Edgematching tiles

Question

Consider a 3×3 grid. Now, look at the patterns which generate 1 to 7 dots around the edges, taking into account rotations and reflections. Turns out there are 49 patterns, as seen in the set below made with index cards and a hole punch.
By hand, I managed to lay out a 7×7 grid where all edges match.

Is it possible to do edge-matching on a cylinder or torus with this set?

Answer 1

A torus isn't possible because where 4 tile corners meet, you'll either have 0 or 4 dots. But the total number of corner dots isn't a multiple of 4.

A cylinder is possible.

Here's a solution where left/right wrapping is correct, but top/bottom wrapping has two errors:

... ... ... ... ... ... ... 
X.X X.X X.X X.X X.. ... ..X 
X.X XX. .X. ..X XX. .XX X.X 

X.X XX. .X. ..X XX. .XX X.X 
X.X X.X X.X X.X X.X X.. ..X 
X.X X.X X.X X.. .XX X.. .XX 

X.X X.X X.X X.. .XX X.. .XX 
X.X X.. ... ... ... ..X X.X 
XXX XXX X.X X.X X.. .X. ..X 

XXX XXX X.X X.X X.. .X. ..X 
X.X X.X X.. ..X X.. ... ..X 
XX. ... ..X X.X XX. .X. ..X 

XX. ... ..X X.X XX. .X. ..X 
X.X X.. ... ..X X.X X.X X.X 
XX. .XX X.. .X. ..X XX. ..X 

XX. .XX X.. .X. ..X XX. ..X 
X.. ..X X.X X.X X.. ..X X.X 
XXX X.. .X. .X. ..X X.X X.X 

XXX X.. .X. .X. ..X X.X X.X 
X.. ... ... ..X X.. ... ..X 
X.. ... ... ... ... ... ..X 

Answer 2

If you take Tom Sirgedas' cylindrical solution above and mirror-swap his upper right and lower right tiles, you get a "solution" that will tile a torus w/ a one-ninth tile defect:

                                        |
                                        |
                                        V
     -----------------------------------------------------------------------
     | .  .  . | .  .  . | .  .  . | .  .  . | .  .  . | .  .  . | .  .  X |
     | X [A] X | X     X | X     X | X     X | X     . | .     . | . [B] X |
     | X  .  X | X  X  . | .  X  . | .  .  X | X  X  . | .  X  X | X  .  X |
     -----------------------------------------------------------------------
     | X  .  X | X  X  . | .  X  . | .  .  X | X  X  . | .  X  X | X  .  X |
     | X     X | X     X | X     X | X     X | X     X | X     . | .     X |
     | X  .  X | X  .  X | X  .  X | X  .  . | .  X  X | X  .  . | .  X  X |
     -----------------------------------------------------------------------
     | X  .  X | X  .  X | X  .  X | X  .  . | .  X  X | X  .  . | .  X  X |
     | X     X | X     . | .     . | .     . | .     . | .     X | X     X |
     | X  X  X | X  X  X | X  .  X | X  .  X | X  .  . | .  X  . | .  .  X |
     -----------------------------------------------------------------------
     | X  X  X | X  X  X | X  .  X | X  .  X | X  .  . | .  X  . | .  .  X |
-->  | X     X | X     X | X     . | .     X | X     . | .     . | .     X |  -\
     | X  X  . | .  .  . | .  .  X | X  .  X | X  X  . | .  X  . | .  .  X |    \->
     -----------------------------------------------------------------------
     | X  X  . | .  .  . | .  .  X | X  .  X | X  X  . | .  X  . | .  .  X |
     | X     X | X     . | .     . | .     X | X     X | X     X | X     X |
     | X  X  . | .  X  X | X  .  . | .  X  . | .  .  X | X  X  . | .  .  X |
     -----------------------------------------------------------------------
     | X  X  . | .  X  X | X  .  . | .  X  . | .  .  X | X  X  . | .  .  X |
     | X     . | .     X | X     X | X     X | X     . | .     X | X     X |
     | X  X  X | X  .  . | .  X  . | .  X  . | .  .  X | X  .  X | X  .  X |
     -----------------------------------------------------------------------
     | X  X  X | X  .  . | .  X  . | .  X  . | .  .  X | X  .  X | X  .  X |
     | X [C] . | .     . | .     . | .     X | X     . | .     . | . [D] X |
     | X  .  . | .  .  . | .  .  . | .  .  . | .  .  . | .  .  . | .  .  . |
     -----------------------------------------------------------------------
                                        |
                                        /
                                       /
                                       |
                                       V

The four "corners" of the above meet thusly about the defect:

                       :
                 +-----------+
           :     | X   X   X |
     +-----------+           |
     | X   .   X | X  [C]  . | ...
     |           |           |
 ... | .  [D]  X | X   .   . |
     |           +---+-------+---+
     | .   .   . |   | .   .   . |
     +-----------+---+           |
         | .   .   X | X  [A]  X | ...
         |           |           |
     ... | .  [B]  X | X   .   X |
         |           +-----------+
         | X   .   X |     :
         +-----------+
               :