University of Rome La Sapienza
Dipartimento di Matematica
Piazzale A.Moro, 00185 Rome ITALY
Serious does not mean, as one might thing, full of practical applications. On the contrary, Hardy states that:
Naturally, we students developed with this idea in our heads. But once we had become mathematicians and then teachers in our turn, we found ourselves facing a situation that was completely different from what we expected, at least in Italy. First of all after an increase in the number of university students during the 1960s and 1970s, including students of mathematics, the numbers evened out, and in particular the number of mathematics students began to decrease. This has meant that new teaching positions were obtained increasing often in faculties other than mathematics, a typical case being Architecture, where there has been an enormous and irrational increase in the number of students and teaching positions during the 1980s and 1990s. This phenomenon has continued, at least as far as the number of teaching positions goes, with the adoption of the 3 + 2 system, that is, three years for an initial or "brief" degree plus due years for a specialization. Further, during recent years the attitude of mathematicians with regard to the so-called "applied mathematics" has changed. Today we can confidently affirm that there are no more prejudices regarding the relationships between "pure" and "applied" mathematicians.
And yet, as recently as 2004 some mathematicians saw going to teach a mathematics course in a faculty of architecture as a kind of "punishment".
This attitude is brought on by at least two causes related to the little space that mathematics courses have in the architecture curriculum: there is an increasing tendency to reduce the number of classroom hours, and a parallel tendency to reduce drastically the arguments treated.
I believe that the ideal mathematics course in the architecture curriculum is, for the large majority of architects, a course in "recipes" -- to paraphrase Robert Musil in The Man without Qualities, on the opinion of engineers regarding mathematics -- which are to be applied without questioning why. The obvious corollary to this idea is that it would be better if the architects themselves taught these recipe courses without troubling the mathematicians to make some derivative or some integral. Although it is true that the mathematics courses serve as technical courses for architects, it is also true that the attitude of a great number of the students and professors of architecture is basically that in the end it is the engineers who have to deal with structures.
It is certainly very difficult to collaborate with other professors in non-mathematics courses for architecture given that the majority of these are ignorant (and prefer to be ignorant) of what could be done in a mathematics course. I recall the first year that I began to teach in architecture at La Sapienza, the University of Rome, in 1996. In presenting the courses the dean of the faculty praised the architect as creative and artistic, describing the courses as a kind of support for future architects for observing, gathering, feeling, almost sensing in the air the new tendencies in art and architecture. Architects as creators. How then can that arid discipline mathematics be of use?
By chance I taught for a year in 1992 at the IUAV in Venice before transferring to the University of Rome. After having taught for several years I posed myself the question, partly out of boredom of having always taught the same things in the same way, of how things could be changed radically. I left the architecture faculty and entered that of industrial design, hoping to find more imagination. In any case, I believed that the best thing I could do was not the write yet another book on lessons and exercises in advanced calculus and analytical geometry (although obviously the great advantage of writing such books is that hundreds of students are "obliged" to buy them, to the great satisfaction of the authors) but instead to try to make comprehensible that mathematics has an enormous cultural value, that it can change our way of thinking and therefore the way that architects design in ways that they perhaps cannot even imagine.
The idea was born out of the project "Matematica ed arte" in 1976, and then in 1996 became the much more vast "Matematica e cultura" [Emmer 2002, 2003, 2004a etc]. Taking as a point of departure the ideas expressed in that dean's presentation of the courses, my ambition was to make it understood that among the many things to remember, observe, and understand there had to be mathematics as well. Not only because mathematics "is the essence of spirit", but because mathematics can be an inexhaustible font of ideas and suggestions, not only of "recipes". Besides, it can be an extraordinary "school of adaptation" for problems that have not yet been encountered.
I did not want, however, to look at the questions "in abstract" (abstraction is one of the great defects attributed to mathematicians by those who do not understand that this is instead one of their great merits) [Osserman 1995]. Therefore I wanted to start with a concrete example in which the relationship between mathematics and culture had profoundly changed our way to looking at the world around us, and therefore also the architect's way of thinking and acting. The theme was that of space, of the mutation of our idea of space, using as the perfect guide the extraordinary book Flatland by E. Abbot, a book published in 1884 but timely today if only one will penetrate the surface and not consider it merely as a somewhat trite criticism of Victorian society. Having made an animated film by the same title [Emmer 1994], I had had the experience of "designing" the space that the book talks about, as well as the characters, the city, and the universe described by the hero, the Square. This is the reason by the book that I wrote on this theme is entitled Mathland [Emmer 2004b].
In the present paper I would like the refer to some of the arguments that this book discusses that I believe are of interest to both student and practicing architects. This is a brief reading of man's adventure of thought in the reign of relationships between mathematics and culture. The example that I have chosen is that of the idea of space, how this idea and the perception of the space around us has changed up to the point where it has arrived to the form of virtual architecture.
Fig. 1. Anamorphosis Architects, Athens, Greece, "Project for the Museum of the Hellenic Wolrd"(2002) © Anamorphosis architects
There was a great emphasis in this project on the spatiality of the construction, a large continuous space in transformation with curved lines that wrap into a spiral, with at its center the exhibition of the Classical period of Greek civilization. This building was in some sense the beginning and the (temporary) end of a dialogue that began with Euclidean geometry thousands of years ago, a geometry that was the basis, together with Greek philosophy, of the formation of Western civilization as we know it today. It shouldn't be forgotten that the influence of many other civilizations, first of all the Islamic, permitted Europe to rediscover the forgotten Greek civilization.
There are some questions to investigate in order to understand at least in part how philosophical, artistic, scientific elements -- in a word, culture -- contributed over the course of centuries to the synthesis of a project such as that for the museum of Greek civilization. It is a sort of voyage into Western civilization of the past 2000 years and more, with an emphasis on cultural aspects related to geometry, mathematics, and architecture.
SPACE IS MATHEMATICS
These are the words of Galileo Galilei, written in Il Saggiatore, published in Rome in 1623. Without mathematical structures it is not possible to comprehend nature. Mathematics is the language of nature.
Let us jump forward several centuries. In 1904 a famous painter wrote to Emile Bernard,
Art historian Lionello Venturi commented that he didn't see any cylinders, spheres and cones in the work of Cézanne (as this the painter we are talking about), and therefore his sentences expressed the ideal aspirations of an organization of forms that transcended nature, nothing more.
In the same years in which Cézanne painted, or rather some years earlier, the panorama of geometry had changed since Galileo's time. In the course of the second half of the nineteenth century geometry had changed profoundly. Between 1830 and 1850 Lobachevskij and Bolyai constructed the first examples of non-Euclidean geometry, in which Euclid's famous fifth postulate on parallel lines was no longer valid. Not without doubts and opposition, Lobacevskij would call his geometry (which today is called non-Euclidean hyperbolic geometry) "imaginary geometry", in as much as it opposed the common meaning of the term. Non-Euclidean geometry would still remain for some years marginal with respect to the rest of geometry, a sort of curiosity, until it was incorporated into mathematics as an integral part by means of the general concepts of G.F.B. Riemann (1826-1866). In 1854 Riemann gave his famous lecture to the University of Gottingen entitled Über die Hypothesen welche der Geometrie zur Grunde liegen ("On the Hypotheses which Lie at the Foundations of Geometry"), which was not published until 1867. In his presentation Riemann set forth a global vision of geometry as a study of the variety of any number of dimensions in any kind of space. According to Riemann's ideas, geometry did not even necessarily deal with points or space in the ordinary sense, but with groups of n ordinates. In 1872 Felix Klein (1849-1925) became professor at Erlangen and gave his inaugural address, known as the Erlangen Program, in which he described geometry as the study of the properties of figures having characters that were invariant with respect to particular groups of transformations. As a consequence every classification of groups of transformation became a codification of various geometries. For example, Euclidean plane geometry is the study of properties of the figures that remain invariant with respect to groups of rigid transformations of the plane made up of translations and rotations. Jules Henri Poincaré affirmed that:
Also due to Poincaré is the official birth of that sector of mathematics that today is called topology, with his book Analysis Sitûs, a Latin translation of a Greek name published in 1895: "As far as I am concerned, all the various research in which I have been involved have led me to Analysis Sitûs (literally, Analysis of position)." Poincaré defined topology as the science by which we know the qualitative properties of geometric figures not only in ordinary space but in space with more than three dimensions as well.
If to all this we add the geometry of complex systems, fractal geometry, chaos theory and all the "mathematical" images discovered (or invented) by mathematicians in the last thirty years using computer graphics, it is easy to understand how mathematics has contributed in an essential way to changing over and over our idea of space, both the space in which we live as well as the idea of space itself.
That mathematics is not merely a "kitchen recipe", but has contributed, when it has not actually determined, the way we have of conceiving space on earth and in the universe. What is lacking is the awareness of mathematics as an essential cultural instrument. This explains the great delay in comprehension and therefore in coming up with new ideas that mathematicians have experienced for decades.
This is so in particular with regards to topology, the science of transformation and invariance. One example is the design of Frank O. Gehry for the new Guggenheim Museum in New York, a design that is even more stimulating, even more topological than the Guggenheim in Bilbao (Fig. 2).
Fig. 2. Frank O. Ghery, "Project for the New Guggenheim Museum in Manhattan", courtesy of © Keith Mendenhall for the Gehry Partners Studio
Certainly, the cultural leap is noteworthy; the construct using techniques and materials that permit the realization of an almost constant transformation, a sort of contradiction between the finished construction and its deformation. It is an interesting sign that one begins studying contemporary architecture using even such instruments as mathematics and science make available, instruments that are cultural as well as technical.
It is worth underlining how the discovery (or invention) of non-Euclidean geometries and of higher dimensions, beginning with the fourth, is one of the most interesting in terms of the profound repercussions that many ideas of mathematicians will have on humanistic culture and art. Every good voyage requires an itinerary, one with the elements that will be utilized in order to give a sense of Space.
The first element is without a doubt the space that Euclid delineated, with the definitions, axioms, and properties of objects that must find a place in this space, a space that is perfection, Platonic space. Man as the genesis and measure of the universe is an idea that has come down through the centuries. Mathematics and geometry must explain everything, even the form of the human being. The Curves of Life was the title of the famous book of 1900 of Cook, who certainly never imagined how true it could be that all forms are found in nature, even those that give rise to life, some mathematic curves. From D'Arcy Thompson's book On Growth and Form of 1914 to catastrophy theory of René Thom, to complexity and the Lorentz effect and dynamic non-linear systems.
The second element was freedom: mathematics and geometry seem to be an arid reign. One who has never dealt with mathematics, has never studied mathematics in school with interest, cannot begin to comprehend the deep emotion that mathematics can stimulate. Nor can he conceive that mathematics is an activity that is highly creative, nor that it is the domain of liberty where it is not only possible to invent (or discover) new objects, new theories, new fields of research activity, but that it is even possible to invent problems. And since mathematics does not always require huge economic resources, it can rightly be said that it is the reign of freedom and fantasy. And certainly of rigor. Of correct reasoning.
The third element to reflect on is how all these ideas are transmitted and assimilated, perhaps not completely understood and only vaguely listened to by various sectors of society. Architect Alicia Imperiale has written in a chapter entitled "Digital Technologies and New Surfaces" in the book New Bidimensionalities , "Architects freely appropriate specific methodologies from other disciplines. This can be attributed to the fact that ample cultural changes are being verified more quickly in other contexts than in architecture." She adds,
Further, Imperiale says,
Does the computer resolve all problems?
The fourth element is the computer, the graphic computer, the logical and geometrical machine par excellence. The idea realized by the intelligent machine that is capable of facing problems that are very different if only it is possible to make them comprehend the language being used. This was the general idea of one mathematician, Alan Turing [see Hodges 1991], which was carried to term under the stimulation of a war. A machine constructed by man, in which has been inserted a logic, that as well constructed by man, conceived by man. A very sophisticated machine, irreplaceable, not only for architecture. In short, an instrument.
The fifth element is progress, the word progress. If we consider non-Euclidean geometry, new dimensions, topology, the explosion of geometry and of mathematics in the twentieth century, can we speak of progress? Of knowledge, without a doubt, but not in the sense that new results cancel old ones. Mathematicians used to say that "Mathematics is like pork, nothing should be thrown away, and sooner or later even the things that appear to be most abstract and even senseless will become useful". Alicia Imperiale writes that topology is effectively an integral part of the system of Euclidean geometry. What escaped the author was what concerns what the word space means in geometry. That is, words. Where instead changing geometry serves to confront problems that are different because the structure of space is different. Space is in its properties, not the objects contained therein. Words.
The sixth element are words. One of the great capacities of
humanity is to give a name to things. Many times in "naming"
words are used that are already in current use. This habit sometimes
creates problems because one gets the impression when hearing
these words of having understood or at least have listened to
the things being spoken of. In mathematics this has happened
often in recent years with words such as fractals, catastrophe,
complexity, and hyperspace. Symbolic words, metaphors. Even topology,
dimensionality and sequentiality are by now part of the everyday
vocabulary, or at least that of architects.
FROM TOPOLOGY TO VIRTUAL ARCHITECTURE
These are words of Courant and Robbins in the famous book What is Mathematics?
Poincaré defined topology as "the science that permits us to know the qualitative properties of geometric figures not only in ordinary space but in space of more than three dimensions." Topology therefore has as its object the study of geometric figures that when subjected to profound transformations so that they lose all of their metric and projective properties, as for example form and dimension, nevertheless remain invariant, that is, geometric figures that maintain their qualitative properties. Figures constructed at will of deformable materials come to mind, which cannot be lacerated or welded; there are properties that are conserved even when a figure like this is deformed in any way possible.
In 1858 the German mathematician and astronomer August Ferdinand Möbius (1790-1868) described for the first time in a paper presented to the Parisian Academy of Science a new surface in three-dimensional space, which is today known by the name "Möbius strip". This new surface has interesting properties. One consists in the fact that if one follows its longest axis with a finger, one eventually returns to the point of departure without ever crossing over the edge of the strip; the Möbius strip has only one side, not two, one external and the other internal as is the case, for example, of a cylinder. While in the case of the cylindrical surface, one can follow with one's finger the upper edge of the cylinder and never arrive at the border of the lower edge, in the case of the Möbius strip, one can follow the whole thing and return to the point of departure, that is, it has only one edge. All of this has important consequences from a topological point of view; among other things, the Möbius strip is the first example of a surface on which it is not possible to fix an orientation, that is, a direction of travel.
Courant and Robbins wrote further:
The key phrase is "geometric intuition". Obviously mathematicians over the years have seen that topology has been brought into the context of more rigorous mathematics, but the aspect of intuition has remained. It is indeed these two aspects, that of deformation that yet preserves the properties of geometric figures, and that of intuition, which play a central role in the idea of space and form that, beginning in the nineteenth century, has come down to us today. Some of the ideas of topology would be intuited through the decades, first by the artists, and then much later by the architects. It is worthwhile to tell the story of the discovery of a topological form by one of the great artists of the twentieth century, a form that, though discovered by the artist, already existed in the world of mathematical ideas. The artist is Max Bill, a great artist and architect, who passed away in 1994 (Fig. 3).
Fig. 3. Max Bill in his Zürich studio (1981) . From the film "The Moebius Band", ©M. Emmer
This is how Bill, in an article entitled "How I began to make surfaces with single sides" tells the story of how he discovered the Möbius strip (Bill call his sculptures in the form of Möbius strips "endless ribbons"):
The interesting thing to note is that Bill believed that he
had discovered a completely new form. Still more intriguing is
that he discovered (invented?) it while playing with a strip
of paper, in the same way that Möbius had discovered it
many years before!
This is what Alicia Imperiale writes in the chapter entitled "Topological Surfaces":
This is the role of topology, as seen by an architect:
Naturally some of the words and ideas are deformed as well as they pass from a strictly scientific context into one that is artistic and architectonic, when seen with a different viewpoint. But this is not actually a problem, not is it meant to be a criticism. There are ideas that circulate freely and everyone interprets them in his own way, trying to gather, as topology, the essence. In all this the role of computer graphics is essential, as this permits the insertion of that variable of deformation-time that would be incapable of being conceived let alone constructed.
With regards to Möbius, Imperiale continues:
Fig. 4. Möbius House by © Ben van Berkel (UN Studio/van Berkel & Bos), 1993-1997
The Klein bottle, another famous topological form, according to van Berkel, "can be translated into a system of canalization that incorporates all the elements it meets and makes them fall into a new kind of internally connected integral organization"; of particular note are the terms "integral" and "internally connected", which have precise meanings in mathematics. But this is not a problem because "the diagrams of these topological surfaces are not used architecturally in a way that is rigorously mathematical, but constitute abstract diagrams, three-dimensional models that permit architects to incorporate into architecture differentiated ideas of space and time."
Max Bill had written something analogous in 1949 regarding the links between art, form and mathematics:
In modern art as well artists have made use of regulating methods based on calculation, since these elements, along with those of a more personal and emotional nature, have formed equilibrium and harmony to every plastic work. Such methods, however, became increasing superficial, according to Bill, since, aside from the exception of the theory of perspective, the repertoire of methods used by artists had stopped growing by the time of ancient Egypt. The new fact arrived with the beginning of the twentieth century:
It was then Mondrian who more than anyone else distanced himself from the traditional concept of art. Mondrian wrote,
It is the opinion of Bill that Mondrian exhausted the remaining possibilities of painting: "I am convinced it is possible to evolve a new form of art in which the artist's wok could be founded to quite a substantial degree on a mathematical line of approach to its content" [Bill 1993: 5].
Further, these mathematical representations, these restricted cases in which mathematics is plastically manifest undoubtedly have an aesthetic effect, adds Bill. And here is the definition of what a mathematical concept of art has to be:
In order to be convincing, Bill must provide examples that are interesting from his artistic point of view, that is, examples that recall the mystery of mathematical problematics such as the "ineffability of space, the moving away from or coming closer to the infinite, the surprise of a space that begins in one part and ends in another, that is at the same time the same, the delimitation without exact limits, the parallels that intersect, and the infinite that returns to itself". In other words, the Möbius strip.
As we have said, architects as well, if with some delay, also became aware of the new scientific discoveries in the field of topology, and more than design and construct, they began to reflect. In a 1999 doctoral thesis, Giuseppa Di Cristina writes:
In "The Topological Tendency in Architecture", the Preface to a volume on the theme of architecture and science, Giuseppa Di Cristina explains,
This is what Stephen Perrella, of the most interesting "virtual" architects today, has to day about Architectural Topology:
In these observations ideas on geometry flow together with those of topology, computer graphics, space-time. The cultural nexus in the course of the years has functioned: new words, new meanings, new relationships.
Without all this, the design of a museum of the Hellenic world would be inconceivable. A culture that arose in that place thousands of years ago is celebrated in that same place with a highly symbolic building of the story of the culture of the Mediterranean.
It would be nonsense to fail to relate this fundamental aspect of the link between mathematics, culture and architecture to students of architecture, to those future architects who will be responsible for the space in which the generations of tomorrow will live.
Translated by Kim Williams
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