### MIT - FOUR-DIMENSIONAL STRUCTURES

#### ON MY TIME IN THE MIT

To continue exploring new technologies that could be used in the sphere of art, and to progress in the study of multidimensional geometry, I moved to the Center for Advanced Visual Studies in the MIT in October of 1975. In this centre I came into contact with noteworthy artists such as Giorgy Kepes, Otto Piene, Jürgen Claus, Mark Mendel, Otto Frei, Dale Eldred, Peter Campus, Walter de María, Charlotte Moorman, Nam June Paik, etc., with whom I came to co-operate in activities like the Sky Art Conference.

I soon met the lecturer David Brisson of Braun University, a painter and sculptor with a considerable mathematical and specialist theoretical education in polytopes with whom I established a good friendship and fervent admiration. His teachings were for me a consistent conceptual base on which I have been able to support later projects related to the development of my works.

Art taken also to mean a means of knowledge, as well as its symbolic and expressive capacity, is one aspect that is fostered in centres such as the one we are dealing with, where an investigative tendency is stimulated, through symmetrical art close to science.

Thus, certain concepts like multidimensional geometries appeared to me then through different, complementary points of view, some of which I must mention. There were, for example, those of mathematicians and geometry specialists like H. S. Coxeter, whose research is a reference in the study of polytopes. Also the lecturer Tom Banchoff, specialist in surfaces and multidimensional spaces, who in the sixties worked on visualizing the complexity of these problems with the aid of computers and films.

Another similar vision, but with the twist of lifting these concepts to the level of expressiveness, is the work of the aforementioned artist David Brisson. Some artists have approached these ideas with varying degrees of luck, like the cubists who debated these questions, and before Malevitch among others influenced by theosophists and thinkers like Ouspensky. Or Dalí in his painting “Crucifixion” (Corpus Hipercubicus). Authors such as Edwin A. Abbott in his intelligent novel “Flatland: A Romance of Many Dimensions”, written in 1884 and since then a classic.

The Science of multidimensional Geometry, together with the studies carried out by G. Cantor & Hausdorff on the Theory of Sets, represents one of the most modern aspects of this discipline in the history of the world of science. It was the German mathematician Grassmann (1809 – 1877) who systematically presented it for the first time in his work “Linealen Ausdehnungslehre”. He worked with Helmholz, assimilating the work of Riemann naturally, above all from 1854, as a support for his theories. The Englishman Cayley and Frenchman Cauchy worked on various problems related with multidimensional topics at the same time as Grassman but did not come to propose them as a system. The Swiss teacher L. Schäfli (1814 – 1895) wrote his “Theorie der vielfachen Kontinuität” between 1850 – 1852. But as with Grassman, the incomprehension of his contemporaries meant that his work would not be published till after his death in 1901, thanks to the efforts of his disciple J.H. Graf. Later, the North American Henry Parker Manning of Brown University published his “Geometry of four Dimensions” in 1914. Since that decade of the last century, this mathematical discipline has been unanimously accepted. Also, according to Albert Einstein’s Theory of Relativity and its concept of the inseparable unity of space and time, it is well to consider the time factor as the fourth coordinate, the occurrence that implies its spatial definition “perpendicular” to the other three Cartesian coordinates that establish the position where this occurrence happens. The space obtained from four dimensions is called “Minkowsli’s Space”, for which he gave his simplified description of the basic forms of the Theory of Relativity and the formulae of Lorenz, expressed in terms of coordinates. However, Minkowski’s geometry of space-time is not Euclidean while our intuitive approximation is, which without doubt may help us to clarify this concept, as in the reading of the aforementioned “Flatland”. The geometry proposed by the aforementioned authors began a new fundamental basis in spatial imagery that should affect the world of representations in classical composite and proportional systems.

This new vision of the Universe necessarily contemplates, no longer possibly but certainly, an extra-dimensional range that goes beyond the known time dimension that can go back 13 billion years in the past or infinitely into the future. The concepts that come from this debate, its methodological tools and the context in which technological and scientific research moves, are very much being taken into account by a growing number of artists involved in the new techno-cultural germination in which we live. My intention during that brief but intense period was to study the possibilities of convergence between the various disciplines described above. On one hand I studied the networks that could allow spatial relations to be established beyond the third dimension to support a formal flow capable of exploring new spaces. On the other hand, I was interested in methodologies, systems and techniques that used natural energies like the sun, wind, tides, and all those procedures related to possible artistic intervention that are not aggressive to nature.

I continued experimenting in this way and generating projects with lasers as a tool to draw in the sky in certain conditions favourable for their visibility. All this came out in the form of flying structures that were technically very simple but of limitless conceptual possibilities.

JOSÉ MARÍA YTURRALDE