Plea for Euclid*
PART(3)
Bernard Cache
www.objectile.net
Let's now take another example at the scale of the territory. Paul Virilio has rightfully emphasized the importance of speed in the perception of territory. Hence the interest in isochrone maps, sorts of deformed territories where, for instance, Bordeaux would appear much closer to Paris than Clermond-Ferrand because the first town benefits from a fast train link. These isochrone maps certainly give shape to the perception of geographical space by train travellers. We could even go further and use a 3D curvature to manifest the coexistence of fast tracks with slower means of communication. We would then get a kind of Riemann surface with a tunnel directly linking Paris to Bordeaux, while the slower means of communications would be inscribed on the outer distended surface of the tunnel. All these phenomena are certainly topological, since it is a question of modified distances which have an impact on order and continuity. But morover, it is a spatial representation of distances measured in time. This is very different from the Beaubourg example where we proposed a spatial structure for an architectural space. And indeed, topology can be very usefull to analyse phenomena the dimensions of which are not restricted to the three dimensions of space. But note that those folded surfaces of isochrone maps or even isochrone tunnels can very well be represented in 3D Euclidean space. Those new representations substitute to the more traditional one and it is rather a sign a richness that Euclidean space can house several types of representation.
As regards multidimensional phenomena, insofar as we want to give an easy intuition of them, the best geometric vehicle remains 3D Euclidean space. Not that we want to repeat Kant's error saying that Euclidean space would be the unique form of spatial intuition, but we cannot avoid the fact that there is a highly positive feed back between our Euclidean intuition and the experimental behaviour of physical space. In his dialogue with Bertrand Russel, Henry Poincaré who, certainly, cannot be suspected of empiricism, would conclude that "Euclidean geometry is not true but is the fittest" (4), and this for two reasons. First because it is objectively the simplest, just like a polynome of the first order is simpler than a polynome of the second order, and then because it suits rather well the behavior of solid bodies and light. Poincaré would even go so far as to justify the limitation of our experience to three dimensions by relating it to the combination of the two dimensional surface of our retina with the single independent variable of our muscular efforts to focus on object according to their distance. In a world where accommodation and convergence of our eyes would require two different efforts, one would certainly experience a fourth dimension in its visual space. But perhaps the best argument in favor of Euclidean geometry is to remind of the satisfaction of mathematicians when the same Poincaré provided an Euclidean model of Lobachevsky's geometry. By contemplating arcs of circles which axiomatically behave as lines, one could at least figure out what it was all about.
Hence, one is perplexed when one hears of non-Euclidean interfaces in cyberspace. When one goes into telecommunication laboratories working on interfaces of the so-called virtual space, one can only be struck by the pregnancy of very traditional spatial metaphors: like the double-pitched house or the village, at any rate, interfaces much more "heimat" than "cyber". But Euclidean space has nothing to do with these traditional archetypes of space. We believe, and our daily experience in computer-aided design and manufacture confirms, that we won't invent any new architectural space without going deeper into Euclidean geometry. Sure we can think of multidimensional topologies, which will then exceed the capacity of 3D Euclidean space. But multidimensionality is not the exclusive privilege of topology, there also exist Euclidean hyperspaces as well as projective hyperspaces. As far as architectural practice goes, we think that what has be to be thought and drawn is the way in which we use, translate, or plunge multidimensional spaces of all kinds into 3D new Euclidean figures. Just as an example, I will mention that Objectile uses everyday mathematical functions with a great number of parameters in order to design 3D surfaces. We then work within multidimensional parametric spaces, although the output is plain euclidean 3D. I would say that the essential part of our work is not to create "multidimensional topological non-Euclidean virtual spaces" but to design interfaces between parametric hyperspaces and 3D Euclidean figures.
By the way, one must be convinced that analytical tri-dimensionality is not entirely outdated in the history of Euclidean geometry. Sure, the principle of locating one point on the plane by two segments was already known to the Greeks, but the real use of co-ordinates could only come out of relations established between those two numbers. Nicolas Oresme who died as the Bishop of Lisieux in 1382, was probably the first one to really introduce rectangular co-ordinates (which he called longitude and latitude) and to establish relations like the equation of the straight line (5). He went further by extending his system of co-ordinates to 3D space and even to a kind of four dimensional space, but he lacked the analytical formalism which would be invented by Descartes (1596-1650) and Fermat (1601-1663). The bi-univocal correspondence between points of the Euclidean space and three-numbers sets (x,y,z) would later be called Cartesian space, although real analytical geometry would only show up in the 18th century. Thus, it is only in 1700 that the equation of the sphere was first written. Up to that point, analytical geometry was limited to the plane and ignored 3D space. As for Descartes himself, he kept on mixing analytical procedures and geometrical methods that he always used for elements of the first order ( straight lines and planes). In his Introductio in analysis infinitorum (1748) Leonard Euler was the first to establish the principle of equivalence of the double axis: x and y; before him, the abscissa was privileged over the ordinate. And it is not until 1770, that Lagrange wrote the equation of the straight line and plane in a system with three equivalent co-ordinates. Fully isotrope 3D space is, then, only 230 years old and originates somewhere between Claude Perrault's Ordonnance des cinq espèces de colonnes selon la Méthode des Anciens (1683) which made the architectural orders a convention and Durand's Précis des leçons d'architecture (1802-1805) which based architectural composition on an abstract system of axis. The grid in itself is certainly a very ancient and fundamental element which can be found in the castrum romanum as well as in the architectural orders with the vertical column and the horizontal entablature. Not to mention the striking formal analogy between the Christian Cross and the Cartesian system of X-Y axis. But it took a very long time before this formal element assumed its current status of abstract grid and we suspect it might take a bit longer before we draw all the consequences of this "cartesianisation" of Euclidean space.
The same slow-moving history happens with projective geometry the integration of which into CAD software is only starting nowadays. At about the same time that Descartes initiated analytical geometry, a French architect, Sieur Girard Desargues de Lyon, had the idea to transpose the methods of perspective in geometry and it took some 200 years before the Napoleoniac officer Poncelet benefited from his captivity in Russia to synthesize projective geometry. In its Brouillon projet d'une atteinte aux événements des rencontres du cône avec un plan (1639), Desargues had made a unified system out of the four conical curves. To be sure, the circle, the ellipse, the parabola and the hyperbola were already known since antiquity. The Greek Appolonius had already considered them as sections of a cone whose basis was a circle. Nevertheless, these curves were studied in isolation, each being thought to constitute a specific case, with primacy given to the circle as the perfect figure. Likewise, in architecture, Borromini would still use arcs of circle to compose the ovals of San Carlino alle Quattro Fontane instead of drawing real ellipsis, which had been reintroduced into the Renaissance by Baldassare Peruzzi (6). In 1604 Kepler had been the first to use the concept of the point at infinity, a concept which enabled him to describe at once the closed finite figures of the circle and the ellipse as well as the open infinite parabola and the double hyperbola. But Kepler still thought of geometry as a symbolic system and the continuum which linked together the conicals was still oriented toward the perfection of the circle. Only Desargues would do away with any symbolism of the figure and draw all the consequences of the point at infinity. The circle became just one particular case of the various conicals, which admitted of being transformed into one another by projections and sections. Parallels would now intersect like any other line and converge toward this point at infinity which is now considered just as another point on the plane. This secularisation of infinity would not be particularly welcome. Apart from a few exceptions like Pascal and Mersenne, Desargues would find considerable opposition amongst artists, metaphysicians and mathematicians. The Académie de Peinture led by Le Brun would opposed this approach to perspective where things should be drawn according to geometrical rules rather than following empirical perception. Hence the reluctance to accept that a circle seen from an oblique angle transform into an elliptical figure with its own consistency. Hence also the emphasis on the optics of the "perspectiva naturalis" where the remoteness of objects is rendered by fading colors rather than by the rigorous geometrical constructions of the "perspectiva artificialis". For their part, the metaphysicians could not accept that the infinite, considered as a divine attribute, be assimilated to common points of the plane. In reaction, Guarino Guarini, architect but also Abby of the counter-reformist Theatines Order, would invent methods to draw complex figures only with finite points. But even mathematicians like Descartes would show some reluctance to considering parallels as converging lines. Apart from a few disciples like La Hire, the work of Desargues would fall into oblivion and it is only at the end of the 18th century that projective geometry would raise a new interest with Monge, Carnot, Chasles and especially Poncelet and von Staudt.
Poncelet would develop in space what Desargues had demonstrated in plane. He enunciated the fundamental principal of duality by asserting that every theorem involving points and planes has a shadow theorem which can be deduced by swapping the words "point" and "plane". Later on, the nine axioms put together as the postulates of projective geometry by Hilbert would give the principle of duality its most general form. Swapping "points" and "planes" enables us to replace the three first axioms by the three following. As for the three remaining, the swap do not alter the postulates. But stepping back from axiomatics, we can say that projective geometry studies the effect of the two types of homographic transformations (projections and sections) on the variety of Euclidean space to which has been added points at infinity. Poncelet would use these projections and sections to develop theorems already demonstrated for simple figures like circles, and extend them to all conicals or curves of the second order. This general method of considering curves of order "n" would pave the way for spaces of curves. For instance, points, circles and straight lines form a four dimensional space in which they are assimilated to circles of zero, finite and infinite radius, allowing one to manipulate these three elements with same mathematical methods. And according to the principle of continuity, Poncelet would also assume that theorems established for finite and real figures also apply to infinite and imaginary cases. Thus, the extension of euclidean space to infinite points is complemented by another extension to imaginary points. This allows, for instance, the radical axis defined by the points of intersection between two circles to keep on existing as an imaginary axis when the circles no longer overlap.
Poncelet was very attached to spatial intuition and wanted to develop projective geometry as an autonomous discipline, independent of algebraic analysis. But the evolution of mathematics would go the other way. Projective geometry would benefit from the same algebraic methods which applied to Euclidean space with cartesian geometry. All points of projective space would be associated to four numbers X, Y, Z, U, the fourth one enabling the addition of points at infinity to Cartesian space, and those numbers would be either real or imaginary to take into account imaginary solutions in equations of intersections. Projective geometry would then become an algebra which developed for its own sake, mathematicians forgetting the spatial significance of their investigations. Meanwhile CAD softwares appeared in 1960's and their structure remained strictly Cartesian. Suffice it to say that the twin brother of CATIA was called EUCLID (7). As far as technical applications are concerned, such as architecture, the digital age is still deeply Euclidean and will probably remain so for all the good reasons we have rehearsed. For instance, as CAD software become parametric and variational, designers can start to implement topological deformation into Euclidean metrics, which means that you can now stretch a model, and still maintain control of its metric relations. What will probably happen is that, one day or another, CAD software kernels will benefit from the extension of Euclidean space within projective geometry. For several years already, a French and Canadian company has started developing the most advanced CAD software kernel on the basis of these ideas and their first real time application in due this year. Not by chance is this company called SGDL, which stands for Sieur Girard Desargues de Lyon: since this team of software developers still find time to peruse mathematical writings several centuries old. They have already developed a single interface which enables one to manipulate points at infinity and thus unify the various primitives, each of which is the object of a dedicated function in traditional software. The various primary forms that constituted the Roman lesson of Le Corbusier in his book Vers une architecture: the cylinder, the pyramide, the cube, the parallelepiped and the sphere, together with more sophisticated figures like the torus, are no longer isolated archetypes but appear as particular cases in the continuum of the quadrics. And more generally, one can notice in the field of mathematics a strong return to geometry initiated by eminent figures like Coxeter. So much so that future CAD software, instead of becoming non-Euclidean will rather benefit from all the extensions of Euclidean space into Arguesian geometry. As such they will integrate the concepts we have mentioned above: points at infinity, imaginary points and multidimensional spaces of curves and surfaces (8).
One last parenthesis about the fundamentals of geometry before considering how the digital age could bring us the real flavour of Euclidean space rather than dissolve it. While examining how the Thirteen books of Euclid have been reworked throughout the history of mathematics, we first looked at the postulates and their axiomatisation by Hilbert. This led us to the fact that the primary terms of geometry should no longer be subject to definition because their signification would, in fact, be a consequence of the relations established by the axioms. But, up to now, we haven't said a word about the common notions by which Euclid referred to arithmetic and logic. Hilbert's axiomatic geometry also relies on basic notions of arithmetic and logic, since there is a hierarchy of axiomatic systems by which geometry includes an arithmetic, which in turn includes a logic. As a direct consequence, geometry suffers from the same flaws as its presupposed axiomatics. Thus geometry is spoilt by the same incompleteness that Gödel demonstrated about arithmetic, which means that there are true propositions which cannot be proven as such and remain undecidable within the system itself. But what Gödel demonstrated is that this is a general flaw of all powerful axiomatics, which doesn't specifically affect Euclidean geometry. For instance, arithmetic suffers from the same flaw and, to our knowledge, very few people take advantage of this incompleteness for claiming for a non-peanean arithmetic. To our knowledge, there is, for instance, no such thing as a "non-peanean finance", even among people manipulating billions of dollars across the planet. Moreover, in 1937 Gentzen re-established the coherence of the theory of numbers by using one single principle referring to an intuition exterior to the formal system. Incompleteness can be mitigated by a limited recourse to intuition. Then, we have already seen to what extent modern physics have validated Euclidean intuition, and we now realize that even its logical structure should not be call into question anymore than logic itself. So whatever the angle we are looking from, the more than 2000 years old work, Euclid's Thirteen books, still appears as the best available theoretical and practical tool to deal with figures in space.
HOME > PART(1) > PART(2) > PART(3) > PART(4)
Notes
(4) Henri Poincaré, La science et l'hypothese, Paris, 1902.
(5) P. Duhem, Dominique Soto et la scolastique parisienne (Annales de la Faculté des Lettres de Bordeaux, Bulletin hispanique, 1911, pp: 454-467).
(6) See Wolfgang Lotz: Die ovalen Kirchenräume des Cinquecento.
(7) CATIA just bought EUCLID while we were writing this text.
(8) SGDL is now able to handle not only the quadrics, but also the cubics and the quartics.
["Plea for Euclid" is published both by Arch'it and ANY Review. The italian translation is by Marco Brizzi. By Bernard Cache, MIT Press published "Earth Moves", available at Amazon.com]
Back to: EXTENDED PLAY
