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Fergus,

My argument for not FOLLOWING the convention for using 'orthogonal' and
'diagonal' on hex grids was not based on the idea that they were not
CONVENTIONS,l but that they did not have the same FULL MEANING as on the
chessboard (crossing edges at right angles, but also moving along paths
that are at right angles), which in turn did parallel the more
comprehensive meanings used in mathematics (as opposed to the less
specific 'at right angles' dictionary entry). Both Dickens and Parlett
were well aware of the existence of hex games such as 'Hexagonal Chess'
and gave definitions that covered both -- they were certainly not out to
fight the convention, but simply to reflect it.

One point of my usage (edge-paths, rather than orthogonals; point-paths
rather than diagonals) is that it avoids the terminology problem for
pieces that demonstratably move exactly like chess Rooks, chess Bishops,
chess Knights, or any chess piece) in games on a hex-tiled board. The
conventional chess pieces follow the paths that reflect conventional chess
patterns; the hex pieces simply follow different paths. The baggage of
definition by analogy from chess (orthogonals into hex-orthogonals;
diagonals into hex-diagonals) disappears if the partial analogy is not
followed.