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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 **R**^{4} 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, x^{2} + xz, yz)

where x^{2} + y^{2} + z^{2} = 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