We know that the only regular polygons that can tile the 2D plane are triangles, squares, and hexagons. One way of seeing this is that, if we try to place regular pentagons (for instance) around a point, we are forced to have overlaps (see picture):
However, if we ignored this and kept wrapping around, after placing 10 pentagons in a "spiral shape" around this corner, we would eventually return to our starting pentagon. If we "unflattened" the tiling, we would see a surface like this, with pentagonal chunks:
In 3D this surface would have to pass through itself, but in 4D we could do it without self-intersections. What is known/studied about the geometry of the resulting 2D surface, or Riemann surfaces that would be equivalent to this? What if we changed it to regular septagons, or regular polygons of any number of sides?