Single Comment

Fergus,

In statistics the term 'orthogonal' (once the surface is scratched)
rests on the geometric sense like it does elsewhere in mathematics --
always consistent with 'at right angles'. For example, orthogonal
comparisons are comparisons with sums of cross-products of zero,
equivalent therefore to uncorrelated, hence represented in a
multidimensional space as vectors with a cosine of zero, placing them at
right angles. 

Regarding 'diagonal' movement in 'cubic' multidimensional space,
there's no reason to consider the space as having anything but the pieces
and a set of potential resting points (think 'Zillions'). Two-D Bishops
ride in a line like they do through collection of two-coordinate systems
-- no established convention is violated by calling that a diagonal move.
If it wasn't for those pesky polygons from geometry, we could give
extended meanings to 'diagonal' for the lines along which N-dim
'Bishops' rider (triagonal, tetragonal, etc.) just like the rec math
folks did for polyominoes, polyiamonds and polyhexes. Given the conflict
with geometry terms looming for N>3, tri-diagonal, tetra-diagonal, etc. do
seem a little more sensible.

On hexagonal boards a conflict with standard chess terminology was (I
suspect) not originally envisioned by game designers. Since standard chess
pieces, fairy pieces and pieces more-or-less designed for hex grids are
also possible, it seems (IMO) that there's little merit in straining and
twisting the language to preserve an inappropriate set of analogies that
(among other things) make Glinski's formulation of 'Hexagonal Chess'
seem like THE way to describe hex grid movement. (But YMMV.)