Poincarésfääriä

Poincarésfääriä refers to a topological space, also known as the 3-sphere. It is the three-dimensional boundary of a four-dimensional ball. Topologically, it is equivalent to the set of all points in four-dimensional Euclidean space that are a unit distance from the origin. This can be visualized by considering the surface of a regular sphere in three dimensions, which is the boundary of a ball in three dimensions, and then extending this concept to a higher dimension.

The Poincaré sphere is a compact, connected, and simply connected manifold. Simply connected means that any closed loop within the space can be continuously shrunk to a point. This property is a key distinction from the three-dimensional torus, for instance.

This object is named after the French mathematician Henri Poincaré, who extensively studied its properties and its relationship to other topological spaces. It plays a significant role in fields like topology and cosmology, particularly in theories exploring the potential shape of our universe. While challenging to visualize directly due to its higher dimensionality, its mathematical properties are well-understood and have been crucial in advancing topological theory.