Steiner's Surface

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Steiner's surface is a local embedding of the projective plane into 3-dimensional space that was useful for the construction of an eversion of the sphere, and is interesting in its own right. Ironically, it was discovered by Werner Boy when he was tasked to prove that such an immersion did not exist.

The surface faithfully embeds local neighborhoods of the projective plane, but it is not a global embedding, as there are self intersections. It is known that one cannot globally embed the projective plane into 3-dimensional space.

In the animation above, the projective plane is indicated in the bottom left, represented as a disk with diametrically opposite points identified with each other. The animation shows the image of a circle of points as the circle grows steadily toward the boundary of the disk. We can see that the map defines a local embedding because the circle of points don't trample on their immediate neighbors as they get mapped to the surface, and the mapped points are constantly moving across the surface as the radius increases.

When the points reach the outer rim, diametrically opposed points map to the same point of the surface, which shows that the map is well-defined on the projective plane.

Steiner's surface has a three-fold rotational symmetry and a single point where it intersects itself three times, which you can readily identify in the demonstration. It is also unorientable.

It appears as a halfway model in our discussion of sphere eversion for \(n=3\). That is, it appears in the sphere eversion precisely at the moment where the outside of the sphere becomes the inside and vice versa.

The equations for the parametrization of Steiner's surface that I used to generate the visualization above were discovered by Bryant and Kusner and can be found in the Wikipedia article

The surface is quite complex and can be better understood by viewing various cross sections of the model by adjusting the clipping parameters. In doing so we see that at the top there are three spherical caps that each merge into the other two and combine to form a single cup at the bottom.