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<P>I have recently been attempting to adapt as many chess pieces as possible to a succinct description applicable to any board, particularly square and hexagonal. In doing so, I have liberally used the terms orthogonal, diagonal, radial, and hippogonal, defined as follows (before reading this discussion):</p> <ul> <li>Adjacent: Sharing either a side or a vertex. <li>Orthogonal: Directions in which spaces share a side. <li>Diagonal: Directions in which spaces share two orthogonally adjacent spaces, but are not themselves orthogonally adjacent. <li>Hippogonal: A jump one step orthogonally, followed by one step diagonally onward. <li>Radial: Includes orthogonal and diagonal directions, but not hippogonal. </ul> <p>The odd part is that there are no diagonally adjacent cells on a hexagonal board under these definitions, and no cells diagonal to the center circle of <a href='http://www.chessvariants.com/43.dir/diplomat/diplomat-chess.html'>Diplomat Chess</a>, but it provides a framework for using as crazy a board as you desire, while remaining true to the way those terms have most often been used.</p> <P>I have not been including specifically three-dimensional pieces (so most of the pieces described here are out), so I have not included triagonal directions. That term to me describes the [1,1,1] direction (uniformly triaxial), without any sense of jump length. Thus, spaces in a triagonal direction should each have three orthogonally adjacent cells that are one step diagonally from the next and previous.</p> <P>I have recently run across Chatelaine in literature, exactly as Gilman describes, so I would not be uncomfortable with using it as a chess piece. However, the piece described has been used elsewhere, particularly in Shogi variants, so I might not include it in my collection.</p>