A self intersecting sphere

(View this page in Romanian courtesy of Alexandra Seremina from azoft)

A bitangency of an immersion s of a closed surface M in R4 is a pair of points (p, q) such that s(p) does not equal s(q), and the line segment spanned by s(p) and s(q) lies in both the tangent plane of s at p and the tangent plane of s at q. For a surface in four space, we expect a finite number of bitangencies.

The pictures are of a projection of an immersion of the sphere in four space, given by the map:

(x, y, z) -> (x, y, x2 + xz, yz)

where x2 + y2 + z2 = 1. In these pictures, we are projecting down the first axis.

This immersion intersects itself exactly once, namely when (x, y, z) is (0, 0, 1) and (0, 0, -1). The colorization of the surface is proportional to the value of x (i.e., the surface's height in the first axis direction), so you can tell that there is a double point along the middle double curve (this curve is actually covered four times, as one can imply from the banded picture), This surface also has four bitangencies, two of which are represented by the black lines, two of which are double pinch points, occuring at both ends of the middle double curve.

One of the questions we can ask is the relationship between bitangencies of a surface in four space and the bitangencies of its projection down into three space. Any bitangency in four space must project to a bitangency in three space, and will be part of a two-dimensional set of bitangencies. In order to determine whether a pair of points (p, q) is a bitangency of the surface in four space, we have to project the surface down two vectors, such that the plane spanned by these two vectors is transverse to the tangent planes at p and q. If the two projected surfaces both have bitangencies bewteen p and q, then (p, q) is a bitangency of the surface up in four space. In addition, if we are lucky enough to project down the secant line of a bitangency, then one of the tangent directions at each point will collapse, and there will be pinch points on the projected surface at both p and q. Furthermore, these pinch points will occur at the same point in three space. Therefore, if our projected surface has two pinch points at the same place, then the corresponding pair of points is a bitangency. In this case, we only need one projection to find a bitangency. Our projected surface has two of these double pinch points.

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Last updated 7/19/99