Plea for Euclid*
PART(2)
Bernard Cache
www.objectile.net
At about the same time non-Euclidean geometry was discovered, Evariste Galois (1811-1832), established the theory of groups that Klein would later apply to geometrical transformations. Klein would go so far as to define the various geometries by the group of transformations which leaves invariant certain properties of geometrical figures. For instance, translation, rotation and symmetry form a first group of transformation, the group of movements which transform geometrical figures without affecting distances nor angles in these figures. This group of movements defines what is called: "metric geometry". Now, if we forget the distances and concentrate on the "shape" of the figures defined by the angles between elements, we come upon a new transformation which is the scaling. Translation, rotation, symmetry and scaling form a wider group of transformations, the group of similitudes which defines Euclidean geometry. Continuing in this fashion, we come upon a third group of transformations, which assimilate the circle, the ellipses, the parabola and the hyperbola as sections through the same invariant cone. This corresponds to the common notion of perspective that a circle should be drawn as an ellipse when seen obliquely and it has been an important debate in architecture: to decide whether our eye is able to recognize circles perceived on the slant (3). Juan Caramuel de Lobkowitz went so far as to propose columns of oval section in order to correct for perspective effects in his counterproposal for the colonnade of St. Peter in Rome. So, getting rid of distances and angles, if we focus on what is called "position properties" as opposed to "metric properties", we can add projections and sections to the group of similitudes. We then get the group of homologies which defines projective geometry. And finally, if we do away with position properties, and only look at the continuity of the figures and at the order in which their elements are linked together, just as if figures where made of an elastic material which can be stretched and deformed, but not torn, we encounter another group of transformations : the homeographies which define topology.
So what clearly appears in Klein's theory of transformation as in Hilbert's axiomatics, is that there is a hierarchy organizing the various geometries. Between Euclidean and non-Euclidean geometries (those of Riemann and Lobachevsky), there is only a question of specifying the form of the Parallel postulate. It is a bifurcation between geometries of the same level. Whereas, between Euclidean geometry, projective geometry and topology, there is a relation of inclusion in terms of both number of axioms and relevant geometrical properties left invariant by a certain group of transformations. Euclidean geometry requires more axioms and more structured properties. Projective geometry and topology can be more general only to the extent that they deal with looser transformations and objects. As such topology enables one to focus on fundamental properties from which our Euclidean intuition is distracted by the metric appearances. Because topology doesn't register any difference between a cube and a sphere, it focuses on what is left, order and continuity, and makes obvious the difference between the sphere and the torus. But, of course, order and continuity are also essential to Euclidean geometry. Euclidean geometry includes topology. Topology is less than Euclidean geometry. Common misunderstandings result from the fact that topology focuses on properties which typically lead to complex interlaced figures, or we would say, which appear all the more difficult to draw since perspective is no longer taught to the general public. Let us take an example which should appeal to architects. The Moebius strip has now become common place in contemporary architecture, although in most projects this remains more a rhetorical figure than a geometrical structure. But there is one well-known building which actually has a rather complex topological structure that has been overlooked, at least to our knowledge. This building is the Beaubourg Center in Paris. This scaffolding-building, currently enveloped by more scaffolding for maintenance, is certainly not original by virtue of its machinic image, which is but a revival of the utopic drawings of Archigram. Nor does it fully function as a urban machine, since no advantage has been taken of possible connections with the underground passing alongside the eastern facade. However, the use of this building induces a very specific experience for all the people whose destination lies above the first floor. In this case, one would enter by the main entrance - provided he knows where it is located - and, so doing, he would not pass directly to the inside but to an intermediate kind of situation, where he remains in an kind of exterior space although being already in the interior of the building. This mixed situation arises from a series of conditions, such as the vastness of the hall, the stains of rainwater on the gray carpet, or the streams of people in coats heading chaotically to escalators where, strangely enough, one would find himself in the inverse mixed situation. The escalators obviously take people back to the outside, suspending them in the air while they contemplate the Parisian skyline, but in the meantime, even if one is again confronted by the weather, one would find oneself more sheltered than one was before in the main hall. The narrow dimensions of the tube, its circular section, and the stillness of the people standing on the mechanical steps while starting to undress, contribute to create a kind of cozy atmosphere which can even become oppressive. And finally, it is only when one enters the rooms of the museum or of the library that one really feels inside, freed from this tension between interior and exterior. So, recapitulating this experience, we would say that one comes from the outside to enter in an external interior and then proceed into an internal exterior before finally getting inside. This spatial experience has the topological structure of a Klein bottle. To what extent this topology has been taken into account in this particular architecture is another question, but the fact is that this structure could be built and actually exist in Euclidean space. Moreover, the Klein Bottle can take many shapes.
One single topological structure has an infinity of Euclidean incarnations, the variations of which are not relevant for topology, about which topology has nothing to say. New topological structures can be incarnated in Euclidean space as squared figures as well as curved figures. Topology cannot be said to be curved because it precedes any assignment of metrical curvature. Because topological structures are often represented with in some ways indefinite curved surfaces, one might think that topology brings free curvature to architecture, but this is a misunderstanding. When mathematicians draw those kind of free surfaces, they mean to indicate that they do not care about the actual shape in which topology can be incarnated. In so doing, they should open the mind of architects and allow them to think of spatial structures before styling them as either curved or squared. And, of course, as soon as it comes to actually making a geometrical figure out of a topological structure, we enter into Euclidean geometry; that is, the design of complex curvature is essentially Euclidean. One should not think of Euclidean geometry as cubes opposed to the free interlacing of topology. On the contrary, it is only when variable curvature is involved that we start getting the real flavor of Euclid, that suddenly the trivial concept of the unique parallel starts playing tricks on you. Willingly or not, architects measure things, and this implies a metric. To be sure, we may want to remind ourselves of the formless topological background common to all saturated geometries, be they Euclidean or not. But this reminder demands accurate work on curvature. Only by mastering the metrics, can we make people forget Euclid.
Notes
(3) Cf: Architectura Civil by Juan Caramuel de Lobkowitz, see also Claude Perrault's critics of optical corrections.
["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]
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