Comments/Ratings for a Single Item

Charles,

While dictionary definitions provide a rough guide to the meaning of words
have, they (of course) only tell us part of the story. Case in point: Why
are the Rooks commonly said to move orthogonally when a Bishop's lines of
movement are also in orthogonal directions? 

One historic and important role of orthogonal lines in mathematics and its
applications is in the measuring of distance. While the King may have the
title, the Wazir is the natural 'ruler' of the chessboard. Start on c1,
move up three Wazir moves, then four Wazir units to the right to g4 and
(using the Pythagorean theorem) you can calculate that you are five
'Wazir units' away from your starting point. This also works with a
'Ferz' -- but on one color only (from c1 three Ferz units NE, four Ferz
units NW puts you at b8, five 'Ferz units' from the starting point).    
 

So both the Wazir or Ferz could be used to measure (Euclidean) distances.
The difference is the Wazir directly measures ALL the whole unit distances
that come up in talking about the square grid of the chessboard. So it
probably was more natural to think of the Wazir/Rook as THE orthogonal
directions on the chessboard.

Of course this isn't 'the way it happened' and it isn't the only way
it could have turned out based on the dictionary definitions, but the
convention for usage is not paradoxical, contradictory or especially
confusing.

Fergus,

In one of 12-12 comments ('As it turns out, the dictionary ...') you
brought up statistics and suggested that a different meaning
('specialized sense') was being given to orthogonal by statisticians. I
responded by indicating that these statistical senses were not different
in their root meaning. You criticized this as involving equations between
sets of coordinates rather than geometry. 
 
Many fields of mathematics are extensions of the concepts of classical
(Euclidean) geometry, in a variety of directions  -- including quite a few
fields which tend to deal with planes and vector spaces. The consistency
with the geometric sense of orthogonal (where orthogonal lines are lines
at right angles to each other) is maintained, even when not superficially
apparent. 

How does one get three 'orthogonal' paths to pass through a point in a
plane? We could Humpty-Dumpty up a meaning of orthogonal that directly
conflicts with the established meaning, but that doesn't impress me as
exactly dazzling the world with our brilliance. Choosing our terms more
carefully to convey our intended meaning seems like a better way to
communicate.

Tony P.,

Your last comments just leave me puzzled about what you're talking about.
So I have no response.

Putting that aside, I have now come up with a sense of orthogonal that
works with nonstandard boards AND refers to right angles. I hope it will
please you. Orthogonal lines of movement from a space are those radial
lines of movement which intersect the edges of the space at right angles,
or when this is impossible, at the points where the intersection comes
closest to forming right angles.

Fergus,

I didn't bring up statistics or any of the mathematical disciplines with
roots in geometry.

Are you sure you mean geometry? The orthogonal == right angle usage comes
from geometry.

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.

Mark,

Latin Squares are typically used when experimental plans involve
'treatment ordering' or 'incomplete block' (nesting of subjects under
some combination of treatments) designs where there is a possibility of
correlation between treatment and assignment. The 'orthogonal' is the
sense of 'uncorrelated' (== zero cosine == 'right angle'), meaning
that there no overall correlation or covariance is introduced between
treatment and assignment (which would otherwise 'confound' a treatment
effect, making it indistinguishable from the assignment or ordering
effect).

(Just a rough sketch from memory; if this sounds like Greek to you, rest
assured that the things called Greco-Latin Squares serve the same
'orthogonal' master).

Orthogonal is used in the study of Latin Squares to mean two Latin Squares
like the following:
1 2 3 4
2 1 4 3
3 4 1 2
4 3 2 1
and
1 2 3 4
3 4 1 2
4 3 2 1
2 1 4 3
which are orthogonal because when you combine the symbols in each cell,
all possible ordered pairs result:
11 22 33 44
23 13 41 32
34 43 12 21
42 31 24 13
Sets of orthogonal Latin Squares are useful in the design of scientific
experiments, or for generating 'magic' squares.

Anyway, this is a technical usage of the word orthogonal that may be
grounded in the 'at right angles' meaning, but if so I think it's very
tenuous. So I feel it gives more aid and comfort to those of us who
believe drafting orthogonal to use the way we do in rule descriptions is
okay.

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.)

Here are my thoughts on some of the suggested terms for directions.
Biaxial, Triaxial, Uniaxial: Acceptable alternatives to orthogonal &c.
for
Square and Cubic but still require a conventional rule for Hex.
Cornerwise, Edgewise, Facewise: Could be ambiguous as 2d games could be
considered a special case of 3d ones with a dimension of 1.
Diapleurol: Should be diapleural, but with that change is interchangeable
with orthogonal as interpreted below.
Lateral: Suggests 'along a line' - which on a Hex board surely means
the
non-orthogonal direction!
Orthogonal: If interpreted as at right angles to the cell boundaries,
corresponds exactly to the CV usage.
Pointwise: Could be interpreted as 'forward or backward orthogonal', as
the line of movement projects to players' notional viewpoints (i.e.
confounding all ranks to one) as a point.
Vertexal: The authentic adjective from vertex is vertical, and the last
thing that CVs need is another meaning for that word.

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.

Fergus,

You seem to have confused a diagonal and something sort of like a
diameter.

A diagonal of a polygon is any line joining two nonadjacent vertices.

A diagonal of a polyhedra is any line joining two vertices not in the same
face.

Other than these two uses, a diagonal line pretty much just means a
slanted line.

For a standard chessboard-like tiling with squares each square has two
diagonals and they line up to form longer lines -- hence THE diagonals of
a chessboard. It doesn't work with hexagons.

I actually think it falls into the 'utinam logica falsa tuam philosophiam
totam suffodiant' category. ;-)

So, we might compress 'bi-diagonal' to 'biagonal'?  And the difference
between the simple 'diagonal' and this new 'biagonal' is so slight
that it might either be confused or considered a typo.  And therefor
could
then be considered interchange-able.  Which will lead us back to the
common use of the 'diagonal'.  d=inv(b)  :p

Here are my conclusions on this matter. First, diagonal is a geometric term
that has nothing at all to do with specific kinds of changes in
coordinates. On a 2D board, the meaning of diagonal is unambiguous. It
describes movement that runs through opposite corners of a space. In 3D
and higher dimensional games, it becomes ambiguous, because there are
different kinds of diagonal movement. In a 3D game, you can distinguish
between diagonal movement that runs through the vertices of 3D spaces,
what Parton calls vertexel, as well as diagonal movement that runs through
opposite corners formed by two edges instead of three. It is appropriate
to distinguish these two kinds of diagonal movement as bi-diagonal and
tri-diagonal. Contrary to what I said earlier, tri-diagonal does not mean
triaxial, and it does not mean triaxial and diagonal. Rather, it describes
the geometric property of movement that runs through opposite vertices of
a polyhedron. It is a useful term for any multidimensional game beyond 2D.
For 4D and up, we can add tetra-diagonal, penta-diagonal, etc.

As distinguished from tri-diagonal movement, bi-diagonal movement runs
through opposite corners formed by only two sides. The Bishops in Chess,
Raumschach, and Hexagonal Chess are all bi-diagonal movers. Thus, Bishop
is an appropriate name for the piece which has it in both Raumschach and
Hexagonal Chess. Gilman has described a property shared by the Bishops in
Chess and Raumschach but not Hexagonal Chess. What he has described is the
property of uniform biaxial movement, not the property of bi-diagonal
movement.

My main point has been that there are two different methods of describing
piece movement, and each method should have its own terminology.
Orthogonal, bi-diagonal, and tri-diagonal are all geometric terms.
Confusion has resulted, because the mathematical method has not had its
own terminology, and people who have used it have tried to redefine the
geometrical terms in terms of coordinate math instead of in terms of
geometry. Case in point is Gilman, who was using diagonal to mean
uniformly biaxial. The mathematical method is a perfectly valid way of
describing piece movement. It just needs its own terminology, which is why
I have proposed the terms uniaxial, biaxial, and triaxial.

As for the word triagonal, I have no problem with the concept behind it,
but I do think that compressing tri-diagonal to triagonal obscures its
meaning. Instead of contrasting triagonal with diagonal, which is like
contrasting British with European, we should contrast tri-diagonal with
bi-diagonal. Tri-diagonal is not a kind of 3D movement that merely
resembles true diagonal movement. Rather, it is a specific kind of
diagonal movement and should be more clearly acknowledged as such.

As for orthogonal movement, it can be understood as straight movement that
never passes through corners. All types of diagonal movement pass through
corners, and all types of angular movement pass through corners, but
orthogonal movement never passes through corners.

Apologies to Mark Thompson and Tony Paletta, I meant Tetrahedral Chess. I
had no idea that there was a Tetragonal Chess. Tetragonal was my coinage
for the direction whose minimum distance is twice the Orthogonal's, and
was evidently playing on my mind. If it is the consensus that Triagonal
should go, so will that term.
Returning to the question of what is what -gonal, I interpret Orthogonal
as mean passing through 1st-degree boundaries (between two cells) at
right
angles to them, not (necessarily) to each other. This extends easily to
boards which are 3d, Hex, or both. Diagonal moves go through 2nd-degree
boundaries (boundaries between 4 1st-degree ones), at 45º to the
1st-degree ones. On a Cubic board, Triagonal moves go through 3rd-degree
boundaries (between 8 2nd-degree ones). The non-orthogonal Hex radial
actually goes ALONG 1st-degree boundaries, and so is not exactly the same
as Square/Cubic Diagonal OR Cubic Triagonal. On that basis there could be
a case for calling the direction Parallel! However it can appear on the
same board as the S/C Diagonal and so should surely have a different name
from that.

Charles,

3D Hex-based games present some really tough issues, partly because
there's no 'natural' generalization of a hex into a regular solid
(e.g., stacking boards gives a kind of hex prism) so our ability to use
analogies -- whatever they might be -- are somewhat strained.

One way to get a handle around SOME 'higher dim' chess is to think in
terms of areas -- maybe planes, maybe not -- with sets of paths defined
for within area moves and for between/among area moves. Essentially not
using coordinate geometry ('grid-like' games), but much closer to
'graph theory' - points and directed sets of paths between points. This
may or may not help in the evolution of your thinking.
  
BTW Mark Thompson's game is 'Tetrahedral Chess'; 'Tetragonal Chess'
(modest 'hexoid' game) is one of mine.

L.,

Don't have a problem with your usage in 3D. Orthogonal is standard,
diagonal matches the 2D Bishop's move, and triagonal doesn't jar with an
established term in a situation where the use of diagonal requires a short
term to make a distinction. My objection was and is to 'triagonal' on a
hex-tiled plane.


Fergus,

I still am in agreement with that other guy who posted under your name
somewhat earlier. I don't generally recommend edge/point terms for square
boards because they are not needed. On the other hand, I (recently)
avoided the terms orthogonal and diagonal in describing movement in
'Canonical Chess' variants on a rotated square-tiled board since it
would have been both ambiguous and confusing.

On a 'normal' chessboard (including Xiang Qi board, etc.) the terms
orthogonal and diagonal have had their meanings established by long and
frequent usage, and the terms are easily understood (translated) by people
who simply know what the words mean in other contexts. On hex-tiled boards
the orthogonal/diagonal terms carry neither the same established meaning
nor the same 'chess knowledge' implications.

I was considering 3 kinds of 3d board. There is the cubic-cell one, on
which the Bishop/Unicorn distinction is well established. There is the
board of several hexagonal-cell boards with three Rookwise lines on a hex
board and a fourth at right angles to them, which can also be viewed as
square-cell boards joined on the skew. On this there can be square-board
Bishops which can reach any cell, and the hex piece commonly called a
Bishop, which is of little use as it is bound to a third of a single hex
board! Then there is the form of board used in Mark Thompson's Tetragonal
Chess, which can also be viewed as an assemblage of square-or
hexagonal-cell boards. On such a board both pieces can be used with
workable moves, and it would make sense to call the hex-derived one
something different.
One characteristic of the hex piece is the length of its shortest move,
which is root 3 times the Rook's - exactly the same as a Unicorn on a
cubic board. As the cubic- and hex-board root-3 riders can never occur on
the same kind of board, at least within 3 dimensions, it seemed logical to
confound them.

I like the terms 'orthogonal', 'diagonal' and 'triagonal' for the
directions of 1-axial, 2-axial and 3-axial in cubic space.  They have the
same syllabic beat.  They are sufficiently different to avoid confusion
with one another.  And they make a nice matching set.

I plan to continue to use them.  Even if a few might think this 'wrong'
or 'un-educated'.  I know what they mean, and many others do also.  I
quess we will just have to tolerate those who are unable to accept them.