The loop in Figure 14a appears to be a simple loop, encircling neither of the Klein bottle's axes. In fact it is not. The simple loop appearance is an artifact of representing a Klein bottle with a simple grid. This loop is a cylinder loop as can be seen when it's morphed to the shape in Figure 14f.
[Technical note: In Figure 14a, following the portion of the path near the top of the grid from left to right, it wraps around half of the cylinder - (4, 5, 6, 7, 8, 9, 10, 11, 12).
Continuing from right to left near the bottom of the grid, the path goes on to encircle the remaining half of the grid -
(12, 13, 14, 15, 16, 1, 2, 3, 4) - as read from the bottom edge scale. So the loop in 14a is seen to completely encirle the cylinder (vertical) axis, consistent with
The grid in Figure 15a represents a Klein bottle with a window cut from it. If you take the window and stretch it outward, you get something like the grid in Figure 15b,
which has two "portals". The perimeter cells of the portals are highlighted in green.
The portals are considered to be coincident. When a cell in the perimeter of one portal is claimed, the corresponding cell in the perimeter of the other portal is also claimed. See Figures 19b and 20b.
[Technical note: Figure 16 demonstrates the transformation between the two morphs represented in Figure 15. As you can see, the two windows in Figure 15b
are connected by an S shaped tunnel. If the tunnel is shortened to length zero, the two windows coincide.]
WIN CONDITIONS ON ALTERNATIVE GRID
In Figures 17a and 17b Red wins by forming a simple connecting path.
In Figures 18a and 18b Red wins by forming a cylinder connecting path.
In Figures 19a and 19b Red wins by forming a mobius connecting path and a mobius loop, two separate paths.
In Figures 20a and 20b Red wins by forming a double mobius connecting path.
PERIMETER CELLS (ALTERNATIVE GRID)
Cells in a perimeter segment can be used as part of a path, just as with the square grid representation.
Figure 21 shows a mobius loop which includes perimeter segment cells.
Copyright (c) March 2009 by Mark Steere