George Duke wrote on Mon, Mar 8, 2010 04:41 PM UTC:
http://www.chessvariants.org/index/listcomments.php?subjectid=ChessboardMath10.
Triangle/Triangle = Square/Square, Tetraktic in threes and fours like DNA.
Triangles afford readily natural movements of Queen analogue, but other
fundamental movements will differ according to interpretation, as in
Hexagons, such as throughout hexagonal Glinsky versus McCooey once at ''Pieces'' of Man & Beast 14 above:
http://www.chessvariants.org/index/displaycomment.php?commentid=25203.
Gilmanesque interpretation can have a field day, but one should
carry the day. In triangles, Knight is logical piece-type leaping through one side and one vertex in the same direction. So Queen and Knight go to differing exclusive triangles from a starting square. On 6^3 Queen from corner reaches 19 of the 36 spaces and centrally Queen 22 of 36. Naturally Queen subdivides into Rook and Bishop. Unlike squares, triangular Rook and Bishop move in the same direction, but to different alternating cells. Put a board in front of you to follow logic of symmetry-separation for piece-types Knight, Bishop and Rook. From any starting cell, R, B, & N are mutually exclusive, as in squares. There is next the expected fourth fundamental, Falcon, fully present, having two destination-type cells but one being paired for three altogether, the two being two-way and the other solitary (central) one three-way. The larger the board, the lower percentage of triangles Queen reaches. The generic designer would not employ actual Queen as piece-type on 36 or 25, since she is too strong when reaching over half the cells. Start using Queen on 64 triangles, then ideally 81 and 100(10^3). By the same token, on 64 triangles (8^3), no one should be greatly interested in the exterior movement mode of piece-type B, which perfects elementary non-trivial Tetraktys (3^3, 9 cells).
ChessboardMath11, now for triangles, has isolated comment with 5x5x5 triangles sketched,
http://www.chessvariants.org/index/displaycomment.php?commentid=25200, and continues CM10; the topic of triangles is in both -10 and -11.
