Examples of 3-dimensional Geometric Manifolds

In this talk, we consider certain combinatorial and geometric aspects of 3-dimensional manifolds. A 3-dimensional manifold is a space that locally looks like Euclidean space, but globally the object can be much more complicated, and it exists in a higher dimension which cannot be directly visualized. Here we investigate the geometries that can arise, and we begin by explaining the 2-dimensional case of hyperbolic surfaces. In the 3-dimensional case, we explore several classical examples using the method of face pairings for construction. Our approach is to analyze this construction combinatorically, and then visualize the manifold using a computer program. We go on to investigate a new family of 3-manifolds introduced in 2022. Our research provides an amazing world of thought behind how our universe could be designed. The universe takes on the shape of a 3-dimensional manifold and depending on its shape it could be possible to look into the night sky and see another Earth!