Thanks Joe Joyce and Ben Reiniger. Great there are more than one way to win a matt, schachmatt. Here is another argument the same as or overlapped with yours.  TRIANGLES OF THE MIND.  Of course the complete triangular grid is already there inside the square. Just take Betza's first step, rotating the utility board 45 degrees. Do not perform Recta-hex-ing, stop at Betza's first sentence.  Give each Square/Diamond a top-half and bottom-half. That object then has all the features of equilateral triangular grid linked, http://www.chessvariants.org/index/displaycomment.php?commentid=25282, since different angles 90, 45 and line lengths topological equivalence dissolves. Then require each piece entered into the space be positioned close to either an ''up-vertex'' or a down-vertex slot, so the two dimensions, conjuring infinitesmals, do not ramify to anything as significant as ''triangle.'' Each cell to play, insofar as it has minimal practical room for piece-marker, must needs have/be an up-vertex area or else a down-vertex and never both. No triangles are involved or ever required to be drawn or even pictured. Play within squares from vertex area to vertex area. Keep looking at squares only and play Chess (Variants). That way the Anti-Triangular board is made only of the everlasting same 64 2^6 squares (and Checkers 100 10^2 squares for enlargement). __________________________________Since we are only using squares, the regular 64 of them, movement has to be carefully defined.  Each move begins and ends into up-vertex slot or down-vertex slot, never something like phony ''side-vertex.''  The latter does not exist.  Remember to keep the board with diamond orientation. Thus 64 ''squares'' are 128 vertex-locales, more than enough. Each up-vertex is unique, so it has a number like '9', Cell 9.  From '9' there are four equidistant other up-vertices lettered 'a' to 'd' CW from top.  They are a square side-length away.  The first one of them, 'a', is actually top-off-center right. And so on with down-vertex locations we want to fulfill from a starting cell.  Play from up-vertex cell to down-vertex and vice versa, and up to up and down to down, completing all the possibilities relatively nearby, for defined piece-types.  Each sub-space/cell is as that infinitesmal building towards a point(corner of Square) and has no dimension except created by convenience in marking. Bi-colouration of the board may tend to distract and can be eliminated. So long as the divisible, exactly-delineated space is elementary/fundamental, Rules then can achieve the full degree of decadent over-refinement customary to CVers. Watch your step. These proofs are not semantic; is not all space everywhere one-dimensional-acquired directionality? Summary: Hexagonal -> Rectahex -> Square -> Triangle (reversible).  The only trick is the Rectahex slide a la Betza.
http://www.chessvariants.org/index/displaycomment.php?commentid=19657