One of goals of the World Mathematical Year is to emphasize the role and the importance of this millenary science and its applications in the Contemporary World and Civilization. Beyond its role in Science, in Technology, in Economics and in many other fields, is not too much to accentuate its cultural importance in our conception of space-time and in its relations with the Art.

To communicate and to popularize Mathematics is one of the great and difficult challenges facing not only mathematicians but also mathematics teachers. If to show is to communicate, a mathematical exhibit, dealing with abstract elements, such as number, shape, probability or algorithm, which is often difficult or even impossible to realize physically, should stimulate and rouse the imagination, the discussion and the curiosity of young people and the general public.

Using geometrical intuitions and images or movement of lines, surfaces or other physical objects, it is possible to evoke mathematical notions such as length, area, volume, symmetry, space and... dimension.

This exhibit of high visual quality that is currently realized, advertising, recreating and expanding a virtual exhibit that exists in the cyberspace in http://www.math.brown.edu/~banchoff/art/PAC-9603/, allows the visitor to relate Mathematics, Art and Computer Graphics, showing the potential of new information technologies and of computational geometry for the communication of Mathematics.

The notion of dimension, that our intuition easily captures when we think of the sequence point-segment-square-cube, is a mathematical concept of deep meaning and of great consequence, not only in our conception of space-time, but also in many other fields of the human activity. For example, degrees of freedom or independent variables are notions that go far beyond the simple three dimensions (length, width and height) of the physical space in which we live.

The exhibit Beyond the Third Dimension takes the title of the book of Thomas F. Banchoff, a classic of the "Scientific American" collection, evoking geometrical objects, computer graphics and the concept of "Dimension". This exhibit is composed of twelve surfaces and small movies representing aspects and properties of virtual geometrical objects in three and four dimensions and also two new pieces on the hypercube.

The contents of this exhibit, essentially created in 1996 and rebuilt in virtual form in the following year, was designed by Thomas F. Banchoff, professor of Geometry at Brown University, USA, in collaboration with some of his students, mainly Davide P. Cervone, currently at Union College, in the state of New York, who added the pieces about the cube and the hypercube, as well as a short introduction to higher dimensions.

The exhibit currently shown is much more that the bilingual version of Surfaces Beyond the Third Dimension, that was shown in 1996 at the gallery of the "Providence Art Club". It is also composed by the panels and the physical support recreated with art by the "Atelier Henrique Cayatte". But, above all, the exhibit is constituted by the mathematical objects which can be visualized by computer graphics, the virtual elements that were recriated by Davide P. Cervone and completely processed by computer by "Arte Numérica".

The itinerant version of this exhibit, that the medieval town of Obidos and the modern city of Funchal show in a concerted response to the challenge that the Information Society presents in the dawn of the new millennium, is just a mirror and an advertisement of the authentic exhibit "Beyond the Third Dimension", that may be found in cyberspace and seen in the CD-ROM of this catalogue and in http://alem3d.obidos.org/.

The Centro de Matemática e Aplicações Fundamentais of the University of Lisbon takes part in the World Mathematical Year 2000, by promoting this initiative in the field of communication and popularization of mathematical sciences, in the framework of its project Matemática em Acção, in collaboration with the Town of Óbidos, the Department of Mathematics of the University of Madeira and the Sociedade Portuguesa de Matemática, and with the support of the Programa Ciência Viva of the Portuguese Ministry of Science and Technology.

**José Francisco Rodrigues**

Universidade de Lisboa