Cache, Bernard; The Table and the Territory; Bliss Conference; 19 May, 1994 Geometry, architecture and the graphic arts have often had recourse to the three basic forms of geometry: the square, the equilateral triangle, and the circle. On several occasions in history, it has seemed possible to reduce all that is visible to a composition of these three elements. However incredible it may seem, it has proved possible to find a certain scheme, enabling the diversity of substances to be reduced to the composition of a small number of elements. As far as the visible is concerned, no system of this kind has ever really been followed up, and the basic geometrical forms have tended to remain in the state of symbolic figures. The appearance of the Computer Assisted Design (CAD) systems with the rise of multimedia and Electronic Highways, are coming more and more to shape our perception of the visible. In fact, however, for a long time we have already been like the robot in Terminator II, whose artificial vision is based on a system of recognition of shapes. It is these very CAD systems that oblige us to make explicit the way in which we conceive forms in general; in the last resort, they also offer us a system of basic elements. Thus, beside the basic classical forms, and the transformational vectors, we have witnessed the appearance of new elements - approximate curves - starting with the BÊzier, followed by various kinds of Spline and now the NURBS. We can note that, while these three basic elements - square, circle and triangle- seem to obey a kind of shared legality, each of them has its own character, which demands explication in more explicit and more highly differentiated figures. If we have the intention of basing a systems on a small number of elements and the pretension of reducing all that is visible to no more than three basic images, it is advisable to avoid any redundancy or overlapping between these three elements, to ensure that the systems is based on three foundations which are as distinct and separate from one another as possible. The square seems to be well established and deserves to appear as the regular polygon par excellence. What distinguishes the circle from the square, is the existence of a degree of curvature. The triangle’s particularity lies in the equality of its angles. Regularity, curvature, angularity - these are the three concepts that we retain from the three classical basic elements. The next step is to find the matching visual figures for these concepts. As far as regularity is concerned, there is no problem since we have said that the square seemed to be a figure that was perfectly adequate to the concept of equivalence. As far as curvature is concerned, we have to find a general figure of any degree of curvature. Moreover, We shall maintain the principle that a general figure of curvature must include on inversion of sign, enabling the transition from concavity to convexity. That is why we shall retain as the figure of any degree of curvature this linear element that the mathematicians, called inflection. Finally, as far as angularity is concerned, we propose to assign it the general figure of the vector, which constitutes a basic element in mathematics and physics. Instead of the square, we shall refer to the figure of regularity as a frame, the frame is the interface between the painting and every inch of architecture. The question of the frame also impinges on photography and the cinema, not to mention sculpture and crystallography. As for inflection, this word exceeds its mathematical context, since it refers to changes of tone or accent in the voice, concerning regional dialects, grammatical conjugations, and emotions. Finally, the vector. The category of the vector is just as large as that of the frame or of the inflection. The vector is the current mathematical formalization of an ensemble of images that we perceive intuitively as forces or movements. Inflection, vector, and frame constitute categories that go far beyond simple mathematical formalism. The latter is only there to provide us with a guideline leading us rigorously. Without metaphor, from one image to another. It may even allow us to invent new images or new modes of articulation between them. There is in all of the great atomists an element of surprise, like a point of return of their doctrine, where the point is converted into line or surface. Take Lucretius: his magnificent poem begins with those black atoms which fall in the white void, limits of the visible and the invisible. And then we come to Book IV, where everything is given a new foundation, not on the binary opposition of matter and nothingness this time, but on the full acceptance of the
situation in which we find ourselves, amid the simulacra. The same thing happens in Leibnitz, the second atomist. At first, it is as if the world is constituted of elementary monads, the limits of the visible and the invisible, without doors or windows. But this all leads to the point where the world is composed not of atoms like grains of sand, but of folds as in roofing tiles. Then the element is no longer the point but the fold. The logical regression to the invisible point as limit leads us back to the world of perceptions and simulacra. In practice, the architect is confronted with the site where building is to take place. Let us consider this site graphically. Here is the relief line: This relief line is presented as an irregularly curved line. It is and abstract line as long as we refrain from imposing on it the traditional marks of the landscapes in which we live. Most of the time, the result is something like this. Architecture leaves its mark on the hilltops or in the valleys. More generally it is always extremes that attract our attention when we are dealing with a continuum: the steepest gradient, the minimization of tensions, the maximization of profit. Under the pretext of reinforcing the identity of the site, the hills are reinforced with a tower, and roads are traced in the river valleys. To devise a new reading of the site, to discover a spirit of the place that reveals an aptitude to change identity, that is what leads us to devote our attention to the mathematical features of singularity. If we generally only apprehend the peaks and valleys of a landscape, it is because we undergo the mark of the vector of weight, which determines our everyday space and allows the designation of the highest and lowest points. If you suppress this vector, if you place yourself in a space without weight, without high or low, without right or left. What is intrinsic remains in the singularity: the inflections. That is why inflection is such a basic image for us. This little element of curvature is situated at the crossroads of several approaches to the visible. The inflections of any line mean placing oneself simultaneously in a space of singularity and in a geography beyond territorial identities. Traditional architecture sets the building on top of the hill, while modern architecture looks for the slope in order to frame an inflection. Inflection seems to constitute a characteristic element of contemporary architecture, as well as of the graphic arts and design. It is interesting to consider how much this element corresponds to this contemporary feeling of weightlessness. Of course, inflections is only a first image. All the same, this is not enough, for while inflection is no longer this negative figure of difference opposing high to low sites, it still falls under the jurisdiction of differential analysis. It is as if we had to go further, to pass to the second derivative and not just consider variation, but a principle of variation. Are we then engaged in a process of infinite regression? Unless we consider the elements of curvature whose function constitutes their own principle of variation. Rest assured, we will not have to look very far. It is sufficient to pass from the general form of inflection to one of its particular forms: the sinusoid. The derivative of the sine function (x) is the cosine function (x), which already has the same form as the sine function but with a delay of a quarter of a phase. If we continue the analysis, we note that the derivative of cosine (x) is -sine (x), i.e. the same form once again, but this time with a delay of the half of a phase; it is therefore easy to calculate that the fourth derivative of the sine (x) is the function sine (x) itself. We are therefore not engaged in an interminable analysis, but we are in a universe of forms that contain the principle of their own variation. There is probably nothing aesthetic about the sine function (x) in itself, but it provides us with on element that enables us to construct the visible. Associated with the calculating ability of the computers, this function is a tool, if we follow Joseph Fourier’s theorem, there is no function, however complicated it may be, which cannot be reduced to a series of sinusoid functions. So the computers and Fourier’s theorem make Leibnitz’s great dream come true: the visible can be mathematized. The sinusoid is the element from which all possible figures can be reconstituted. This would be all the more true in that the sinusoid functions make it possible to compose any form or any color, which are defined by the wave-length of a periodic function. We should treat all these harmonics with prudence. First of all, there is no need for a mathematical function to be beautiful in itself. In the second place, once we pass a certain level of complexity, mathematical writing, just like its result in drawing, loses all relation with intuition. Perception is by no means reducible to information. But inflection had a past. It had been formed in a tradition well before the question of its numerical formula had even been raised. Alois Riegl demonstrated in his remarkable Stilfragen that the very essence of what was to appear as the worst crime in the eyes of the modernists - the vegetal rinceau - had certainly not developed in a sequence of fantasies, and even less that this ornament was intended to represent vegetable species in a travesty of architecture, but that it constituted a series of continuous transformations obeying precise rules which succeeded one another. Far from being the result of individual whims, the ornament conforms to a tradition. The vegetal rinceau is thus neither capricious not representational. Le Corbusier’s free plan, lacked the history. The freedom of the linear trace submits with bad grace to the final effect that one wants it to realize; and activity develops which, out of prudence, only makes use of a restricted number of lines. The baroque era was probably the one that knew best how to experiment with curvature, with its wealth of folds and gathers. That is why inflection plays a major role in it. Inflection is a new type of point; the center of the curve. If we confine ourselves the figure of mannerist inflection, the site of the
centers of the curves immediately assumes the shape of a forked curve with its two branches tending towards infinity in opposite directions, following an asymptote perpendicular to the tangent at the point of inflection. The mannerist inflection could be explained by a center of curvature extended to infinity, but the rococo inflection will imply infinity in its very fold. The rococo is characterized by this vertiginous point where the curve begins to turn around itself until it reverses the direction of rotation, and where, with the radius of the curve at zero, the circle is confused with its center, a maelstrom whose circumference is nowhere but in this point of inversion. It is this way of bearing infinity in the fold which forms the subterranean link between the rococo line and the contemporary fractal line, this fold to a higher power which always comes back and never allows itself to be effected totally. At this point it seems necessary to constitute a nation of the complete image centered on the organization of the relation between inflection, vector and frame. Perhaps the best contemporary example of a complete image has been given us by Francis Bacon. Bacon’s work forms a systems by means of the recurrence of these three elements. The question which incessantly repeats itself in his work is: how to frame the vector to allow the accident to emerge in the inflection? The question has never been put as clearly as in these very beautiful complete images which Bacon hast left us: Figure in movement 1976. Landscape 1978, or even better Jet of Water 1979. They are complete images in which the three elements are related to one another in the most direct manner. This can be represented diagrammatically in the following way. Let s be clear about it, completeness does not therefore assume a normative meaning. Above all, it should not be taken to mean that it is necessarily the right thing to do under all circumstances. It would be necessary to review all the combinations of signs, i.e. the three in isolation, the three pairs, the triplet in its complete image, without forgetting the zero image. However, this is only a first level of analysis, for as soon as one considers that the three elements can be involved in hierarchical relations, a systems must not only contain all the possible combinations but the sixteen arrangements as well. Finally, if we conceive that the function of the frame is to organize the relation of the vector to the curvature, and that this proposition certainly cannot be reduced to the simple institution of a hierarchical relation between these three elements, but to a space of relations with an a priori indeterminate number of dimensions; and if, moreover, we recall that each of the elements it itself only a value in a series, then we see what a vast field this is and how just our three little elements can lead to a systems whose objective sill always be unattainable, that the visible is the object of a mystery is a very old idea. In the Christian West this mystery is called the Incarnation, and the Renaissance painters never stopped questioning it in their paintings of the Annunciation. This theme of the Annunciation provided the opportunity to construct a complete image to a higher power; thus we find each of our three elements in it, but they appear twice - once explicitly, and once implicitly. Thus, the wing of the angel describes a magnificent inflection, pure sign, in opposition to the folds or the Virgin’s robes. Similarly, a first vector links the angel to the Virgin, either through the simple gaze of the angel or through the gesture of his hand pointing towards his interlocutor. Very often, however, this horizontal vector is duplicated by a vertical vector that takes the form of a lily carried by the angel. Finally, the scene is generally set within the framework of a portico, but always a second framing element comes to open the scene onto a mysterious exterior: and impost opening onto the sky. What is the origin of these duplications of the elements? What accounts for this second power? If the Annunciations carry the complete image to the power of two, it is that this theme drives the system of the visible to its limits. Thus the angel is the messenger and the Virgin is the addressee; it is as if the inflection of the angel formed a transmission wave, as if the folds of the Virgin formed the sound or timbre of a membrane. Horizontal gaze, vertical lily; zero and one; on the basis of which all the information can be written; not the author, but the direction which points towards Him; and the portico itself is duplicated in an element that can be called a quasi-frame, for it is condemned to remain in a virginal state, since what it frames is situated out of reach, transcending any attempt to frame it. There is the unitary relation of the inflection with itself, of the angel to the Virgin or of the wave to the sound. There is the unitary relation of the vector to itself, of the vertical to the horizontal, of the lily to the gaze. There is the unitary relation of the frame to itself, of the frame to the quasi-frame, of the portico to the impost. There is the binary relation of the vector to the inflection, of the angel carrying the lily. There is the binary relation of the inflection to the vector, of the Virgin who submits to the gaze of the angel. There are the ternary relations of the two complete images, which the angel and his lily, the virgin and the gaze of the angel each form in his or her arcade.