## Chess Leapers, Riders, and Compound Pieces

Ideas about how a world with more than three spatial dimensions would work - what laws of physics would be needed, how things would be built, how people would do things and so on.

### Chess Leapers, Riders, and Compound Pieces

On a hypercube chessboard, in euclidean geometry, in which all the cells are hypercubes of the same size, an (m1,m2,m3,...,mn) leaper is a piece that moves a total of m1 cells in the x1 dimension, a total of m2 cells in the x2 dimension, a total of m3 cells in the x3 dimension, and so on to a total of mn cells in the xn dimension. Shuffling around the coordinates of the vector of an (m1,m2,m3,...,mn) does not change the leaper. So for instance an (m1,m2,m3,...,mn) leaper is the same as an (mn,m2,m3,...,m1) even if m1=/=mn. Think of how a chess knight is both a (2,1), and a (1,2) leaper. A basic leaper cannot be blocked as there are no cells between its current cell and the cell it lands on, that it could land on.

An (m1,m2,m3,...,mn) chess rider is a piece that moves a whole number of cells along the (m1,m2,m3,...,mn) direction, with the number of cells it can move to being limited only by the size of the board. A chess rider can be blocked but only on cells that are in the same direction as it moves. In standard chess two examples of riders are the rook, and the bishop, with the bishop being the (1,1) rider and the rook the (1,0) rider.

A compound piece is a piece that combines the movement of at least two other pieces. For instance in standard chess the queen is a compound piece of the rook and bishop, and the king can be thought of as a compound piece of the royal ferz and royal wazir with the ferz being the (1,1) leaper and the wazir being the (1,0) leaper.

In chess triangulation is when you return to a previous position, but with it being the other players turn to move. For instance if you have a position with it being black to move, using triangulation you can return to the same position, but with it being white to move. On a 2d chess board, with the previously mentioned requirements for the board, it's possible to triangulate with riders, or compound pieces, but not with simple leapers. You can for instance triangulate with a king because it combines the movement of a ferz and wazir, and similarly you can triangulate with a bishop because it can move a different number of squares each turn as a rider. The same is also true for one dimension.

In higher dimensions there are some simple leapers can be used for triangulation, as they can return to their original cell within an odd number of turns. For instance, in 3d the (2,1,1) leaper can triangulate while only making moves that are equivalent to (2,1,1) moves. For instance it could make the move vectors (2,1,1), (-1,-2,1), and (-1,1,-2) to get back to its original cell in 3 moves. If the number of non 0 components of the movement vector of a leaper is odd, and if one component of the vector is equal to the sum of the other components of the movement vector, or if the movement vector can be split into multiple vectors that all meet the previous 2 requirements, then the leaper can triangulate. For instance the (5,3,2,2,1,1,1,1) leaper can triangulate as its movement vector can be split into the (3,2,1), and (5,2,1,1,1) vectors, which both fit the first two requirements for a movement vector for a leaper that can triangulate.
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anderscolingustafson
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