Non euclidian 4D spaces

[Frisk-256]

I was thinking about the insane geometry that could exist in non-euclidean 4D, so there would obviously be H4 and S4, 4d hyperbolic and 4d spherical. 4D spherical could be tiled by 4d polytopes to make 5d polytopes, and I heard that there are some 120-Cell and 600-Cell(think I got the right shapes) tilings of 4D hyperbolic space. Then, there would be the product spaces(is that what its called?). You could multiply 2d spaces into 4d spaces. there would of course be H2*E2, S2*E2, but then a weird one, H2*S2, 2 dimensions of spherical with 2 dimensions of hyperbolic, that would be weird. There may also be H2*H2 and S2*S2, but I don't know if those would be different than H4 and S4. Then, there would be H3*E and S3*E. Also, I probably got a lot of terminology wrong in this post, Im interested in the topic but don't know much terminology. Anyway, what sort of tilings would exist in these spaces, and is H2*H2 and S2*S2 different from H4 and S4?

[Klitzing]

Take any spherical geometry polyhedron, any euclidean tiling, any hyperbolical tiling.
Then replace all the faces of those by according prisms.
This provide you with a finite (spherical geometry) or infinitely extending slab of height 1.
Next stack multiple such slabs ontop of each other, pile them infinitely.
Thus you will get S2xE1, E2xE1, H2xE1 geometric objects.

--- rk

[Frisk-256]

so catesion products of lower dimensional tilings, Im wondering about more complicated ones.

[mr_e_man]

Note that H1 = E1. So the 4D space H2*H2 contains a 2D subspace H1*H1 = E1*E1 = E2. On the other hand any 2D subspace of H4 is H2. And of course H2 ≠ E2. Therefore H2*H2 ≠ H4.

Similarly, the 4D space S2*S2 contains a 2D subspace S1*S1 (a torus), whereas any 2D subspace of S4 is S2 (a sphere).

(I mean "geodesic subspace", i.e. with no curvature relative to the higher space.)


There are non-Euclidean 3D spaces other than the obvious ones. The "solv" space is shown in this video by ZenoRogue, and other spaces are described here.