Interactive Visualization of Manifolds
**Outer Surfaces View**

**3-Manifolds Inner View**

The second part of the exhibition is designet to allow an intuitive investigation of 2 and 3-manifolds through facilities for interactive visualization.

The first installation (figure below) shows the relationship between a surface geometry and an outer view perspective. It's a projection of three basic surfaces: the sphere, the 2-torus (planar) and the bi-torus with two holes (hyperbolic). The visitor chooses between one of them to carry out the interaction. In the installation, a particle is drawn simultaneously in the chosen surface from the observer's viewpoint. In an interactive table, the visitor can move a control. If the control position is shifted off center, the particle moves in this 'direction' on the surface. As a result, the program draws a geodesic curve segment. Thus, it's possible to draw curves on the chosen surface. After a time, the first part of the curve disappears, as if produced by a trail of a particle with low persistence.

The second installation consists of an open space with a control at its center (fugure below). Just as in the previous installation, the visitor starts by choosing on of the three manifolds (flat, spherical, hyperbolic). The interaction control is supported by an elastic string placed in the middle of the room. Just as in the previous installation, there are many light beams that begin and end in the same particle. Thus, visitors control the path of the particle and, as a consequence, see the same curve repeated in various directions and distances. The curves are drawn as a function of the particle. This installation also simulates the sound related to the geometry of the manifold being shown; for example, using the echo in the respective space.