It was my suggestion that the notation for this game be changed so that the new file to the left of the a-file would be called the @-file, and the new file to the right of the h-file would be the i-file; the point being that now I can read the game score and follow it blindfold!
Treating the new files specially soon inspired a new idea of my own.
Think of a Polynesian canoe, with outriggers: the outriggers are different from the main body of the boat. Similarly, on the Outrigger board, the rules may be a bit different on the new files than they are on the "main board" (the 8x8 central area).
For example, suppose that the central 8x8 board is a Cylindrical Chess board. A Rook moving to the right from h1 goes to a1 (which is adjacent to h1 because the board wraps around) instead of being stopped by the edge of the board.
Using a Cylindrical Outrigger board, the Rook moving rightwards from h1 would have the choice of either wrapping around to a1, or going to i1 and being stopped by the edge of the board.
Digression: Of course, one could also posit a different type of Rook, one that would not have such a choice. If forced to wrap, it would be unable to attack an enemy King that had escaped to the outrigger area. This could be a charming feature of the game.
The Ke1 cannot move to the rear and capture the Ke8, so the game is possible to play. This is a great improvement over old-fashioned Toroidal Chess.
The Ke1 can move to e2 (after the Pawn leaves e2, of course); then it can get to e7 by moving South.
Similarly, when the Knight from g1 goes to f3, it has 12 moves available to it: from f3 to e1, using its choice not to wrap around, or from f3 to e7, moving in exactly the same direction but taking advantage of the opportunity to wrap around the edge.
Even stranger is the realization that there must be an "infinite" square in the same location as each "outrigger" square. You could picture tha outrigger squares as raised up n 3D from the infinite two-dimensional board, and if you picture their inner edges as tilted down to touch the normal board, you get a good idea of how the rules should work.
Note that the Bishops in Concentric Outrigger Four-Cushion Billiards:00 Chess (do I win the prize for the game with the longest name?) would be remarkably powerful in an open position.
Imagine a game withe the standard 8x8 board, and the FIDE-chess pieces, but in addition there are two extra files on the board; these files are empty at the start of the game.
These files are conveyer belts that move one square per ply; from White's viewpoint, the lefthand one moves forward, the right-hand one moves backwards. When a piece is moved onto one of these files, the conveyer belt automatically changes the position of the piece, one square after White's move, one square after Black's move. If the piece does not get off of the conveyer belt in time, it falls off the end and is gone for good.
You must make a complete legal move, you may not depend on the conveyer belt to get you out of check; and so a Bishop may have the opportunity to give check on the opposite-colored squares, but can never actually move to them!
How about a game with concentric outriggers, a special rule for the main board, and a different special rule for each outrigger?