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If you move DIAGONALLY one space, you move one space in two dimensions - i.e. 1 space left, then 1 space up.
If you move TRIAGONALLY one space, you move one space in three dimensions - i.e. 1 space left, 1 space up, then 1 space into an alternate timeline.
If you move QUADRAGONALLY (I think that was intentional) one space, you move one space in 4 dimensions - i.e. 1 space left, 1 space up, 1 space into an alternate timeline, then 1 space into the past.
It's really hard to explain in words, but go into practice and pick "Only Unicorns" or "Only Dragons" and create a bunch of alt. timelines, then try moving the pieces. The game does a really great job of visualizing how the pieces move.
Diagonally is easy to explain because you can think of a square. Move from one corner to the other. The sides of the square can be chosen from the four avaiable ones in the game, up-down, left-right, past-future, timelineUp-timelineDown. So for example a bishop can move diagonally on the board itself, stay on the space and move diagonally through time and timelines or a combination, most common being space-time, one step to the past and one step in one of the directions on the board.
Triagonal movement can be compared to a cube, moving from one corner of the cube to the opposite corner. The three sides of the cube can be chosen from the four available ones just like for diagonal, but with more complexity of course. The king and queen are the only regular pieces that can move this way. An example of a triagonal move would be to step into the past of an adjacent timeline and also move one space orthogonally on the board that they arrive at.
Quadragonal movement is most difficult to compare to anything tangible in our own three dimensional space we live in, mathematicians often speak of a hypercube, which is a four dimensional object like a cube but with four sides that are of equal length and the sides are also perpendicular to eachother. In the game it means that you can use all four of the available directions to move. The queen and the king are the only ones that can do this too. So for example you can move one space to the past, in an adjacent timeline and move diagonally on the board you arrive at.