Single Comment

Tony P.,

As it turns out, the dictionary does agree with you. Nevertheless, at
least with respect to a hexagonal board, the movement I described as
diagonal is still diagonal by this broader definition. Furthermore, when
you draw straight lines through nonadjacent vertices of the hexagons,
every line that isn't a Bishop path runs parallel with a Rook path. Thus,
the hexagonal Bishop moves on all diagonal paths that do not pass over any
two spaces that share a common side.

Well, we have a couple options. (1) We can overhaul our terminology by
doing away with diagonal and orthogonal and replacing them with more exact
terms. (2) We can use diagonal and orthogonal in specialized senses.
I expect there is too much resistance to changing the terminology, and I
think it is common practice in many fields to use common terms in
technical senses. For example, statisticians have their own specialized
use of orthogonal. I propose that we accept technical senses of diagonal
and orthogonal that are specifically suited for describing movement on
both standard and nonstandard boards. Here is what I propose.

Orthogonal movement is the only kind of movement possible on a 1D board.
It moves along a single row of spaces, taking row in the broad sense to
refer to any series of spaces connected by a shared side with each
neighbor, no matter what direction it runs in. A row may be straight or
curved, depending upon the geometry of the spaces, but it may not zigzag.
A row may be understood to exist even when it is ignored for purposes of
coordinates. For example, Hexagonal Chess has rows running along three
axes, but only two axes are used for coordinates.

Diagonal movement, in the specialized sense, can be understood as movement
that runs through nonadjacent corners of spaces without going through any
spaces that share a common side. This is just a slight refinement of the
dictionary definition, so that it remains distinct from orthogonal. This
definition is perfectly adequate for Hexagonal Chess.

In multidimensional variants, we can begin to distinguish between corners
formed by two sides, by three sides, etc. This provides a basis for
distinguishing between different kinds of diagonal movement. With each new
dimension, there would be a new kind of diagonal movement.