Topology is a branch of mathematics dealing with the properties of space - whether it is contiguous, contains holes, and so on.
Cellular automata can be characterized in entirely topological terms, thus they can be studied within the scope of topology; see Cellular automaton § Generalizations and topological characterization.
A surface – e.g. the grid used to play the Game of Life – may be infinite, like a plane or a cylinder, or finite, like a sphere or a torus. A finite surface has to be curved to form a cylinder, (infinite in only one direction), torus or Klein bottle. The image shows how to "roll" and connect the sides in each case:
- Infinite plane: default.
- Cylinder: "rolling" the plane and connecting the opposite sides marked "1".
- Torus: "rolling" the cylinder and connecting the opposite circles marked "2".
- Klein bottle: "rolling" the cylinder, "twisting" it in the fourth dimension and connecting the opposite circles marked "2" and "S"; note that the "S" becomes a "2" after twisting.
- Cross-surface: like the Klein bottle, but "twisting" the opposite sides while creating the cylinder and then "twisting" the opposite circles when creating the cross-surface.
- Sphere: joining adjacent sides, rather than opposite sides as is done for the torus.
- Topology at Wikipedia