Variations on a torus
Why should we be content with just a torus in four dimensions (see the discussion of the Mathematicians torus)? After all, we can view a torus as just the Cartesian product of two circles: S x S, where S denotes a circle; we can play the same game with any two figures in the plane. For example, we could consider the Cartesian product of a circle and a square, a line and an oval, or a triangle and a hyperbola.
While the Cartesian product of two plane figures will be quantitatively different from the torus S x S, it will not necessarily be qualitatively different. Some figures can be continuously deformed into a circle, and so are not qualitatively different a circle. For instance, we can push out the edges of a triangle onto an enscribing circle (as in the diagram on the right); similarly, a square can also be deformed into a circle. In Mathematics, the adjective homeomorphic is used for objects that are qualitatively the same, such as the triangle, square, and circle. Thus the Cartesian product of a triangle and a square is homeomorphic to the torus S x S, since each factor is homeomorphic to a circle.
Observe, however, that a line segment can be continuously shrunken to a point (as in the diagram on the right). This operation of continously shrinking or expanding an object is called homotopy; and two objects that can be deformed in this manner to the same object are said to be homotopic. Thus we would say that a line segment is homotopic to a point; and consequently, the product of a line segment and a circle is homotopic to the Cartesian product of a point and a circle, which in turn is homeomorphic to a circle.
