Comments by TonyPaletta

While I'm somewhat distressed at abandoning amphibian Pawns, I feel
relatively lucky. I don't think I could have fully recovered if I'd
developed Sea Chess but had to leave out the 'Prawn'.

Somewhat toroidal boards, soldier (/Shogi) pawns, bicolored multimove
turns -- that's the spirit. For the time being, though, I'd personally
prefer to stick to a somewhat simpler game (with standard pawns) if it
adequately showcases Amphibian pieces.

Michael,

If you can look at the 3D system as an [x,y,z]-coordinate system, then a
rider that makes a series of consistent unit leaps in two coordinates only
(e.g. --[1,1,0]-rider, [1,0,1]-rider or [0,1,1]-rider) could properly be
called a Bishop -- its a 2D Bishop when there's a choice of planes, and
becomes a regular Bishop when there is only one plane (such as on a flat
board). Generally that's been the piece called 'the Bishop' in
3-(4-,N-)dimensional chess -- a convention to call any other piece the
Bishop would probably be more confusing.

Peter,

Faced with the problem of describing geometric movement on a regular grid
(i.e., intersections of two sets of equally spaced parallel lines) back in
1980 I chose the terms 'edgewise' and 'pointwise' to refer to movement
from the center of one space to the center of another in a line which
bisects a side or an angle, with the continuation of such movement
constituting 'edge-paths' and 'point-paths'. 

This convention works equally well for square- and regular hex-tiled
boards (which are grids or sections of a grid) regardless of their
orientation, while not directly conflicting with a very common
mathematical usage (e.g., orthogonal axes).

Fergus,

English is English, not the sum of its roots. Why distort a word with a
clear and established meaning, and give it a new meaning that directly
conflicts with its established meaning (so that three 'orthogonal' paths
CAN pass through a point in a plane) in precisely the context it is to be
used? Seems a lot harder to 'explain' than edge-paths and point-paths.
It's (literally) a poor choice of words.

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.

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.

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.

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.)

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).

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.

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.

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.

Peter,

Parlett does start his discussion of movement in two dimension with: 
'Before exploring two-dimensional war games it is desirable to ESTABLISH
a terminology of movement and capture, as a surprising amount of
confusion, ambiguity and inconsistency is exhibited in the existing
literature of games.' (emphasis added)

It's unclear to me whether he's trying to (1) describe common usage, (2)
summarize dominant practice (3) prescribe usage or (4) simply provide a
basis for his further discussion so he can write the book. The inclusion
of hippogonal leads and his criticism of Murray me to (3) or (4), but it
isn't that clear.

Pritchard (as encyclopedist, but also as popularizer) tends to go with the
primary source descriptions and is generally descriptive rather than
prescriptive. He (properly) avoids taking positions except where a game
author's conventions are truly strange (and even then, he is seldom
outright critical -- though sometimes revealing a droll wit in the best
tradition of British writers).



All,

Just to summarize some of my main comment lines (personal opinions and
preferences) in this long thread:

(1) I'm not a fan of jargon-for-jargon's sake. If connected to a
specific convention the author feels is necessary in presenting his/her
own work, present the material in a context and do try to be
straight-forward, clear and reasonably accurate in your terminology.  

(2) Personal naming conventions (for pieces, but also for other concepts)
belong inside an author's body of work. This allows you to rethink your
choices, frame your decisions within the context of their use, and present
what you feel is a finished product.   

and (initially least) 
(3) Some existing naming 'conventions' -- orthogonal and diagonal as
used in hexagonal chess, for example -- suggest parallelisms with more
familiar, well-established concepts from chess and mathematics that simply
don't exist. Since the terms don't convey what they appear to convey,
there is a good case to be made for not following those naming
conventions.

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.

Fergus,

It has occurred to me that you were arguing with yourself. I never said or
implied (check) that there was NO USE OF the term orthogonality could be
satisfied in a hex grid -- looking back that seems to have been your
original point that you were looking to refute. 

My point was that a fundamental fact from plane geometry would be
contradicted by any such definition of 'orthogonal movement' -- one that
is satisfied by ' orthogonal movement' in chess: 

at most two orthogonal lines (or paths) 
can be drawn thru a point in a plane.