View in 3-dimensional Spaces is divided into distinct parts, including an introduction to the concepts of Geometry and Topology; two interactive facilities on 2- and 3-manifolds; theorems and quotes related to the subject; and biographies of mathematicians pioneers in this field. The nature of three-dimensional spaces is the theme of "View in 3-dimensional Spaces", an exhibition conducted by scientists and artists from Brazil and France.
The classification theorem of surfaces, which are two-dimensional spaces, is a classical result of Mathematics and was solved more than 100 years ago. The theorem states that closed and orientable surfaces are topologically equivalent to the ball or to the n-torus. Thus, there are three fundamental classes of space: spherical, flat (torus with genus equal to 1) and hyperbolic (torus with genus greater than 1).
In 2002, the Russian mathematician Grigori Perelman proved Thurston's Geometrization theorem on the classification of 3-dimensional spaces. As a result, Perelman solved the famous Poincaré Conjecture, one of the Millennium Problems that were open since 1904.
The exhibition project is part of a high-level scientific research work. In this context, we show for the first time in the history of Mathematics, visualizations of all eight fundamental geometries of Thurston.
In the exhibition, we reveal intuitively this revolutionary advance to the Mathematics field through images, animations and interactive installations, so that the public can glimpse the beauty of the ideas involved. Thus, each work displayed represents one of the concepts that are part of this theory.
Our presentation philosophy is to describe mathematical notions without formulas and with a minimum of text. We favor a sensory approach through audiovisual elements, showing purely geometrical aspects.