01-19-2021, 08:36 PM
(This post was last modified: 01-19-2021, 09:38 PM by xenos1984.
Edit Reason: Wrong idea...
)
Quote:Are you literally thinking about games on a cube, like a representation of a physical object, or would that pattern be repeated to create a more complex game board that is more theoretical?
The original idea is indeed to have games on a cube (or some other non-Euclidean space). In fact, for the cylinder I even made myself a physical chess board - I already had a small travelers' chess board with magnetic pieces, so I just wrapped a checkered paper around a tin can. It actually worked.
Now looking at the infinitely repeated patterns, from a mathematical point of view the same board game in which, e.g., a pawn is on some particular site on the cube, can equivalently be represented by an infinite pattern of unwrapped cubes on the plane, where an infinite number of pawns is on the plane, with exactly one pawn on each site which corresponds to the original site on the cube, provided that all these pawn move simultaneously.
So in other words, the number of playable sites is in any case finite. The infinite pattern is just a visualization, which shows an infinite number of copies of each site.
Quote:If you can suggest additional graph operators that might be useful for such transformations, please suggest them.
I'll think about it! For the cube / rhombile, something like the quadhex board would probably be useful. It's a bit different, because the hexagon is cut differently into thirds. I'm still struggling with the directions, even with the "ordinary" quadhex board without wrapping, but for that I shall come up with its own thread, as it's an independent problem.
Quote:By the way, I tried the subdivided Rhombile board (as shown in the first URL you mention) and got the following: rhombile-2.png
Code:(board (subdivide (subdivide (dual (tiling T3636 3)))))
Not quite the same, and not a playable game board, but pretty :)
Awesome! I wouldn't call it unplayable. I could imagine some variant of Morris or Chinese checkers to be played on the vertices...
Quote:I haven't got around to implementing the projective boards yet -- other demands of the project have taken precedence -- but they are high on the list of features to add so should be done soon.
I had a look at the Projex board. I really like the idea, also because of the topology involved from which follows that exactly one player has a loop that wraps around the doubly connected space. Still I think that the board can be slightly improved, to remove the "irregular" spots where cells appear twice as neighbors (such as the a,f,j,o,s,x). I'll try to come up with a sketch of what I have in mind.
The idea I had does not actually work with hexagonal boards...