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The Torus

The torus is the orientable surface with Euler characteristic = 0. It is the genus = 1 surface formed by adding a cylindical handle to the sphere. There exist an embedding (no self-intersections) of the torus in R^3 and an immersion (self-intersections) which is not regularly homotopic to the embedding.

Description and pictures of the standard embedding of the torus.

Description and pictures of a figure-8 immersion of the torus.

The torus is easily seen to be a surface of revolution: a planar curve is revolved about an axis. In the previous examples, the axis of revolution was displaced from the planar curve. The next image of the torus is obtained as follows: an ellipse centered at the origin in the xy-plane is tilted 45 degrees. It is then revolved around the y-axis, which intersects it in two places.

Description and pictures of the elliptical torus of revolution.

Also interesting is Thomas Banchoff's work on The Flat Torus in the Three-Sphere.

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