bsaucer wrote:How many ways can a surface "curve" or "bend" in 4D? In 3D a surface can be flat, rounded (like a bowl), saddle-shaped, or rolled up like a cylindar. I'm referring to the ways a surface can bend at one of its points, not the topology of the whole surface.
What kind of "surface" are you speaking of? In 4D, you have 2D surfaces and 3D manifolds.
2D surfaces have a LOT more freedom in 4D, and can twist into bizarre shapes such as Klein bottles, or the Real Projective Plane (
very pathological shape! I still can't quite visualize how it is connected). However, 2D surfaces do not bound a closed 4D region (they are as "thin" as strings, in 4D). Klein bottles cannot hold water, unfortunately.
3D manifolds are the 4D analogs of surfaces in our 3D world, and share many analogous properties. But because they have 3 dimensions, they also have more degrees of freedom than surfaces in 3D. Take a cylindrical 3-manifold, for example. You can either twist it around the 3-plane it lies in and join it end-to-end to make a 3D torus, or you can twist it perpendicular to the 3-plane, join it end-to-end, and make a different kind of torus: half the surface of a duocylinder. Or, you can fold it in 4 equal sections by 90-degree angles, to make the "flat" sides of a cubinder. Alternatively, you also have the round side of a cubinder, which you can get by rolling up a 3-manifold in one dimension. Rolling up a 3-manifold in
two dimensions makes a spherindrical surface. Then you also have the surface of a 3-sphere, which is a 3-manifold rounded in 3 dimensions. You can also make spheri-conical surfaces by cutting a 3D "sector" from a 3-ball and rolling it up so that its cut faces join.
The possibilities are fascinating.