🕸Fergus Duniho wrote on Fri, Dec 12, 2003 05:14 PM UTC:
Tony P.,
You say: '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').'
When we're dealing with squares or cubes, the geometry naturally fits
with the coordinate system, and there is indeed no special reason to pay
attention to the geometry when thinking in terms of the coordinate system
will do. In such a context, we can even get away with thinking of
orthogonal and diagonal as meaning uniaxial and uniformly multiaxial. The
problem comes in when we try to apply such thinking to games whose
geometry does not fit with the coordinate system. Hexagonal Chess is a
prime example of this. In this case, the geometry does matter, and it
becomes important to recognize that orthogonal (or straight) and diagonal
describe geometric relations, not equations between sets of coordinates.