This is the basic pattern. Each trapezoid is further divided in rectangular cells to create a board.
This board is obtained by taking the three-player chess and gluing left and right side to each other. Note that in the initial position, the pieces of each player occupy one edge of the board. The pattern has translation symmetry and rotation centers of order 3; this symmetry group is divided out, so that on the final board, each colored cell appears exactly once.
This board has the same topology as the previous one, and is obtained by dividing the pattern by the same symmetry group. However, it is obtained by gluing two three-player chess boards, such that those edges of the board where each player's pieces start are glued to their counterparts from the second board. Note that there are three edges, and following one edge one passes by the pieces of two players.
This board is also composed from two three-player chess boards like before, but before gluing each player's pieces to the second board, the edges are twisted. This board has a single edge, following which one passes each player's pieces twice - once on either (king or queen) side.