Geometry and Topology

This part of the exhibition covers basic concepts of dimension, topology and non-Euclidean spaces, and investigates three-dimensional manifolds. This is shown through educational videos.

# Dimension

The first video begins by explaining the concept of dimension. To illustrate this concept, we use three categories of objects that have their equivalence in all dimensions: balls, simplices, and cubes.

We present these concepts through an animation, which simultaneously shows a ball in each of the four dimensions, then a simplex in each of the four, and, finally, a cube in each of these dimensions.

We use this development progressively in order to illustrate the constructive principle, for example, from a zero dimensional cube to a three dimensional cube (see figure below).

# Manifolds

The second and third videos (see following figure), show simple examples of "manifolds" that are not Euclidean space. These correspond to the torus, the bi-torus, which is an example of hyperbolic space, and the ball (in dimension 1, 2 and 3).

In the case of the torus, the construction is quite simple. The 1-torus is a circle which is obtained by bonding the start and end points of a segment. The 2-torus, which is a ring, is obtained by gluing the opposite sides of a filled square. The 3-torus is obtained by bonding the opposed sides of a full cube. The latter construction is represented in the first part of the video, then we have corresponding constructs for the hyperbolic space (bi-toro) and the sphere.

# 3-Manifolds Inner View

The last video illustrates a 3-manifold inner view and shows its fundamental domain To this end, two manifolds examples are presented: a 3 dimensional torus and a cube formed by mirrors.