fourtorus

A fourtorus is a mathematical object. It is the four-dimensional analog of a torus. A familiar torus is what you get when you take a circle and revolve it around an axis that lies in the same plane as the circle but does not intersect it. This operation generates a three-dimensional shape, like a donut. A fourtorus is formed by a similar process, but in four dimensions. Imagine taking a three-dimensional torus and revolving it around a three-dimensional hyperplane that does not intersect it. The resulting shape is a fourtorus.

Alternatively, a fourtorus can be constructed as the Cartesian product of four circles. If we represent a circle as S¹, then a fourtorus is S¹ × S¹ × S¹ × S¹. This product construction is a common way to define higher-dimensional tori. A one-torus is just a circle (S¹), a two-torus is a surface like the skin of a donut (S¹ × S¹), and a three-torus is formed by revolving a two-torus around an axis (S¹ × S¹ × S¹). Following this pattern, the fourtorus is S¹ × S¹ × S¹ × S¹.

Visually, it is impossible to fully picture a fourtorus in our three-dimensional experience. However, its properties can be studied mathematically. Like lower-dimensional tori, the fourtorus has a complex topology. It is a compact manifold. Understanding fourtori is part of the broader field of topology, which studies the properties of spaces that are preserved under continuous deformations.