"Endless" Chess Idea

     Everybody (even people who dont know how to play chess) knows, what chessboard consists from checked black and white squares.

People who knows the chess game, knows what a lot of different pieces are situated on this board, and they can move on it horisontally, vertically using diffirent rules. All this wonderful gamesplay ends on the borders of the board - it is impossible to move through the borders.
     But it is very easy to eliminate the borders of the board: you should "glue" them vertically - sides a and h:

This manipulation is very simple and effective. Pieces can move according to the same rules, no any conflicts in the rules appear, but you get a lot of additional possibilities. The piece, what dissapeared bihind one of the borders, appears from the other and so on.
     This cylinder board is possible to change to flat surface. In this case you have the following:

You add boards left and right of the original board (so verticals a and h touch each other) and all this configuration continues endless. In all these boards we have the same pieces position, and the same moves, but thouching edges of the boards are not taking in account so you can move from one vertical to another.
     As we could think, it is easy to finish this board transformation by "glueing" 1st and 8th horisontal and so change board to toroid:

In analogy with "one-time" transform this "two-time" glueing can be illustrated by this picture:

Unfortunately, this nice mathematical result is completely useless for real chessplay. If we use standard starting setup, each king is under 4-times check and mate just before the first move.
     If we wish to transform board in both directions, we should find some other way to do this. For example, we can "glue" each border - white and black with itself, so we can make something like globe with parallels and meridians. But here is one problem: we have tho "poles", and at these poles the square structure of board changes. so we cannot move at the poles using the usual rules. So it seems, no way exists to make the appropriate board, but ...

We have the same structure of cylinder board horisontally, but vertically each border thouchs himself but turning from left to right. And magically all the problems disappear! White and black squares are in order, any piece can move as usually, starting position works fine. But the board is now endless, and we can have new endless possibilities in this old game.