Chess has been thought as a 2 dimensional game. The main reason, I think, it is thought as a 2 dimensional game is because chess is played with two coordinates and each coordinate represents one plane. Because of this reason when an additional chessboard is added on top of chessboard, it is viewed as 3 dimensional game. The three dimensional game has three coordinates and there is a up lift or perhaps downward movement for chess pieces.

The most well know example is the 3-D chess played in the movie Star Trek, where players play chess on 3 layers of chessboard.

Is chess really a 2 dimensional game? The knight actually has to jump over chess pieces (especially in the opening) so the knight’s move is not a two-dimensional move like any other pieces. From this point, we know chess can not strictly be classified as a 2 dimensional game. It really is a 3 dimensional game if the knight has to jump over pieces in opening, at this point it reaches another undefined dimension before it lands on the regular chessboard.

On the other hand, if we strictly look at the chess game from algebraic notation’s point of view, then the knight’s “intermediate” move is not recorded. So even though it does reach an “undefined” dimension (nth dimension, where n is undefined), since only two coordinates are used, chess may still be called as a two dimensional game