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.


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).



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.