Plea for Euclid*
PART(1)
Bernard Cache
www.objectile.net
So what are the consequences of this evolution of mathematics for architects? Do we have to banish the five Euclidean postulates as an outdated heritage from the Greeks, not unlike the Moderns dismissed the five orders of architecture? By the way, the Moderns already made allusion to other geometries but with very little tangible results. And nowadays, in our digital age, does the computer compels us to think and live in a multidimensional, non-Euclidean, topological space? Or shouldn't we instead consider the computer as a variable compass, which will open new potentialities within the old Euclidean space? These questions require that we investigate in close detail what happened to geometry, and their answer is all the more important in as much as geometry still remains the very basic tool of architecture.
Euclid wrote his Thirteen books of Geometry some 300 years before Vitruvius composed his own Ten Books of architecture at the turn of the first millenium. This certainly makes Euclidean geometry one the oldest works of science. And moreover Euclid didn't start from scratch. Many of the theorems he used where known much before his time, by the Egyptians in particular, who the Greeks always held in great respect. What Euclid did, what was profoundly original, was to systematize a corpus of what before him remained a collection of isolated theoretical demonstrations and practical solutions. In fact Euclid's work has two faces. On the one hand, it is a description of space both as a form of intuition and physical phenomena, on the other hand, it constitutes one of the first major work of logic. Hence the double evaluation required by the Elements of geometry, both axiomatic container and physical content. As such, the Elements are a first attempt to link together abstract logic and sensual experience, bearing witness to the multiple origins of geometry of which Michel Serres reminds us (1).
So let us start with the physical face of geometry. The dismissal of Euclidean geometry by architects sounds rather surprising when one notices how appreciated it is by contemporary scientists, even by those who cannot be suspected of orthodoxy, such as Roger Penrose. In his book: The emperor's new mind. Concerning computers, minds and the laws of physics Penrose argues that Euclidean geometry comes first in the list of the very few theories which deserve the label "superb" for their phenomenal accuracy. Einstein's theory certainly teaches us that space(-time) is actually "curved" (i.e not Euclidean) in the presence of a gravitational field, but generally, one perceives this curvature only in the case of bodies moving at speeds close to that of the light. Hence, the very limited impact of Einstein's theory on technology. Normally, "over a meter's range, deviation from Euclidean flatness are tiny indeed, errors in treating the geometry as Euclidean amounting to less than the diameter of an atom of hydrogen!". As those familiar with the difficulties created at a building site by the 1/10 millimeter accuracy of numerically controlled components surely know, Euclidean geometry is a more than sufficient approximation of architectural space. Certainly, our experience is that free curvature surfaces in architecture require rigorous accuracy, but we still have a way to go before trespassing the boundaries of Euclid's description. Funnily enough, the fact is that Lobachevsky himself undertook to submit geometry to experimental verification. He, who had plainly assumed the consequences of non-Euclidean geometry, was conscious that geometry was a matter of choice, where intuition had nothing to dictate to logic. Nevertheless, with the physics and experimental accuracy of his time, Lobachevsky concluded that Euclidean geometry was the best model of space. By which we see that our trivial intuition is rather well suited to what we experimentaly build up as a reality of space. As Einstein put it : "What is incomprehensible is that the world is comprehensible".
Now, when it comes to the other aspect of the Elements, following the more logico rather than the more geometrico, things become a bit more complex and require that we look at the organization of the work. Euclid started by three types of propositions: successively Definitions, Postulates and Common Notions, which form the basis of all of his theorems or propositions.
The 23 definitions introduce terms which range from: the "point" as that which has no part, to: the "parallel" as coplanar straight lines which never meet.
These definitions are followed by the famous five postulates:
1- To draw a straight line from any point to any point.
2- To produce a finite straight line continuously in a straight line.
3- To describe a circle with any center and distance.
4- That all right angles are equal to one another.
5- That, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which the angles are less than two right angles.
And then, Euclid introduces some common notions of arithmetic and logic like: "Things which are equal to the same thing are also equal to one another" or "The whole is greater than the part".
In fact, definitions, postulates and common notions will each create their own problems that we will analyse chronologically, as they have been confronted over the course of the history of mathematics, i.e. starting with the postulates.
What puzzled even the earliest commentators is the fact that Euclid himself seemed to shun his fifth postulate since he made no use of it in the first 28 propositions. The first four postulates were easily accepted as the propositions that enable one to construct figures and conduct demonstrations with a ruler and a compass (2).
On the other hand, the postponed use of the fifth postulate led commentators to think that it was not really necessary, that one could either get rid of it, or prove it to be a consequence of the first four postulates. Proclus (fifth century A.D.) already mentioned Ptolemy's attempt at emendation (second century A.D.), and proposed his own method. But for more than two thousand years, all attempts proved unsuccessful until an Italian Jesuit Priest, Girolamo Saccheri published in 1733 a little book with the title Euclides ab omni naevo vindicatus (Euclid Freed of Every Flaw). Saccheri used his earlier work in logic to undertake a reductio ab absurdum, trying to demonstrate that the negation of this unfortunate postulate would lead to contradiction. In so doing, not only did Saccheri not find any logical contradiction, but he actually demonstrated many theorems of what we now know as non-Euclidean geometry. However the geometrical consequences of what he established were so unexpected and so different from general intuition that he felt entitled to conclude that he had come upon propositions "at odds with the nature of the straight line". Saccheri was so anxious to vindicate Euclid that he took these unexpected geometrical results for the logical contradiction he was looking for.
By 1763, up to twenty-eight different attempts to solve the fifth postulate problem were listed and criticized by Klugel. The main result of these investigation had been to produce equivalent forms of the fifth postulate, the most promising of which being that of Playfair, a Scottish physicist and mathematician (1748-1819), who reformulated it as the Parallel Postulate: "through a given point P there is a unique line parallel to a given line". It is under this new form that several mathematicians, working independently at about the same time, will sort out the problem of the fifth postulate. In fact, the German Gauss had preceded the Russian Lobachevsky and the Hungarian Bolyai, each of whom had proved the independence of the Parallel Postulate in 1823, but Gauss didn't dare to publish his discovery fearing the criticism that could be brought on by such a counter-intuitive result. The three of them demonstrated that no logical contradiction would arise with the other four postulates, whatever the number of parallels to a given line are assumed to pass through a given point. Spatial intuition would just have to adapt to each case. The four postulates constitute what is called the "absolute geometry" after which geometry bifurcates. Once this absolute geometry is assumed, you have three options: you can stay within Euclidean geometry and assume that the number of parallels is only one; you can state that there are no parallels which lead to the " elliptic geometry " of Riemann; or, finally, you can postulate that there is more than one parallel, which opens the doors to Lobachevsky's "hyperbolic geometry".
Meanwhile, it is worth noticing that, in so doing, mathematicians actually consolidated the basis of Euclidean geometry. It was certainly no longer the only geometry, but they had proved that the fifth postulate was not only necessary but essential in making the logic of Euclid coincide with general intuition. As for the other geometries, the problem would be the opposite, i.e. : to find intuitive models which would suit their logic. To do so, one would have to jettison the usual signification of terms like: "point", "line" or "plane". For instance in Poincaré's model of hyperbolic geometry "lines" become arcs of circle, and in the spherical model of elliptic geometry "points" are pairs of diametrically opposed points. What becomes apparent is that the problem then moves from the postulates to the definitions. Far from providing an intuitive support to the logical demonstrations, the meaning of primary terms like points or lines is deduced from the system in which they are used. Primary terms are indefinable, just as postulates are undemonstrable. Hence the necessity of tracking every remaining spatial intuition in the Elements of Euclid.
This will be the great achievement of Hilbert in his Grundlagen der Geometrie, published in 1899. Geometry became axiomatic. Hilbert based his system on 21 axioms, which he organized into five groups, the number "five" establishing a continuity with Euclid's work. The first group, projective geometry is composed of 8 axioms which establish the relations between, rather than the definitions of, the concepts of points, lines and planes. For instance, one will find that the proposition: "two distinct points determine a unique line", can be converted in the converse relation "two distinct lines determine a unique point". This principle of Duality was developed by Poncelet who systematised the Projective Geometry of Desargues. The second group, topology, gathers the four axioms of "order" establishing the meaning of "between". If A, B and C belong to the same line and B is between A and C, then C is also said to be between C and A. The third group, congruence, gathers the 6 axioms needed to define geometrical equality. The fourth group holds one unique axiom, which is the famous parallel postulate. And finally, the last group is made up of the two axioms of continuity, including the one known as the axiom of Archimedes.
Combinations of these 21 independent and consistent axioms enable one to generate many geometries. Euclidean geometry is based on the totallity of the 21 axioms and it has been proved that this logical system is saturated, which means that you cannot add a 22nd axiom without creating contradictions in the system. From this perspective again, Euclid's work must be recognized as the most structured geometry. The other geometries can be generated either by the negation or the suppression of one or several of these 21 axioms. Thus, the negation of the uniqueness of the parallel postulated in the fourth group generates non-Euclidean geometries. But one can also investigate non-Archimedean geometry by negating the axiom of Archimedes in the fifth group. And, when it comes to the suppression of axioms, we get a more general but also less structured geometry. Thus Frederic Klein has shown that projective geometry is independent of the Parallel Postulate, which means that there can be both a projective geometry of Euclidean and non-Euclidean geometry. Topology, in its turn, is based on an even more restricted number of axioms. And this is easier to understand when we look at geometry from another angle.
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Notes
(1) Cf Michel Serres: Les origines de la géométrie, Paris, Flammarion, 1993.
(2) As concern architects and designers, it is very interesting to notice the logical and geometrical approach of ornament by Jan Hessel de Groot who imposed himself the restriction to generate figures and ornamental motifs only by 45° and 60° set-squares. In Dreihoeken bij Ontwerpen van Ornament (Ornamental Design by triangles), he exposes his aim as to: 1. demonstrate the simple way that forms take form, and their precise self-determination, 2. provide a tool that can be used to maintain unity in ornamental composition. Cf: Il progetto dell'ornamento. Jan Hessel de Groot, in "Casabella", n.647, luglio-agosto 1997, pp. 64-73.
["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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