Hey all.
Yesterday, I began to realize a whole other area of 4D geometry that proves to be very interesting.
First of all, I was thinking of how the duocylinder's bounding manifolds may be thought of as 3D cylinders bent around in 4D until their ends meet. This is rather suggestive of the approach in topology of gluing faces together; so I wondered what would happen if we took a cube, twisted it around in the ZW plane, and glued its top face to its bottom face. The resulting manifold would have 4 cylinder-shaped holes formed by the +/- X and +/- Y faces of the cube. Filling in these holes with 4 cylinders, we get the cubinder.
Then I was reading up on the n-torus, and was trying to imagine what a 3-torus might look like based on what I've learned of 4D visualization so far. Is it the same shape as the chorage traced out by a 3-ball if it was rotated around a plane outside of it? It seems, from the Wikipedia page, that the 3-torus is rather the trace of a 2-torus rotated around a plane. This would imply that there are at least 2 different types of torii in 4D: a spherical torus, and a toroidal torus! The former has only one "doughnut hole", but the latter, if my intuition is correct, has two such holes. So the two shapes would be topologically distinct.
Not only so; if you take the cubinder (minus the 4 cylindrical "lids") and glue another pair of opposite faces together, you get a cylindrical torus: a shape with 2 torus-shaped cavities which, if covered with two torus-shaped lids, would bound a 4D volume. Is this a third type of torus possible in 4D? (Of course, if you glue the last pair of opposite faces together, you get the 3-torus.)
What I'm curious about, is if any of these shapes are topologically equivalent to the duocylinder. I suspect not, unless I missed something obvious, because the duocylinder has two bounding surfaces that are circular in two disjoint planes.
Of course, things start to get interesting if you make Moebius-strip-like twists with these topological constructions. I'm still trying to picture how the real projective plane would embed in 4D. I'd love to get a good handle on it. (Unfortunately, I'm not sure if I'm ready to deal with 2-manifolds in 4D yet; so far, I've limited myself to 3-manifolds because at least they are more restricted in degrees of freedom and so are easier to comprehend. Nevertheless, I've come to the conclusion that the Klein bottle is a misnomer: a 4D liquid would not be confined in a Klein bottle; it would just spill off its "sides", just like water spills off a Moebius strip. The container must be a closed 3-manifold in order to hold the liquid in place.)