Plea for Euclid*
PART(4)
Bernard Cache
www.objectile.net
From this perspective then, is the contemporary rejection of Euclid a merely academic matter or just another strategy of the avant-garde? Is it a tabula rasa of space that seeks to succeed the tabula rasa of time through which the Moderns sought to rid us of the past? We have seen how the controversy about projective geometry in the Baroque era was already architecture and still philosophy. All the same, we can suspect that when a proposition as errant as non-Euclidean architectural space becomes so widespread, it bears the value of a symptom. And the truth of this symptom seems to indicate that we have lost the will or, at least, the strength and the ability to establish distances, not only in the geometrical sense, but also in the philosophical sense in which Nietzsche used this concept (9). We might find ourselves facing a false choice between the old metaphysical circle and the cyber topological ectoplasm, which choice would result in either the solid sphere where distance is established as a constant surrounding an identified centre or the teratology of inconsistent figures… Just as if curvature could not start to vary without having us fall into indeterminacy.
But let us come back to technical matters. The value of Euclidean space would be of less consequence to us, were we not involved in numerically controlled manufacturing [Cf www.objectile.net]. For a tool path is fundamentally a parallel to the surface to be manufactured. In other words, a machining program generator starts by calculating the set of points at an equal distance from the surface, distance given by the radius of the spherical tool of the router. A machining program is basically the parallel to a free surface, whichever it is. And the concept of parallel, so fundamental in Euclidean geometry, starts creating interesting problems long before we contemplate free surfaces.
Fig. 1-7.
Take all the decisions that a CAD software has to take when it comes to drawing parallels. By definition, the parallel to a point P is the set of points at a given distance from P. It is then a circle, from which we already see that the parallel transforms a figure into another type of figure [fig. 1]. Then comes the case of the straight line. The parallel is easy to draw as long as we consider a infinite line. The parallels are then the two lines transposed on each side of the original line at a given distance. This is the canonical case according to which we generally confuse parallel and translation [fig. 2]. But this misconception will quickly vanish when we consider the finite straight line. Then there no surprise as long as we consider the core of the segment whose parallel can be assimilated to the two transposed segments, but what happens when it comes to the ends? Rigorously speaking, we must add two half circles to the two segments on each side of the original straight line, because they are indeed at the same given distance from the original line. The line then becomes an oblong [fig. 3]. Otherwise, if we want to stick to the intuitive idea of the two segments, we have to add a rule to our definition of the parallel, specifying that we only want the points at a distance that can be determined according to a orthogonal line, meaning where the latter can be calculated [fig. 3bis]. But we will quickly see that this rule will lead us to other problems in the next case : that of the square, since the first result of the parallel will then be eight independent segments [fig. 4].
We now have to distinguish the inside of the square from the outside and make two different cases in order to link together the four isolated external segments and shorten the four internal crossing segments. The shortening of the interior segments is preliminary to the elimination of loops in the most general case of parallels, while the gaps between the four external segments is a direct consequence of our additional rule [fig. 4bis]. If we cancel this rule again, the external parallel will become a square with rounded corners [fig. 4ter]. This result might satisfy many people in the molding industry but it leaves us with a figure different from the original square and its internal parallel. And this might certainly dissatisfy many architects and designers, as well as the primary intuition of the general public. This intuition would tend to extrapolate on the basis of the very special case of the circle where both the inner and outer parallel remain a shape similar to that of the initial circle [fig. 5]. In this case the generation of the parallel creates bigger and smaller circles, and the operation becomes a scaling transformation, which is called "homothetie" in French, based on the Greek etymology "homoios" meaning "similar". And the dissatisfaction might even become greater when we notice that the parallel is generally not a reciprocal transformation. If, coming back to our square and its two parallels, we consider the inner one and generate its external parallel at the same distance used previously, we come upon a rounded square the straight parts of which merge with the sides of the initial square [fig. 6]. Thus, the reverse parallel of a parallel to an initial figure is not necessarily this initial figure. CAD software usually provide several options to solve this problem in the case of polygonal or composite figures: external parallels can be chosen to be rounded, squared or chamfered (10) [fig. 7]. And in the case of architecture software, elements like walls are "intelligent" enough to behave as the majority would expect them to; which of course can be a serious limitation. Similarly when figures are composed of more complex elements, CAD software can always find solutions for extrapolating distorted squares, but we must be aware of the irreversible loss of information each time we come back from an internal parallel to an external parallel [fig. 8].
Fig. 8 - Two parallels to the same curve by different
extrapolation method.
Fig. 9 - Parallels on both side of a simmetrycal curve.
This irreversible loss of information becomes even more blatant in the case of a free open curve as one increases the distance according to which the parallel is generated. The first effect of the parallel is to reduce the radius of curvature of concavities and increase that of the convexities. This is the case of a positive distance in reference to the concavities, the phenomena being inverse in the case of a negative distance [fig. 9]. The parallel breaks curvature symmetry by emphasizing concavity while softening convexity, or vice versa. This is an issue of more than merely mathematical interest for architects and designers, since concavity and convexity are the basic intensive qualities on which sculpture is based [fig. 10]. As Henry Moore wrote: "Rodin of course knew what sculpture is: he once said that sculpture is the science of the bump and the hollow" (11). Convexities and concavities are the mathematical equivalents of the sculptural bumps and hollows. But the parallel does not only break symmetry, it starts creating loops where the distance assumes a value superior to the local radius of curvature on the initial curve [fig. 11].
Fig. 10 - Convex relief, Symmetrical relief, Concave relief.
Fig. 11 - Parallel at a distance superior to the radius
and parallel at a distance equal to the radius.
The parallel transforms simple undulations into strange surfaces, although still Euclidean, or strange because Euclidean should we say, where loops can themselves include other loops [fig. 12]. Architects and designers shouldn't ignore this issue either, since the basic operation carried on by a CAM software is precisely to cancel the loops on the parallel surface which a spherical tool has to follow in order to machinate a relief. It is this elimination of loops which is responsible for the loss of information we have already encountered in the case of the square. On the final relief, on the inverse parallel of the parallel, this loss of information will appear as zones where the tool is too large to get into the concavities and will then substitute its own constant radius for the varying curvature of the initial relief [fig. 13]. We can bet that baroque architects already knew quite a bit about these transformations by which they endlessly created curves out of curves, as in the multiplying of cornices and frames.
Fig. 12 - Fiocchi su di una parallela ad una curva libera.
Fig. 13 - Machinic parallel, final relief, initial relief.
As one can see, applied Euclidean geometry is not that simple. On examining the intricacies of the basic concept of the parallel, one understands that it is far more complex than the false intuition by which we assimilate this operation to usual transformations such as translation or scaling. There still are very basic operations which the best modelers available on the market, like Parasolid, cannot solve directly. Let us mention for instance, the generation of swept shape along a line the radius of curvature of which becomes locally inferior to half the width of the section. As an example of this type of problem, we would very much like to record the exact geometry of the handrail that Desargues drew for the entrance stairs of the Château de Vizille built in Dauphiné in 1653, and then check to what extent it can be drawn with current CAD software. What we are hinting at is that architects can turn the complexity of Euclidean geometry into richness. The examination of a simple operation like the parallel of the square already shows a variety of unexpected cases, but if we think about these difficulties as architects, we can certainly take advantage of them in order to investigate new figures within Euclidean space, figures which we will be able to manufacture and build. At the moment when Euclidean geometry is supported by computers, we can start thinking of the general concept of variable curvature. Circles can be modulated. Parallels do not need to be drawn at a constant distance anymore. Old methods like Guarini's "maniera di condurre una linea ondeggiante" (12) can be pushed further than was previously possible.
Fig. 14.
Figure 14 shows a sketch that Guarini presented in his "Architettura civile". Arcs of circle are drawn from the center located at the intersection of the oblique lines and passing by the intersection of these oblique lines with a variable horizontal line. Guarini explicitly presents his sketch as a general device, which enables to generate a variety of undulating lines. As such it can be drawn on parametric software. The lines thus generated result more or less concave and convex according to the position of the horizontal line in relation to the middle axis of the rectangle. And if, thanks to the parametric software, you push the horizontal line beyond the limits of the rectangle, you then get the strange looped lines we have already encountered. It is no wonder since Guarini's sketch is a method to generate undulating lines by creating parallel at a variable distance. So now think of undulations which don't need to be created out of planar arcs of circles! Think of parallels at a distance which is always variable! Many unexpected figures will then enable us to incarnate complex topologies in Euclidean space. We have only caught a whiff, we haven't really tasted it yet!
Bernard Cache, December 21st 1998
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Notes
(9) Let us also mention that parallelism was the name under which Leibniz and Spinoza discussed the question of the relation between body and soul.
(10) Interestingly enough, we find those three solutions at many architectural scale, starting by that of the urban block built around a square courtyard. The peripheral contour of the block can remain square or become chamfered (Plan Cerda) or rounded.
(11) Henry Moore writings are collected by Philip James in Henry Moore on Sculpture, New York, Da Capo Press, 1992.
(12) See Guarino Guarini: Architettura civile, Lastra III, Trattato III, Capo secondo: Del modo di piegare varie linee curve necessarie all'ortogarfia: maniera di condurre una linea ondeggiante. How to fold various curved lines necessary for orthography: way of drawing an undulating line.
["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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