Mathematics plays a crucial role in uncovering the true forms of the universe through the exploration of multidimensional spaces. Geometric topology, a branch of science that studies various types of spaces in all dimensions, has wide-ranging practical applications. These applications range from creating secure communication networks to deploying satellites using origami principles, analyzing large data sets, and designing new materials.
Initially, our world appears to be a flat plane, allowing us to use city and terrain maps effectively. However, we actually inhabit the surface of a massive sphere with minor irregularities like mountains and valleys. This surface represents a two-dimensional space where movement occurs in two distinct directions: north-south or east-west.
While a spherical surface is one example of a two-dimensional space, another option is the surface of a torus, resembling a bagel. Both the sphere and torus have two independent directions of movement, but their topological properties differ significantly. Unlike on the sphere, certain closed paths on the torus cannot be reduced without tearing the surface.
Throughout the past century, mathematicians have painstakingly cataloged all possible two-dimensional spaces, classifying them based on topological properties, determining unique characteristics for each class, and establishing connections between them. This comprehensive catalog serves as a model for studying higher-dimensional spaces.
Exploring three-dimensional spaces has proven to be much more complex. While progress has been made in recent decades, scientists have yet to achieve a complete classification similar to the two-dimensional case. Nonetheless, this accumulated knowledge assists physicists and astronomers in unveiling the topological structure of the universe, determining whether it is closed like a sphere or infinite like a plane.
Introducing time as a fourth dimension adds another layer of complexity. In physics, an object’s position is described by four numbers: three spatial coordinates indicating its location and a fourth coordinate representing time. To fully characterize the movement of a comet, one must know both its spatial and temporal coordinates. Essentially, these four parameters combine to form a unified spacetime concept, a fundamental idea in modern physics stemming from Einstein’s theory of relativity.