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The mathematics of the Ideal Villa: Palladio and Le Corbusier compared

Colin Rowe’s essay on The Mathematics of the Ideal Villa, Palladio and Le Corbusier compared

Originally published in AR March 1947, this piece was republished online in August 2010

“There are two causes of beauty - natural and customary. Natural is from geometry consisting in uniformity, that is equality, and proportion. Customary beauty is begotten by the use, as familiarity breeds a love to things not in themselves lovely. Here lies the great occasion of errors, but always the true test is natural or geometrical beauty. Geometrical figures are naturally more beautiful than irregular ones : the square, the circle are the most beautiful, next the parallelogram and the oval. There are only two beautiful positions of straight lines, perpendicular and horizontal; this is from Nature and consequently necessity, no other than upright being firm,” SIR CHRISTOPHER WREN.

Palladio’s Villa Capra, called the Rotunda, has, perhaps more than any other house, imposed itself on the imagination of subsequent generations, and as the ideal type of central building, it has become part of the general European experience. Mathematical, abstract, four square, without apparent function, its dry aristocratic derivatives have enjoyed universal diffusion; when he writes of it Palladio is lyrical.

“The site is as pleasant and delightful as can be found, because it is upon a small hill of very easy access, and is watered on one side by the Bacchiglione, a navigable river; and on the other it is encompassed about with most pleasant risings, which look like a very great theatre and are all cultivated about with most excellent fruits and most exquisite vines; and, therefore, as it enjoys from every part most beautiful views, some of which are limited, some more extended, and others which 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 Corbusier’s Précisions is unavoidably reminiscent of this. No less lyrical, but rather more explosive, he is describing the site of his Maison Savoye at Poissy.

“Le site, une vaste pelouse bombée en dome aplâti…. La maison est une boîte en l’aire au milieu des prairies dominant Ie verger. II est à sa juste place dans l’agreste paysage de Poissy. Les habitants venus ici parce que cette campagne agreste était belle avec sa vie de campagne, its la contempleront maintenue intacte du haut de leur jardin suspendu ou des quatre faces de leurs fenètres en longueur. Leur vie domestique sera inserée dans un rève virgilien.”

The Savoye House has been given a fair number of interpretations: it may be a machine for living in, an arrangement of interpenetrating volume and external space, another emanation of space, time and architecture. It is probably all these things; but the suggestive reference to the dreams of Virgil, and a certain similarity of site, solution and feeling put one in mind of the passage in which Palladio describes the Rotunda. The landscape there is more agrarian and bucolic, there is less of the untamed pastoral, the scale is larger, but the effect is somehow the same.

Palladio, writing elsewhere, amplifies the ideal life of the villa. Its owner, from within the fragment of created order, will watch the maturing of his possessions, and savour the piquancy of contrast between his fields and his gardens; reflecting on mutability, he will contemplate through the years the antique virtues of a simpler race, the harmonious ordering of his life and estate will be an analogy of paradise.

“The ancient sages commonly used to retire to such places, where being oftentimes visited by their virtuous friends and relations, having houses, gardens, fountains and such like pleasant places, and above all their virtue, they could easily attain to as much happiness as can be attained here below.”

Perhaps these were the dreams of Virgil. Freely interpreted, they have gathered round themselves, in the course of time, all those ideals of Roman virtue, excellence, Imperial splendour and decay, which make up the imaginative reconstruction of the ancient world. It would have been, perhaps, the landscape of Poussin that Palladio would have longed to penetrate, to roam among the portentous apparitions of the antique: it is possibly the fundamentals of this landscape, the poignancy of contrast between the disengaged cube and its setting in the paysage agreste, between geometrical volume and landscape which has the look of unimpaired nature, which lie behind Corbusier’s Roman allusion. If architecture at the Rotunda forms the setting for the good life, at Poissy it is certainly the background for the lyrically efficient one; and if the contemporary pastoral is not yet sanctified by conventional usage, apparently the Virgilian nostalgia is still present. From the hygienically equipped boudoirs, pausing while ascending the ramps, the memory of the Georgics no doubt interposes itself, and, perhaps, the historical reference adds relish as the car pulls out for Paris.

A more specific comparison that presents itself is that between Palladio’s Villa Foscari, the Malcontenta, and the house which in 1927 Corbusier built for M. de Monzie at Garches. A diagrammatic comparison will reveal the fundamental relationships.

In general idea, as can be seen, the system of the two houses is closely similar. They are both conceived as single blocks, with one projecting element and parallel principle and subsidiary façades. Allowing for variations in roof treatment they are blocks of corresponding volume, eight units in length, by five and a half in breadth, by five in height. In both cases six ” transverse” lines of support, rhythmically alternating double and single bays, are established; but the rhythm of -the parallel lines of support, as a result of Corbusier’s use of the cantilever, differs slightly. At the villa at Garches it is ½ : 1½ : 1½ : 1½ : ½, and at the Malcontenta 1½ : 2 : 2 : 1½. In plan, Corbusier thus obtains a sort of compression for his central bay, and interest seems transferred to his outer bays, which are augmented by the extra half unit of the cantilever; while Palladio secures a dominance for his central division, and a progression towards his portico, which focuses interest there. In both cases the projecting element, terrace or portico, occupies 1½ units in depth.

Structures, of course, are entirely different, and both architects look to structure to some extent as a justification for their dispositions. Palladio employs a solid bearing wall, and of this system he writes…

“it is to be observed, that those (rooms) on the right correspond with those on the left, that so the fabric may be the same in one place as in the other, and that the walls may equally bear the burden of the roof; because if the walls are made large in one part and small in the other, the latter will be the more fit to resist the weight, by reason of the nearness of the walls, and the former more weak, which will produce in time very great inconveniencies and ruin the whole work.”

Palladio is concerned with the logical disposition of motifs dogmatically accepted; but he attempts to discover a structural reason for his planning symmetries. Corbusier, who is proving a case for structure as a basis of the formal elements of design, contrasts the new system with the old. He is a little more inclusive.

“Je vous rappelle ce plan paralysé ‘du maison’ de pierre et ceci à quoi nous sommes arrivés avec la maison de fer ou de ciment armé.
plan libre
façade libre
ossature independante
fenêtres en longueur ou pan de verre
toît jardin
et l’interieur muni de casiers et débarras de l’encombrement des meubles.”

Palladio’s structural system makes it almost necessary to repeat the same plan on every level of the building; and point support allows Corbusier a fairly flexible arrangement; but both architects make a claim, which is somewhat in excess of the reasons they advance. Solid wall structures, Palladio declares, demand absolute symmetry; a frame building, Corbusier announces, requires a free arrangement: these must be, at least partly, the personal exigencies of high style, for asymmetrical buildings in the traditional manner in fact remain standing, and frame buildings of conventional plan continue to give aesthetic satisfaction.

In both houses the principal rooms are on the first floor, linked to the garden by an external feature and flight of steps. The main floor of the Malcontenta shows a cruciform hall, and symmetrically disposed about it are two suites of three rooms each, two staircases and a portico. At Garches the central hall remains, one of the two staircases occupies a similar position, but the other has been turned through an angle of ninety degrees, the entrance hall has been revealed from this level by an asymmetrical well, and the external feature corresponding to the portico becomes partly a re-entrant volume, obliterating a line of support and placed in a less perceptible relationship to the main room. The cruciform shape has disappeared, and a Z-shaped balance is achieved by throwing the small library into the main apartment. There is a subsidiary cross axis at Malcontenta, which is suggested at Garches by the central voids of the end walls. These convey a certain careful character to the plan, but there is no through vista.

The wall at the Malcontenta forms the traditional solid pierced by vertical openings, with the central emphasis in the pediment; and the outer ones have the windows placed towards the extremities of the façade, a device which seems to reinforce the cubic quality of the block. The double bay in the middle is expressed by a single door, or in the rear elevation by a “Roman baths” motif, and carries the upper pediments of the roof. Horizontally the wall falls into three main divisions: base; piano nobile, corresponding to the ionic order of the portico, terminated by a flattened entablature; and a superimposed attic with cornice. The base plays the part of a projecting, consistently supporting solid, upon which the house rests; but while the attic and piano nobile are rusticated, the base is treated as a plain surface. A feeling of even greater weight carried here is achieved by this highly emotional inversion of the usual order.

In the Villa at Garches the exploitation of the structural system has led to the conception of the wall as a series of horizontal strips, alternating void and solid, a system which places equal interest in both centre and extremity of the façade, and is maintained by Corbusier’s almost complete suppression in elevation of the wider spans of the double bays, which are arranged to read as two separate bays. Any system of central vertical accent, and inflection of the wall leading up to it, is profoundly modified. The immediate result in the garden elevation at Garches shows itself in the displacing of portico and roof pavilion from the central position which they occupy in the Malcontenta. They are separated, the one occupying the three bays to the left of the façade, and the other a central position in the solid, but an asymmetrical one in the whole elevation. The diagonal of the staircase forms the balance.

The entrance elevation retains the central feature in the upper storey, but it is noticeable that the further development of this feature within itself is asymmetrical. The downward indication of weights in this sort of façade is impossible; and
to see the central feature, interrupted by the horizontal voids, centrally repeated in the base, would be grotesque. Displacement and breaking up of the feature are again compensated by diagonal relationships; and in the ground floor entrance marquise and service door fulfill these purposes.

The other chief point of difference lies in the idea of the roof. In the Malcontenta it forms a pyramidal superstructure dominated by the temple fronts of the upper pediments, which occur above, and augment the central features of the main wall. Interest and silhouette are provided by the highly romantic chimneys, which possess a mediaevalizing quality, recalling the complicated machicolations of the now disappeared courtyard walls. Garches has a flat roof on two levels, treated partly as enclosure cut out of the block, and scattered with the irregular incident of gazebo, perforation and pavilion. The main plastic elements, the framed terrace of the entrance elevation and the pavilion of the garden front, are placed respectively in symmetrical and asymmetrical relation to the façades below. As at the Malcontenta, they are dominant features in the composition, but in neither case are they placed in direct vertical relationship with the principal features of the lower wall.

Corbusier’s treatment of the base is not continuous. In the cantilevered façades it is affirmed by set-backs or horizontal voids, elsewhere it is not expressed.

Mathematics and musical concord as the basis of ideal proportion was a common belief in Palladio’s North Italian circle, where there was felt to be a correspondence between the perfect numbers, the proportions of the human figure and the elements of musical harmony. Sir Henry Wotton, as Ambassador at Venice, reflects some part of this attitude when he writes:

“The two principal Consonances that most ravish the Ear are, by the consent of all Nature, the Fifth and the Octave, whereof the first riseth radically from the Proportion between two and three, the other from the double Interval between one and two, or between two and four, etc. Now if we shall transport these Proportions, from audible to visual Objects, and apply them as shall fall fittest…, there will indubitably result from either, a graceful and harmonious Contentment to the Eye.”

It was not in fact suggested that architectural proportions derived from musical harmonies, but rather that the laws of proportion were established mathematically and universally diffused. The Platonic and Pythagorean universe was compounded of the simpler relationships of numbers, and such a world was formed within the triangle made by the square and cube of the numbers 1, 2, 3. Its qualities, rhythms and relationships were established within this framework of numbers up to 27; and if such numbers governed the works of God, it was fitting that the works of man should be similarly constructed, and that a building should be a representative in microcosm, of the same process exhibited to a larger scale in the workings of the world. In Alberti’s words “Nature is sure to act consistently and with a constant analogy in all her operations,” what is patent in music must also be so in architecture, proportions are a reflection of the harmony of the universe, their basis, scientific and religious, was quite unassailable. Palladio had the satisfaction of an entirely objective aesthetic.

Corbusier has expressed similar convictions about proportion. Mathematics bring “des verités réconfortantes,” and “on ne quitté pas son ouvrage qu’ avec la certitude d’être arrivé à la chose exacte.” It is, indeed, exactness, precision, neatness that he seeks, the overall controlling shape; and within, not the unchallengeable clearness of Palladio’s volumes, but a sort of planned obscurity. Consequently, while in the Malcontenta geometry is diffused through the internal volumes of the building, at Garches it resides only in the total block and the disposition of its supports.

The theoretical basis on which Palladio rested broke down in the eighteenth century, when proportion became a matter of individual sensibility and inspiration; and Corbusier, in spite of the comforts which mathematics afford him, occupies no such unassailable position. The functionalist theory was, perhaps, an attempt to re-assert a scientific aesthetic with the objective value of the old. Its interpretation was crude. Results can be measured in terms of the solution of a particular process; proportions are apparently accidental and gratuitous. It is in contradiction of this theory that Corbusier imposes mathematical patterns upon his buildings: these are the universal “verités réconfortantes.”

Thus, either because, or in spite of theory, both architects share a common standard, a mathematical one, defined by Wren as “natural beauty”; and within the limitations of a particular programme, it is not surprising that the blocks should be of corresponding volume - 8 : 5½ : 5. Corbusier has carefully indicated his relationships by regulating lines, dimensions and figures, and over all he places the ratio of the golden section, A : B = B : (A+B). Thus he indicates the ideal with which he would wish his façade to correspond, although in actual fact the figures 3 : 5 = 5 : 8 thus represented are only approximate.

Palladio also provides his plan with cryptically explanatory dimensions, and thus the rooms comprising the suites of three can be read as a progression from a 3 : 4 to 2 : 3 relationship. They are numbered 12 : 16, 16 : 16, and 16 : 24.

The façade is divided vertically into four main units, the two central ones being really a single division by their common expression as portico. The horizontal divisions are complicated by the introduction of the order, which presupposes alongside the “natural” proportions, a series of purely “customary” relationships. In fact these horizontal divisions are uneven although, as the figures show, they roughly approximate to a division into fifths-a fifth part to the attic and approximately three-fifths of the remaining wall surface to order and entablature.

Corbusier also divides his façade into four units; but in his case horizontally. The two central units are partly unified by their placing alongside the garden terrace, and could be considered as corresponding to Palladio’s piano nobile. The vertical divisions are in the relationship indicated by the equation (3 : 5), which Palladio uses horizontally. In both cases there are elaborations in detail of dominant, complicated by imposition upon subsidiary system. It is by vertical extension into arch and vault, diagonal of roof line and parapet, that Palladio modifies the geometrical asperities of his cube; and the use of the circular and pyramidal elements with the square, seems both to conceal and amplify the real nature of the volumes. Some of these resources are the prerogatives of solid wall construction, freedoms of the “plan paralysé,” and the introduction of arched forms and pitched roofs is a liberty which Corbusier at Garches is unable to allow himself. In the frame building it is not, as in the solid wall structure, the enclosing walls that are a dominant, but the horizontal planes of floor and roof.

The quality of partial paralysis, which Corbusier has noticed in the plan of the solid wall structure, in the frame building is transferred to the section. Perforation of floors giving a certain vertical movement of space is possible; but the sculptural quality of the building as carving has disappeared, and there can be none of Palladio’s firm sectional transmutation and modeling of volume. Extension must be horizontal following the established horizontal planes; free section is replaced by free plan, paralysed plan by paralysed section; and the limitations in both cases are equally severe; as though the solid wall structure had been turned on its side, the former complexities of section and subtleties of elevation are now transferred to plan.

The shapefulness and spatial audacities of the Garches plan continue to thrill; but it is an interior which seems to be regulated by the intellect only, operating, as it were, inside a stage vacuum. There is a permanent tension between the organised and the apparently fortuitous. To the intellect it is clear, to the senses deeply perplexing; and it seems not to be possible to stand anywhere in it, at anyone point and receive the palpable impression of the whole. Both buildings can be absorbed from without; but from within, in the cruciform hall of the Malcontenta, there is a clue to the whole building, which is crystallized and focused there. At Garches, the theoretical equidistance between floor and ceiling conveys an equal importance to all parts of the volume in between. Allowed a sufficient height, it might be treated as a single volume, but otherwise the development of focus becomes a somewhat arbitrary proceeding. Corbusier accepts this limitation, and accepts the principal of horizontal extension; at Garches the central focus has been consistently broken up, concentration at one point is disintegrated, and replaced by a peripheral dispersion of incident. The dismembered fragments of the central focus become, in fact, a sort of serial installation of interest round the extremities of the plan.

The system of horizontal extension comes up against the rigid bounding lines of the rectangular block, which is fundamental to the programme. Elaborate external development is, therefore, impossible, and Corbusier logically employs the opposite resource, inversion in the place of extension, gouging out large volumes of the block as the terrace and the roof garden, and exposing them to the outer space. Thus the peripheral incident, which replaces the focus, sometimes becomes one and the same with the inversions, which represent an essentially similar feature to Palladio’s vertical extension.

This system of regular diffusion of interest and irregular development of points of concentration, throws into intense relief the geometrical substructure of the building. A comparable process to that in the plan takes place in the elevations, where the horizontal window treatment conveys equal interest to the centre and verge of the façades, and produces similar disintegration of vertical emphasis and displacement of the central feature. Elimination of focus immediately transfers interest to extremities of the block, which acquire a clarity and tautness, as though they were trying to restrain the peripheral incident from flying out of the block altogether.

A specific comparison is less easy to make between the Villa Rotunda and the Savoye House of 1930, the houses which seemed to provoke it. The problem, although at first it appears to be more severe, in actual fact offers a wider range. The emotional impression, concentrated in two fronts at Garches and the Malcontenta, is diffused here through all four, resulting in a more complex internal disposition and a greater geniality of external effect. The structural system of the Poissy house is less clear, and its central character is somewhat discounted by the cantilevered prolongations of what are presumably the east and west façades; and by the “directed” expression of the ground floor, with its porte cochère and utility entrances. There is a noticeable easiness and lack of tension in these façades; but there are analogous developments from the earlier houses in both cases. Such are Palladio’s development of central emphasis in both plan and elevations; and Corbusier’s extended interest throughout his façades and dispersing of focus. The complicated volumes of the roof gardens replace the Palladian pitched roof and cupola; Palladio’s four projecting loggias are replaced within the block as the first floor roof garden, which could also be considered, as the dominant element of this floor, to correspond to the domed saloon of the Rotunda.

Symbolically, and in what might be called the sphere of “customary” beauty, these two groups of buildings are in different worlds. Palladio sought complete clarity of plan, the most lucid organisation of conventional elements based on symmetry, as the most memorable form of order, and mathematics as the supreme sanction in the world of external forms. In his own mind his work was essentially that of adaptation, the adaptation of the ancient house; and at the back of his mind were always the great halls of the Imperial thermae, and such buildings as Hadrian’s villa at Tivoli. He has several schemes of archeological reconstruction of Greek and Roman domestic buildings, based on Vitruvius and Pliny, and incorporating elements, which in Greek and Roman practice would have been found only in public buildings, but which he regarded as general. Rome for him was still alive, and if the ancients had adapted the temple from the house, their large scale planning was no doubt similarly reflective. Development was, therefore, less a matter of innovation, than an extension of ideas already implicit.

Corbusier has an equal reverence for mathematics, and would appear to be sometimes tinged with a comparable historicism. He seems to find a source in those ideals of convenance and commodité displayed in the ingenious planning of the rococo hotel, the background of a social life at once more amplified and intimate. The French have an unbroken tradition of this sort of planning; and one discovers, in a beaux arts utilisation of an irregular site, elements which, if they had not proceeded Corbusier, would have been curiously reminiscent of those suave boudoirs and vestibules. Corbusier admires the Byzantine architecture of the Mediterranean world, and there is also present a purely French delight in the more comprehensible aspects of mechanics… the little pavilion on the roof at Garches is at the same time a temple of love and the bridge of a ship, the detail is precise, the most complex architectural volumes are fitted with running water.

Geometrically, both architects may be said to have approached something of the Platonic archetype of the villa, which the Virgilian dream could be held to represent. The idealisation of the cube house must lend itself very readily to the purposes of Virgilian dreaming. Here is set up the conflict between the contingent and the absolute, the natural and the abstract; the gap between the ideal world and the too human exigencies of realisation receives its most pathetic presentation. The bridging must be as competent and compelling as a well-executed fugue, charged as in these cases with almost religious seriousness, or sophisticated, witty allusion; it is an intellectual feat which reconciles the mind to the fundamental discrepancy of the programme.

Palladio is the convinced classicist with the sixteenth century repertoire of well-humanised forms. He translates this “customary” material with a passion and a high seriousness fitting to the continued validity that he finds it to possess; the reference to the Pantheon in the superimposed porticoes; to the thermae in the cruciform saloon; the ambiguity, profound, in both idea and form, in the equivocal conjunction of temple front and domestic block. These are charged with meaning, both for what they are and for what they signify; and their impression is poignant. The ancient house is not re-created, but there is in its place a concrete apparition of antique virtue, excellence, Imperial splendour and stoicism: Rome is there by allusion, the ideal world by geometry.

By contrast, Corbusier is in some ways the most ingenious of eclectics. The orders, the Roman allusion, are the apparatus of authority, customary, and in a sense universal forms. It is hard for the modern architect to be quite so emphatic about any particular civilisation; and with Corbusier there is always present an element of wit, suggesting that the historical reference has remained a quotation between inverted commas, possessing always the double value of the quotation, the associations of both old and new context. The world of classical Mediterranean culture, on which Palladio drew so expressively, is closed for Corbusier. The emblematic representations of the moral virtues, the loves of the Gods and the lives of the Saints, the ornamental adjuncts of humanism, have lost their former historical monopoly.

Allusion is dissipated at Garches, concentrated at the Malcontenta; within the one cube the performance is mixed, within the other; Roman; Corbusier selects the irrelevant and the particular, the fortuitously picturesque and the incidentally significant forms of mechanics, as the objects of his virtuosity. They retain their original implications of classical landscape, mechanical precision, rococo intimacy; one is able to cease hold of them as known objects, and sometimes as basic shapes; but they become only transiently provocative. Unlike Palladio’s forms there is nothing final about their relationship: their rapprochement would seem to be affected by the artificial emptying of the cube, when the senses are confounded by the apparent arbitrariness, and the intellect more than convinced by the intuitive knowledge, that here in spite of all to the contrary, there is order and there are rules.

Corbusier has become the source of fervent pastiches, and witty exhibition techniques: the neo-Palladian villa became the picturesque object in the English park. Content is different in both cases, and a bad portico is usually more convincing than an ill-executed incident. It is the magnificently realisable quality of the originals which one fails to find in the works of neo-Palladians and exponents of “le style Corbu.” The difference is that between the universal, and the decorative or merely competent; perhaps in both cases it is the adherence to rules which has lapsed.

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