"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.