>For instance, one page shows the 14
> different types of pentagon that will tile a plane.
The Cairo tiling is one of those 14. It has advantages over many
other non-regular tilings in that it is monohedral (all the same
face) and isohedral (every face has the same "relative position" to
the surrounding faces). However, you can't easily start at one Cairo
pentagon and move in a straight line to a row of other cells, as you
can on a square or hexagonal grid. Blobz (= Cairo-ized "Blobs"
or "Ataxx" or "Spot") is a good example of a game that allows this
kind of transplanting, because it doesn't involve pieces moving long
There are plenty of monohedral tilings, though as John Lawson says,
the only regular polygons that will tile the plane are equilateral
triangles, squares, and regular hexagons. I can't think of any games
on other monohedral tilings either, apart from original ones that
haven't been tested (or that haven't passed).
One idea that I haven't done anything with involves a non-convex
equilateral pentagon, angles 36 degrees, 108 degrees, 108 degrees, 36
degrees, and (a reflex angle) 252 degrees. This pentagon has the
property of being able to tile the plane in a great many ways -- you
can hardly put the tiles together in such as way as to make it
impossible to fit another one in. Also, they're mirror-symmetric, so
they could be flipped over without disturbing the array.
I would make a good number of these in stiff cardboard, black on one
side and white on the other like Othello pieces, and play them on a
mat with a large ellipse drawn on it, where they aren't allowed to
lap over the edge of the ellipse. The idea would be that the first
player places a tile with his color showing, then the players
alternately place tiles edge-adjacent to a tile already played, and
somehow a player "captures" pieces of the other color by flipping
them, as in Othello. That's the rule that I haven't figured out.
But if one could find such a rule that would lead to a good game, it
would have the interesting property of being completely abstract and
yet having an infinite number of possible opening moves...