TorusTopologien

TorusTopologien refers to the mathematical concept of a torus and its topological properties. A torus is a donut-shaped surface, topologically equivalent to a sphere with two holes. In mathematics, a torus is often represented as the product of two circles, $S^1 \times S^1$. This means that any point on the torus can be described by two independent angular coordinates. Topologically, a torus is a compact, orientable, and connected manifold. Its fundamental group is isomorphic to the direct product of two copies of the integers, $\mathbb{Z} \times \mathbb{Z}$, reflecting the two distinct ways to loop around the torus without returning to the starting point immediately. The genus of a torus is one, indicating it has one "hole." Torus topology has applications in various fields, including physics, computer graphics, and knot theory, where it serves as a fundamental object for studying more complex shapes and spaces. Different types of tori exist, such as the standard torus, but topologically they share the same fundamental characteristics.