They are seperately colorbound for boards
Colorboundness in higher dimensions?
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For example a bishop is colorbound to Black in one 3d layer, but to white in the next one
There are indeed many versions of area binding, and the familiar board checkering that reflects the binding for Bishops and other diagonal movers is just one of those. In two dimensions inversion-symmetric pieces that have only 4 moves tend to be area bound, sometimes to very strange subsets of the board. Take for instance the 'Righthanded Chiral Knight', which is the piece that can only make the 4 Knight jumps consist of a single orthogonal step, followed by a diagonal step bending 45 degrees to the right. It can only reach squares on a lattice that covers 1/5 of the board. So this is a 'color alternator' that suffers area binding in 2d.
In one of the variants I designed (actually it was just an army for Chess with Different Armies) I used a piece I called Dragonfly, which moves along files like a Rook, and has the four most-sideway moves of the Knight. (So from e4 it would cover the e-file, c3, c5, g3 and g5.) This piece is bound to the odd or even files, and can even force checkmate on a bare King if it can prevent the latter from reaching the edge file it cannot attack.
I made a spreadsheet to evaluate different aspects of pieces, particularly fundamental leapers and sliders, in boards that have however many dimensions.
However, one of the issues I was running into is the inability to reliably predict which fundamental leapers are colorbound and which are not.
To make matters worse, not every bound piece necessarily stays on the same color. The Threeleaper, (3,0)-Leaper, in 2D is bound to 1/9th of the board but it's an alternating-color piece. The (1,1,1)-Leaper or slider, which I've sometimes heard called the unicorn or meteor, can only reach 1/4th of all squares despite alternating colors. Here is what that looks like with the slider. Orange dots are where it can slide to. Pink circles are where it could eventually reach.
Insofar as I can tell:
In order to avoid being colorbound, an elementary leaper must:
A. Not share a common factor greater than 1 between all of their nonzero components. The degree of additional boundness is the factor to the dimension power.
B. Have a mix of even and odd components after common factors are divided out. Otherwise, it is further bound by two to the number of components minus one.
C. Have an odd Manhattan move length once the common factor is divided out. Even move length causes single colorboundness if no colorboundness was incurred during stage 2.
So for example, the 1,1,1 "meteor" piece is bound to 1/4th the board in 3D. But the 1,1,1,0 piece in 4D is perfectly capable of dancing its way to any square it so desires, much like a Knight would, because zero is even. In addition, it goes from having 8 directions of movement to 32.
Yet the 3,2,1 and 3,2,1,0, which I have decided to call the Cazeb because they combine elements of a camel and a Zebra, are both strictly colorbound. Albeit with ridiculous 48-directional and 192-directional move patterns, respectively, making them the strongest elemental leaper in the 3rd dimension on an 8x8x8 board and tied for the strongest in the 4th dimension.