## Sunday, June 21, 2020

### [ppfrcurf] Permitting hard sides on the fundamental polygon

Identify pairs of edges in a square: go out one edge and reappear through the other edge in the pair.  Orientation of the edges matters.

If all edges connect through to another edge, then there are 4 classic topologies:

• sphere
• torus
• real projective plane
• Klein bottle

If 2 edges are hard boundaries, then there are 3 additional possibilities:

• hemisphere
• cylinder
• moebius strip

If all 4 edges are impenetrable boundaries, then we have the plain old

• square

What if we permit mirrors, gluing an edge to itself?

Inspired by the various topologies examined for squared squares.  Sphere and hemisphere are missing on that site.  We could also consider fundamental polygons in which adjacent matching edges are connected the wrong way, e.g., AABB.  Small discs will behave funny at corners, but because the problem is posed on a grid, we don't need the good behavior of an actual manifold.

In https://mathoverflow.net/questions/172784/do-all-combinatorially-distinct-fundamental-polygons-correspond-to-surfaces , the answer by Gary Kibble is useful.