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4D Visualization Part 2
Posted:
May 26, 1993 5:05 PM
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"I think the best way to understand four-dimensional
space is by analogy with three dimensions," says John
Sullivan of the Geometry Center. "This often gives a good
idea of what happens in 4D, but the problem is that
sometimes you miss things.
"For example, I was talking to my Differential Geometry
class about the fact that in 3D, there is only one
direction tangent to a line and one direction normal to a
plane. This means that by specifying a point on the
sphere, you can uniquely specify a line by its tangent or a
plane by its normal. In 4D, there is still one direction
tangent to a line and one direction normal to a three
dimensional hyperplane; however, a two dimensional
object in 4D has two normal directions. I think in some
ways this makes a 2D surface in 4D is harder to understand
than a 3D hypersurface.
"It is possible to describe a two-dimensional plane in 4D
by a wedge product of two vectors in the plane. This
product lies in a six-dimensional vector space, and by
normalizing, one has a vector in the five-sphere. By
analogy with 3D, since any vector on the two-sphere
specifies a plane in 3D, one would think that any vector in
this five-sphere would give a plane in 4D, but in fact this
is not true. Only a vectors in a subset of the five-sphere
called the Grassmanian actually specify planes in 4D.
"The reason not all vectors in the five-sphere specify
planes is related to another point missed by analogy; in
3D, all rotations fix a line. Even if you rotate around one
line and then rotate around another line, there is some
line in between which is fixed by the combined rotation.
However, in 4D, there are many rotations that do not fix
lines. Take, for example, a rotation in the x-y plane
followed by a rotation in the z-w plane. Under this
combined rotation, rather than tracing a circle, the
orbit of a point might be dense on a torus."
Aside from all these pathologies, Sullivan has gained
quite a bit of understanding by analogy. For educational
purposes and for fun, he has written a program which does a
stereographic projection of 4D regular solids into 3D,
where they become soap bubble clusters. This can best be
understood by analogously considering the
stereographic projection of, say, a cube or
dodecahedron on the two-sphere into the plane.
In his research on minimal surfaces, Sullivan has never
actually done work which specifically applied to four
dimensions. All the proofs work in arbitrary
dimensions. However, even in this case, he always draws
the picture in 3D. "I am just careful when I write the
proofs to say things which are true in all dimensions."
In contrast to Sullivan, parts of Ken Brakke's work on
minimal surfaces is specific to 4D. I will give details of
their research in another article.
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