole e le va tio n .On th e o th e r h an d , th e en tran ce fron t a t G arche s re ta in s what cou ld b e regard ed

as th e an alogu e o f Pa llad ia 's u p pe r p ed im en t. T h is is th e cen tra l e lem en t o f th eu pp e r s to ry; b u t th en it is a lso no ticeab le , in sp ite o f itssvmm e trica l p os ition ,th a t th e fu rth e r d eve lopm en t o f th is e lem en t w ith in itse lf is n o t symm e trica l. N ordoe s it p rom o te symm etry in th e facad e as a who le ; and , th ough it is re spond ed toby th e large cen tra l w in dow of th e en tran ce h all, s in ce th e ho rizon ta l gash es o fth e w in dow s ac t to p roh ib it any exp lic it lin kin g o f th ese two m an ife s ta tion s ,th e re en sue s in th e e leva tion som eth in g ve ry like th at s im u ltan eou s affirm ationand d en ia l o f cen tra lity wh ich is d isp layed in th e p lan . T hu s a cen tra l focu s iss tip u la te d ; its d eve lopm en t is in h ib ited ; an d th e re th en occu rs a d isp lacem en t an da b re akin g up o f exac tly w ha t Pa llad ia wou ld h ave p resum ed to b e a no rm ativeempha s i s .A no the r ch ie f po in t o f d iffe ren ce lie s in th e in te rp re ta tion o f th e roo f. A t th e

M alcon ten ta th is form s a pyram id a l su p e rs tru ctu re wh ich am p lifie s th e vo lum e ofth e hou se (P la te 9); wh ile a t G arch e s it is con s titu ted by a fla t su rface , se rvin g asth e floo ro f an en clo su re , cu t ou t from -and th e reby d im in ish in g-th e hou se 'svo lum e . Thu s , in th e on e bu ild in g th e beh avio r o f th e roof m igh t b e d e scrib ed asad d itive an d in th e o th e r as su b tractive ; b u t) th is im portan t d is ti n ction ap art, b o throo fs are th en fu rn ish ed w ith a varie ty o f in cid en t, regu lar o r ran dom , p ed im en to r p avilion , wh ich a li ke en te r in to im portan t-th ough very d iffe ren t-re la tion sh ip sw ith th e ve rtica l su rface s o f th e walls b e low .


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8 The Mathematics of the Ideal Villa

That m ath em atics and m usical con cord were th e bas is o f id eal p roportion was acommon be lie f of th e circ le s in wh ich Palladio moved . Here the re was fe lt to b e aco rre spond en ce b e tween the pe rfec t n um bers , th e p roportion s o f the hu~.an fig-u re and th e e lem en ts o f m u sical harmony:" and Sir ' H en ry W otton , as B ritish am-bassado r to V en ice at a s ligh tly la te r date , re flec ts som e part o f th is a ttitud e whenh e w rite s :Th e two prin cip al Con sonan ces th at m ost ravish . th e ,Ear are ,?y th e con sen t of a llNatu re , th e Fifth and th e Octave; whereo f the firs t riseth rad ica lly, from th e Pro-portion b e tween two and th ree . Th e o th e r from th e doub le In te rval, b e tw een on eand two or b e tween two and fou r, e tc . Now if we shall tran spo rt th e se Propor-tion s, from aud ib le to vis ib le Ob jects , an d app ly th em as shall f~1I fitte s t ... ,th e re w ill in dub itab ly re su lt from e ith e r, a grace fu l and harmonious Con t en tmen tto th e Eve .?

It was not, in fact, suggested th at arch itec tu ra l p roportion s were d e rived frommusical h arm on ie s , bu t ra th e r th a t th e law s of p roportion were estab lish ed m ath e -m atica lly and eve rywhere d iffu sed . T he un ive rse of P la ton ic and Pyth agoreanspecu la tion was com pounded of th e sim p le r re la tion sh ip s o f num bers , and su ch acosm os wasfo rm ed w ith in th e triangle m ade by th e squ are and th~ cub e of th enum bers 1) 2 ,3 .. A lso , its qu alitie s , rhythm s , and re la tion sh ip s were es tab lish edw ith in th is fram ework of num bers up to 27; and if su ch num bers gove rn ed th eworks of God , it w as con sid e red fittin g that th e works o f m an shou ld b e sim ilarlycon stru cted , th a t a bu ild ing shou ld b e a rep re sen ta tive , in m icrocosm , of th e p ro -cess exh ib ited at a large r scale in th e work ings of th e world . I n A I b e rti's words :"Nature is sure to act consistently and with a constant analogy in all her opera-tlons";8 and , the re fo re , what is p aten t in m u sic m u st a lso be so in arch itec tu re .Thu s, w ith p roportion as a p rojec tion of the h arm ony of th e un ive rse , its b asis -bo th sc ien tific and re lig iou s-was qu ite un assa ilab le ; and a Pallad io cou ld en joyth e satis fac tion s o f an aesth e tic b e lieved to b e en tire ly ob jective .

Le Corbu sie r h as exp re ssed s im ilar convic tion s abou t p roportion . M ath em aticsbr ing "des verites recontortantes,' and lion ne quitte pas son ouvraqe qu'avec 1 0certitude d'etre arrive a la chose exacte";9 bu t if it is in d eed exactness wh ich LeCorbu s ie r seeks, w ith in h is b u ild ings it is n o t the unch allengeab le c lar ity o f Pal-lad io 's vo lum es wh ich one fin d s. It is , in s tead , a type of p lann ed ob scu rity; and ,con sequen tly, wh ile in the M alcon ten ta geom etry is d iffu sed th roughou t the in -te rn al vo lum es of th e en tire b u ild ing, a t Garch es it seem s on ly to re s id e in theb lock as a who le and in the d isposition of its su pports .Th e th eo re tica l p os ition upon wh ich Pallad io 's position re s ted b roke down in

th e e igh teen th cen tu ry when p roportion becam e a m atte r o f in d ivid ual sen s ib ility


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9 Th e M athem atics of th e Id ea l V illa

and p rivate in sp ira tion ;'? and Le Corbu sie r, in p ite of th e com fort wh i 'h m th -em atics afford h im , s im p ly in te rm s of h is location in h is to ry c n c upy no uchunassailab le position . F unction alism was, p e rhap s} a h igh ly Positivis tic tem p t toreasse rt a scien tific aes th e tic wh ich m igh t possess th e ob jective valu of th e o ld ,and th e u ltim ate ly Platon ic-A ris to te lian critique . Bu t its in te rp re ta tion was cru de .Resu lts m ay be m easu red in te rm s o f p rocess , p roportion s are apparen tly acci-d en ta l and gratu itou s; and it is in con trad iction to th is th eory th at Le Corbu sie rim poses m athem atical p atte rn s upon h is bu i Id in gs . T hese are th e un ive rsal IIveritesrecontortontes. J1Thus, e ith e r b ecau se of or in sp ite of th eory both arch itects share a common

standard , a m ath em atical one , d e fin ed by W ren as "natu ral" beau ty; and , w ith inlim ita tion s of a particu lar p rogram , it shou ld th e re fore no t b e su rp ris ing that th etwo b locks shou ld be of corre spond in g volum e or th at bo th arch itects shou ldchoose to m ake d idactic advertisem en t of th e ir adhe rence to m athem atical fo rm u-lae . O f th e two-and , p e rhap s , characte ris tica lly-Le Corbu sie r is th e m ore aggres-s ive ; and at Garches he care fu lly ind icate s h is re la tion sh ip s by an apparatu s ofregu lating lin es and figu res and by p lacing on th e d raw ings of h is e levation s th eratio of th e gold en section , A : B = B : (A + B) (F igu re 2 ).Bu t, if Le Corbu sie r 's facades are for h im the p rim ary demonstra tion s of th evirtu es of a m ath em atical d isc ip lin e , w ith Pallad io it wou ld seem that th e u ltim atep roof o f h is th eory lie s in h is p lan . T h roughou t h is Quattro Iibri, P alla dio c on sis t-en tly equ ip s bo th h is p lan s and e levation s w ith th e ir num erical. apo loge tic (Plate8); bu t th e cryp tic little figu res wh ich he append s to h is d raw ings seem always tobe m ore convin cing, or at leas t m ore com prehen sib le , 'when th ey re la te to th ep lan . And th is is , p oss ib ly, to be unde rs tood , for in a hou se such as th e M alcon -ten ta th e p lan m ay b e seen as an exh ib ition of 'n atu ral ' b eau ty, as th e pu re th in g,ab s tract and un com p licated ; bu t th e facad es are , of n ecess ity, ad u lte ra ted (thoughscarce ly to th e ir d e tr im en t) by an in tru s ion of 'cu s tom ary' m ate ria l. T he facadesb ecom e com p licated , th e ir s tric t P la ton ic ra tionale m ay be u ltim ate ly vitiated byth e trad itional p resen ce , in th is case , o f th e Ion ic orde r wh ich possesses its ownration ale and wh ich in evitab ly in troduces an alte rn ative system of m easu rem en t(P late 11).The con flic t b e tween th e 'cu s tom ary' d em ands of th e ord e r and a se rie s o f 'n at-

u ral ' re la tion sh ip s m igh t b e assum ed to be th e sou rce from wh ich th e facades ofth e M alcon ten ta de rive . T hey are suggestive , evocative , bu t th ey are no t eas ily orto ta lly su scep tib le to m athem atical regu lation ; and , th e re fore , it is again towardPallad io 's p lan that one reve rts . P rovid ed w ith exp lanatory d im ension s , th e two


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10 The M athem atics of th e Id eal V illa

F igu re 2 Garches, e levations.

-


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11 Th e M ath em atics of th e Id eal V illa

su ite s com p ris in g th ree room s each can b e read as a p rogre ss ion from 3 : 4 to a2 : 3 re la tion sh ip . T hey are num be red 12 : 16, 16 : 16 an d 1.6 : 24.And he re , on th e p ar t o f Le Corb u s ie r an d Pallad ia , w e h ave to recogn ize , if n o t

d u p lic ity , at le as t w ish fu l th in king; b u t, if th e ra tio o f 3 : 5 = 5 : 8 is on ly an ap -p roxim ation to th at o f th e gold en sec tion , and if th e id ea l m easu rem en t of Pa l-lad ia 's room s doe s n o t con cu r w ith what is th e ir ac tu a l size;" th is is to b e ex-pec ted and it sh ou ld no t b e con s id e red u se fu l to en la rge upon th e se in con s is -ten c ie s . In s te ad it sh ou ld b e con s id e red m uch m ore op por tu ne to exam in e Pal-lad io 's p re fe rence fo r th e tr ip le d ivis ion and Le Corbu s ie r 's p rop en s ity to d ivid eby fou r.A t th e M alcon ten ta , as a lready no ticed , th e facad e s are d iv id ed ve rtically in to

th re e p rin c ip a l fie ld s , th ose o f th e po rtico an d th e flan kin g walls , an d hor izon tallyth e sam e s itu a tion p reva ils in th e sequen ce , b asem en t, piano nobile, a ttic ; b u t atG arche s , in sp ite o f th e com parab le s tru c tu ra l patti, i t is a lw ays a s itu a tion if n o tof on e , a t le as t o f two o r, a lte rn a tive ly , o f fou r fie ld s o f in te re s t w ith wh ich weare p re sen ted . T hu s in th e en tran ce e leva tion , i t is a bu s in e ss of fou r and on ewh ich p reva ils ; and , in th e gard en facade , th is b re akdown b ecom es a m atte r o ffou r an d two .

Bu t, in bo th h ou se s , th e re a re e lab oration s in de tail o f th e d om in an t sch em awh ich b ecom es com p lica ted by its in te rp lay w ith a su b s id ia ry sys tem . Th a t is : i t isb y ve rtica l ex ten s ion in to arch and vau lt , d iagon al of roof lin e an d p ed im en t th a tPa llad io m od ifie s th e geom e trica l asp e ritie s o f h is cube; an d th is u se o f th e c ircu -la r an d pyram id al e lem en ts w ith th e squ are seem s both to concea l and to am p lifyth e in tr in s ic seve rity o f th e volum es . How eve r, th e arch , th e vau lt, an d th e pyra-m id are am ong th e p re roga tive s o f so lid wall con s tru c tion . Th ey are am ong th efre ed om s o f th e trad it iona l p lan , th e lip/an paralyse"; and th e in trodu c tion o farched fo rm s an d p itch ed roofs is a lib e r ty wh ich a t Garch e s Le Corb u s ie r is u n -ab le to a llow h im se lf. F o r in th e fram e bu ild ing it is obviou s ly n o t) as in th e so lidwall s tru c tu re , th e ve rtica l p lan e s wh ich p redom in a te . Rath e r it is th e ho rizon ta lp lan e s of floor an d roo f s lab s (P la te 12); and , th e re fore , th e qu al ity o f para lys iswh ich Le Corb u s ie r n o ticed in th e p lan of th e so lid w all s tru ctu re is , to som e ex-ten t, tran sfe rred in th e fram e bu ild ing to th e sec tion . Pe rfora tion o f floors , g iv inga cer ta in ve rtica l m ovem en t o f sp ace , is p o ss ib le : b u t th e scu lp tu ral qua lity o f th eb u ild in g as carv in g h as d isap p eared an d th e re can b e no th in g of Pa llad io 's firmsectiona l tran sm u ta tion an d m od e lin g o f volum e . In s te ad ] fo llow ing th e p red om -in an t p lan e s o f th e s lab s , in th e fram e bu ild in g exten s ion an d e lab o ra tion m u stoccu r h o rizon ta lly . In o th e r word s , fre e p lan is exch anged fo r fre e sec tion ; b u t th e


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12 The M ath em atics o f the Idea l V illa

:. ting as 'those of the old ; and , aslim ita tion s of the new syste . m are qui te as exac I . h h f_. . h b t n ed on its s id e W it t e o rm er com -though th e so lid wall s truc tu re as een u r ., .- . f ,. ( ow transposed to plan there m ayplexit ies of section and su btle ties 0 e eva IOn n . . , '. ._ f P '1 dio's cho ice of p lan and Le Corbusier s choice ofbe here som e reason or a a I . . .I . . bei th documents in each case m ost illu s tra tive of e lem en tarye e va tio n s as, e rng ne 'J '

mathem atical regu lation . . .Th e spatial audacities of the Garches p lan con tinue t~ th r ill; bu t It m ay som :-tim es seem to be an in ter ior wh ich is accep tab le to th e In te llect a lone-to the In -te llect ope ratin g from with in a stage vacuum . Thu s th ere is at Garches a perm a-nen t ten sion b etween the o rgan ized and th e app aren tly fo rtu itou s. Concep tu ally)all is c lear; b u t, sen suously, a ll is deep ly pe rp lexing. The re are s ta tem en ts of ah ierarch ica l ideal; th ere are coun te r s tatem en ts o f an egalita rian one . Both housesm ay seem to be app rehensib le from withou t; bu t) from w ith in , in the crucifo rmhall of the M alcon ten ta, there is a c lue to the whole bu ild ing; wh ile , at Garches, itis n eve r possib le to stand at any poin t and r-ece ive a to tal im p ress ion . For atGarches the necessary equ id is tance be tween floo r and ce ilin g conveys an equalim portance to all parts of the volum e in between ; and thu s the deve lopm en t o fab so l u te focus becom es an arb itrary, if not an im possi b le , p roceed i ng. Th is is thed ilemma propounded by the system ; and Le Corbusie r respond s to it. H e accep tsth e p rincip le of ho rizon tal exten sion ; th u s, at Garches cen tral focus is con sis ten tlyb roken up , concen tra tion at anyon e poin t is d is in tegra ted , and the d ism em beredfragm en ts of the cen ter b ecom e a periph era l d ispersion of inciden t, a ser ia l in s tal-lation of in te re s t around the extrem ities of the p lan .Bu t it is now that th is system of horizon tal exten sion wh ich is conceptually

logica l com es up again st th e rigid boundary of the b lock wh ich , almost ce rta in ly,is fe lt to be perceptually requisi te ;'? and ] consequen tly, w ith ho rizon ta l exten -s ion checked , Le Corbusier is ob liged to em p loy an opposite re sou rce . That is , bygouging ou t large volum es of the b lock as te rrace and roof garden , he in trodu ces acon trary im pu lse of energy; and by opposing an exp losive m om en t w ith an im -p losive ~ne , by in troducing invers ive gestu res alongside expansive ones, h e againm akes sirn u ltaneous use of con fl icti hg stra tegies .. By its c~m plexitie s , th e re su ltan t system (or sym b iosis of system s) th row s in toIn ten se re lie f th e e lem en tary, geom etri ca l su bstructu re of th e b ildu . d. " U I lng, an , as aseque l] the pe flphe~al in ciden t wh ichsubstltu rss for th e Pallad ian focus can alsobecome com pounded with th e inve rs ions (o f te rrace and roof garden ) wh ich rep -resen t an essen tially an alogous developm en t to Pallad ' 's s t f . . Jten sion . . . 10 s strategy 0 vertical ex~


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1.3 The M athematics of the Ideal V illa

F in ally, a com parab le p rocess to th at wh ich occu rs in p lan takes p lace also inth e e levation s, whe re th e re is the sam e regu lar d iffu s ion of value and irregu lardeve lopm en t of po in ts of con cen tra tion ; and h e re , w ith th e horizon ta l w indowsconveying an equality to bo th th e cen te r and ve rge of the facades , a d is in tegra tionof focus wh ich is n eve r com p le te cau ses a b risk oscilla tion of a tten tion . Here , as inth e p lan , th e re is no th ing re s idu al, n oth ing passive , n o th ing slow m oving; and th eextrem itie s of th e b lock, by th is m eans , acqu ire an ene rge tic c lar ity and tau tness ,as th ough th ey were trying to re s tra in the p e riph e ral inc id en t from flying ou t o fth e b lock altoge th er ,A de ta iled com parison is le ss easy to su sta in b e tween the two houses wh ich )

in i t ia lly, seem ed to invite th e i r lin ki n g toge th er: th e Savoye Hou se and th e V i l IaRo tonda; and , con ce ivab ly, th is is b ecause n e ith e r o f the se bu ild ings is so en tire lycond en sed in its s truc tu re and its em otion al im pact as are , re sp ective ly, th e earlie rGarches and the la te r M alcon ten ta . Th e Savoye Hou se and the Ro tonda are bo thm ore fam ou s: b u t th ey are also , in each case ) m ore obvious ly Pla ton ic and easy totake . Possib ly th is is b ecause they are bo th in th e round ; and th at, th e re fo re , whatis concen tra ted in two fron ts a t Garch es and th e M alcon ten ta is h e re d iffu sedth rough_fou r, re su ltin g in far greate r gen ia lity o f exte rn al e ffect. B u t, if th e re is ano ticeab le easin ess and lack of ten sion to b e found in th ese facad es , th e re arean alogou s d eve lopm en ts to th ose in th e o the r hou ses . Su ch are Pallad io 's con ce rn , .b o th in p lan and e levation , w ith cen tra l em phas is and Le Corbu sie r 's d e te rm in edd isp e rsa l o f focu s . A t Po issv, ju s t p ossib ly, th e com p lica ted vo lum es of the upp e rroo f garden rep lace th e Pallad ian p itch ed roof and cupo la; and again , ju s t p ossi-b ly, Pallad ia 's fou r p ro jec tin g loggias are sub sum ed w ith in th e b lock as the en -c losed te rrace wh ich , a l te rn ative ly I as the dam in an t e lem en t o f th e piano nobile,cou ld also b e con sid e red to co rre spond to th e dom ed salon of th e Ro tonda.Bu t, sym bo l ica lly and in th e sphe re o f 'cu stom ary' b eau ty I Pallad ic's and Le

Corbu sle r 's b u ild ings are in d iffe ren t world s . Pallad io sough t com p le te c larity ofp lan and the m os t lu c id o rgan ization of conv n tion al e lem en ts b ased on sym -m etry as th e m ost m em orab le form of o rde r , an d m ath em atics as th e sup rem esanction in th e world o f fo rm s. In h is own m ind h is work was essen tia lly th at o fad ap ta tion , th e ad ap ta tion of th e an cien t hou se ; and , a t th e b ack of h is m ind werealways th e great h alls o f th e Im peria l the rm ae and su ch bu ild ings as Had rian 's villaa t T ivo li. H e had seve ra l sch em es of arch aeo logical recon stru ction of G reek andRom an dom estic b u ild ings, b ased on V itruviu s and Pliny, inco rpo ra tin g e lem en tswh ich in G reek and Rom an p ractice wou ld h ave been found on ly in pub lic bu ild -


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14 The M athematics of the Ideal V illa

in gs bu t wh ich h e regard ed as gen e ra l. In d eed , Rom e fo r h im was still . su p rem e lyalive ; an d, if th e an cien ts h ad ad ap ted th e tem p le from th e hou se ) th e ir la rge scalep lann ing was, no doub t, s im ilarly re fle c tive . .No to riou s ly, Le Corbu s ie r has an equal reve ren ce fo r m ~th e~~tlcs and h .e wou ld

I som e tim es to b e tinged w ith a com parab le h is tO riC ism ...F o r h is p lan sappe araso ,.} . . .'h e seem s to fin d at leas t on e sou rce in those id eals o f convenance and comrnodtted isp layed in th e ingen iou s p lann ing of th e Rococo ho te l, th e b ac.kground of asocia l life a t on ce m ore am p lified and in tim ate . T h e F ren ch , u n til recen tly) pos-se ssed an unb roken trad ition of th is so r t o f p lann ing; and , th e re fo re ) on e m ayoften d iscove r in a Beaux A rts u tiliza tion of an irregu lar s ite , e lem en ts wh ich ifth ey had no t p reced ed l.e Corbu sie r m igh t seem to b e cu riou s ly rem in iscen t o f h isown h igh ly su ave vestib u le s and boudo irs . Le Corbu sie r adm ire s th e Byzan tin eand th e anonym ou s arch itec tu re o f th e M ed ite r ran ean world ; and th e re is a lsop re sen t w ith h im a pu re ly F ren ch d e ligh t in th e m ore ove r t asp ects o f m echan ic s .T h e little p avilion on th e roof a t Garch es is , a t th e sam e tim e , a tem p le o f loveand th e b ridge of a sh ip . T h e m ost com p lex arch itec tu ra l vo lum es are fitted w ithru n n ing w ate r.G eom e trica lly, b o th arch itec ts m ay be said to h ave app roach ed som e th ing ofth e Pla ton ic arch e typ e of the id eal villa to wh ich th e fan tasy of th e V irgilian

d ream m igh t b e supposed to re la te ; and th e rea liza tion of an id ea wh ich is rep re -sen ted by th e hou se as a cub e cou ld also b e p re sum ed to lend itse lf ve ry read ily toth e pu rpose s o f V irgilian d ream ing. F or h e re is se t u p th e con flic t b e tween th eab so lu te and th e con tingen t, th e ab s trac t and th e n atu ra l; and th e gap b e tw eenth e id eal world and the too hum an exigen cie s o f rea liza tion h e re rece ives its m ostp ath e tic p re sen ta tion . T h e b r idging m ust b e as com pe te n t and com pe llin g as th econ stru c tion of a w e ll-execu ted fugu e ; and , if it m ay be ch arged ) as a t th e M alcon -ten ta w ith a lm os t re ligiou s se riou sn ess , o r , as a t G arch es , im bued w ith soph is ti-ca ted and w itty a llu s ion ) its su ccessfu l o rgan iza tion is an in te llec tu a l fea t wh ichreconcile s th e m ind to what m ay be som e fundam en ta l d iscrep an cie s in th e p ro -gram . .

A s a con s tru c to r o f arch ite c tu ra l fugu es) Pallad ia is th e convin ced class ic is t w itha s ixteen th cen tu ry rep e rto ry of we ll-h um an ized fo rm s; and h e tran s la te s th isrece ived m ate ria l w ith a p ass ion and a h igh se riou sn ess fittin g to th e con tin u edvalid ity th~ t h e fin d s it to posse s s .. T h e re fe ren ce to th e Pan th eon in th e sup e rim -posed p ed im en ts o f th e M alcon ten ta , to th e th e rm ae in its cru cifo rm salon th eam b igu ity, p ro found in bo th id ea and fo rm ) in th e equ ivocal con ju n ction~f tem -p le fron t and dom es tic b lock; th e se are ch arged w ith m ean ing] bo th fo r what th


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ted iou s ly am using exh l b ition tech n i ques ; bu t it is th e m agn ificen tly~eal ized qu al-ity o f th e origin als w h ich on e ra re ly fin d s in th e works of n~o~~all.ad lan s~nd ex-pon en ts of 'le s tyle Corb u .' T he se d is tin c tion s scarce l~ requ ire m Slsten ce ,a .nd~odoub t it sh ou ld on ly be sen ten tiou sly sugge sted th at, In th e case of th e derivativeworks , it is p e rhap s an adhe rence to (ru les ' wh ich has lap sed .

Addendum 1973Though a para lle l o f Sch in ke l w ith late Co rbu m igh t n o t b e so reward ing as th ecom parison o f early Corbu an d Pallad ia , m uch th e sam e argum en ts as.th ose su r-fac ing in th is ar tic le m igh t qu ite well b e foun d d eve lop ing th em se lve s IJ , fo r th eV illa M alcon ten ta, one we re to su b s titu te th e B erlin A lte s M useum and , fo rGarches , th e Pa lace of th e A ssem b ly a t Chand igarh . Illu stra tion s (Plates 13~16)m igh t su ffice to m ake th e p o in t: a conven tion al classica l. parti equ ipped w ith tra-di t ional poche and m uch th e sam e part! d is to rted an d m ade to p re sen t a com pe ti-tive varie ty o f loca l ge stu res -p e rh ap s to be und ers tood as com pen sation s fo r tra-di t ional poche,A critic ism wh ich begin s w ith ap p roxim ate con figu ration s and wh ich th en p ro -

ceed s to id en tify d iffe rence s, wh ich seeks to es tab lish h ow the sam e gen era l m o tifcan b e tran sfo rm ed accord ing to th e logic (o r th e com pu lsion ) of sp ec ific analyti-ca l (o r s tylis tic ) s tra teg ie s , is p resum ab ly W olflin lan in orig in ; and its lim itation sshou ld b e obviou s .. It cannot se riou s ly d eal w ith ques tion s o f iconograp hy andcon ten t; it is p e rh ap s over syrn metrical: an d , b ecau se it is so dep end en t on doseana lysis , if p ro trac ted , it can on ly im pose eno rm ou s s tra in upon both its con -s ume r and p rodu cer. Howeve r, if one wou ld not like to im agin e one se lf COI1~fron ted w ith th e resu lts of an in ten sive critica l workou t on th e materiel providedby th e A lte s M u seum and th e Palace o f th e A ssem b ly, th is re se rvation shou ld no tb e und erstood as dep reciating th e lim ited valu e o f su ch an exercise . F or th e twobu ild in gs in cite com parison and can a lso , bo th of th em , stim u la te fu rth e r p aralle lw ith ce rta in p roduc tion s of M ies van d er Rohe . Bu t, if n orm al in tu ition m igh tsugges t so m uch , a W olflin ian s ty le of c ritical exercise (though p ain fu lly b e longin gto a period c. 1900) m igh t s till po ssess th e m e rit o f ap pealin g p rim arily to what isvisib le and o f, th e reby, m aking th e m in im um of p re ten ce s to e rud ition and th eleas t p o~s i b le num b~r. ~ f re fe rence s ou ts id e itse lf. I. m igh t" in o th e r word s , p ossessth e m en ts o f acce ss ib ility-for tho se who are w i I Iing to accep t th e fatigue .


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17 The M athem atics o f the Id eal V illa

Notes1 Isaac W are , The Four Books of Palladia'sArchitecture, London , 1738, p. 41.2 Le Corbusier , Precisions sur un etat pre-sent de l'architecture et de l'urbanisme. 'P aris, 193 0, pp. 136-38.3 W are , p. 46.4 W are , p. 27.5 Le C orb usie r, Precisions, p. 123.6 For th ese p articu lar ob se rvation s I amh igh ly in deb ted to Rudolf Wit tkower Archi-,tectural Principles in the Age of Humanism,London , 1949.7 Sir Hen ry W otton , The Elements of Archi-tecture, pub lished in John Eve lyn , Parallel ofthe Ancient Architecture with the Modern ,3rd ed ') London , 1723, p . xv. .8 G iacomo Leon i, Ten Books on ModernArchitecture by Leon Battista Alberti, 3rded ., London , 1755, p . 196.9 Le Corbusie r an d Pie rre [eannere t, Oeuvrecomplete 7970-7929, 3rd ed ., Zu rich , 1943,p . 144. These rem arks re fe r to Garches .10 "The b reak away from the laws of h ar-m on ic p roportion in arch itectu re" is exten -s ive ly d iscu ssed in W ittkower (see n . 6), b u tth e p aralle l d isin tegration o f the Platon ic-A risto te l ian cr itical trad ition is som ewhatm ore lacon ically ob se rved by Logan PearsallSm ith : "There are great you th s too whoseach ievem en ts one m ay envy; the boy Davidwho slew Golia th and B ishop Berke ley whoann ih ila ted , a t th e age o f twen ty five , in1710, the exte rnal world in an octavo vo l-um e; and th e young David Hum e, who, in1739,. by sweep ing away all th e p .rop s of th ehum an unders tand ing, des troyed for eve rand ever all p ossib ility of know ledge . II Lo-gan Pearsall Sm ith , All Trivia, London ,1947, p . 159.

IIFor the actu al ra th e r than th e id eal in -te rn al m easu rem en ts of th e M alcon ten ta eeO ttavio B erto tti Scarn ozzi, Les batiments ettes desseins de andre palladia, Vicenza ,1776-83,12 It is poss ib le to suppose that the r ig idboundarie s of Garches were cons id e red to bep e rcep tually necessary. The house is p re -sen ted as on e of 'th e fou r com position s ' inOeuvre complete 7970-7929, p. 189; and , inPrecisions, p. 73, Le Corbus ie r w rite s o fGarches : "Pou r s ' i rnposer a l 'at tent ion, pou roccup er pu issam en t l'e sp ace, il fa lla itd 'ab ord un e su rface p rem iere d e fo rm e par-fai te, puis u n e exaltat ion de la p latitude dece tte su rface par l'apport de que lges saillie sou d e trous faisan t in te rven ir un m ouvem en tavan t -arri ere . II


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19 The M athem atics of th e Id ea l V illa

P la te 4 V illa M alcon ten ta .

Pla te 1 V illa Cap ra-Rotonda, V icenza .And rea Pallad io , c . 1550.Pla te 2 V illa Savoye, Po issy. Le Corbu sie r,1929-31.P la te 3 V illa Malcon te nta (V illa F oscari},Malcon ten ta d i M ira . Pallad io , c . 1550-60.


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21 Th e M athem atics of th e Ideal Villa

Plate 5 V illa Ste in , Garches . Le Corbus ie r,1927.Plate 6 V illa Ste in .Plate 7 Villa Ste in. Plan. Dla te 8 Villa Malcon t en t a . Plan.


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Plate 9 V illa M aJcon ten ta . Aeria l view.Plate lO Vilia Stein. Axonom etric view.Plate 11 V illa M alcon ten ta. Facade .


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24 The athematic of the Ideal ilia

Plate 12 Pro je ct, M aison Dom ino. LeCorbus i e r I 1914.

Plate 1.3 Altes Museum, Berlin. KarlFriedrich Schinkel, 1823. .-----. -- - - ~ - - - - -r ----- _


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25 The Mathematics of the Ideal ilia

ES : ; u : ; a n ~ . . . . ._. ._. . . . . . ._ . . . . . . . . . T. ~ . . . . " . . " . . . .~ . .. . ~ - I . . ~ II I( iii 1\ It . . . . - = 1 .!~ j : L " I- .iii. . . . . . -il . . . ., ~ - 1-- . . . .t . ' t. '. .t 1 III i. . . . . . . ._. -. . . . . . . .I - . . 4 1 1 I1 \ iii . '- ..;-, _ ' .. ' . III1

L _ . ,. " 1 '.I_ . r


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26 The M athematics of th e Ideal V i I1a

Plate 15 Palace of th e Assem bly, Chand i-ga rh . Le Corbu s ie r , 1953- .


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27 The M ath .em atic s o f th e Ideal Villa

a

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Pla te 16 Palace o f th e A ssem b ly. P lan .

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The Mathematics of the Ideal Villa