If you find a geometric pattern generalized for 2, and 3 dimensions, can you say it'll work for N dimensions?

Question

I think I'll start with an example. Let's say that in 2D, you define an infinite 2D domain between two parallel lines. In 3D, say you know this translates to defining an infinite 3D domain by an infinitely long right prism. Can you say that in 4D+ this translates to a prismatic polytope that is infinite in the normal direction of a hyperplane?

Answer 1

There are several examples in geometry where a pattern that persists in lower dimensions does not extend to higher ones. For example, take the following sequence of Schäfli symbols:

• $\{5\}$ is a pentagon
• $\{5,3\}$ is a dodecahedron
• $\{5,3,3\}$ is the 120-cell
• but $\{5,3,3,3\}$ is not a finite polytope. It is a tiling of 5D hyperbolic space.

For further examples, look up Borsuk's conjecture and the sausage conjecture.