Comments/Ratings for a Single Item

I found this article an interesting read.  The proposed names were both
descriptive and imaginative.  Although there will be little chance that
they will supplant previous accepted common titles for some pieces. 
There
are a few which may find some use.

As to the word 'triagonal', it is a compression of 'tri-diagonal' and
has become the accepted descriptive for the move in 3D Chess which
involves the change along three axis.  It has been in use for many
decades, and even appears in several dictionaries.  Albeit noted as a
vulgar form.

What dictionaries does triagonal appear in? It is not in Webster's 10th,
and it is not listed at dictionary.com. Although it does appear in the
OED, where it is described as an erroneous formation of trigonal, the only
definition given for it is triangular. The word is new to me, probably
because I don't play 3D games, and my objection to it arose from
Gilman's erroneous contrast between triagonal and diagonal, which
suggested that triagonal movement is not diagonal. As he said, hexagonal
boards 'have a triagonal but no diagonal.' Well, if that is what he
truly believes, then he is unaware of what you have just told me, that
triagonal is just a compression of tri-diagonal, for anything that is
tri-diagonal is still diagonal. My objection to the word stands because of
the confusion it can cause.

While I'm on the subject, I'll mention that this confusion has also been
engendered by our regular misuse of the word orthogonal, of which I have
also been guilty. The word orthogonal really means at right angles. The
directions that a Rook can move on a chessboard can be described as
orthogonal, because they are really at right angles to each other. But it
is just a relation between the two directions, meaning the same thing as
perpendicular, not an independent quality shared by each direction. The
most accurate word I can think of to describe the quality shared by the
directions Rooks can move on square and hexagonal boards is lateral.

Anyway, our misuse of the word orthogonal and our correct use of diagonal
has led to the mistaken notion that gonal is a proper root for use in any
neologism that describes an axis of movement in Chess variants. Gonal
comes from a Greek word for angle. Diagonal movement goes through angles.
So-called orthogonal movement goes through sides. To use common English,
we could speak of corner-wise movement and side-wise movement. For 3D
games, we could speak of corner-wise movement, edge-wise movement, and
face-wise (or side-wise) movement. When we use these plain English terms,
we can see that triagonal and diagonal are the same thing. They are both
corner-wise movement. In Raumschach, the Unicorn is the true 3D
counterpart of the Bishop, and it is the Bishop of Raumschach, not the
Bishop of hexagonal Chess, that does not move diagonally. It is called a
Bishop, because some of its movements through the cubic spaces of
Raumschach look like 2D diagonal moves from a flatland perspective, but
it, and not the Unicorn, is the truly novel piece in Raumschach.

For clarification, the purpose of these articles is threefold:
(1) as a forum for developing new names - I welcome feedback from anyone
who can better my efforts, particularly on the mixed-range pieces;
(2) as a reference for my own variants, to avoid a lot of explanation
repeated on several pages - I have already shortened the Tunnelchess by
referring to Shield Bearers;
(3) as a resource for game inventors who, like myself when I discovered
the site, have no clear ideas of their own.
Regarding Triagonal, it came to me instictively and on finding it already
in use I decided to stick to it. Orthogonal was a term I picked up from
the CV pages. If a term does not appeal to me, such as Hippogonal, I do
not use it. If there is a general preference for naming directions after
their commonest piece (Unicornwise, Rookwise, Knightwise &c.) I may
switch
to that usage.
Regarding what is a Bishop I note that the square- and cubic-board pieces
commonly called Bishop both move in multiples of root 2 times the minimum
distance between squares and (if at least one dimension is even) are
bound
to half the board. The hex piece is bound to less and moves in multiples
of root 3 times the minimum distance - like the 3d Unicorn.
Regarding Michael Howe's fear of involvement in a 'history of
disagreements', comments on the CV pages are like a muti-player variant
-
a world of changing alliances. As a Brit I find Fergus Duniho's British
Chess un-British, but I have actually defended his Yang Qi. I note that
the strongest defence of this page has come from someone I have
quarrelled
with in the past.
Finally a point on my mixed-range pieces. I thought of trying to make
them
all start with Dragon but got stuck and gave up. For the record,
Chatelaine is a generic for the lady of a stately home. Anyone else
puzzled by names, or able to think of better ones, feel free to comment.

I found this item a complete waste of time. Why should one person's list
of names for chess units be of interest -- when totally unconnected to any
significant body of work or original contribution? If anywhere, a list of
idiosyncratic piece name proposals belongs in an obscure discussion
forum.

I am stunned by the lack of editorial standards implicit in adding this
type of material as a 'contribution'. Gee, I thought any CV designer
would have a huge list of unused piece names. 

I generally agree with FD's points concerning the use and abuse of
language. Although I do find affected pseudo-learned illiteracies amusing
as all heck, as well as a great time saver when reading.

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.

I wrote the following offline, and it is not a response to anything since
my last comment. I will come back later and look at what has been written
since.

This discussion has helped me see more clearly that there are two
alternate methods for describing movement across a board, each equally
valid and each useful for boards on which the other isn't. One method
describes movement in terms of the geometrical relations between spaces,
and the other describes movement in terms of the mathematical relations
between coordinates. On the usual 8x8 chessboard, not to mention any 2D
board of square spaces, these two approaches converge. The two main
geometrical relations on a square board are diagonal and lateral. A
diagonal direction is one that goes through opposite corners of a space,
while a lateral direction is one that goes through opposite sides of a
space. The mathematical relations between coordinates concern how many
axes change in the movement of a piece. In Chess, a Rook's movement
changes its place on only one axis, while a Bishop's movement changes its
place on both axes. In terms of these mathematical relations, the Rook's
movement can be described as uniaxial, and the Bishop's can be described
as biaxial. More specifically, the Bishop's movement is uniformly
biaxial. The Knight's movement is also biaxial, for it too changes its
place on both axes, but it does so unevenly, moving across one axis more
than it does the other. In Chess, a Rook's movement is both lateral and
uniaxial, and the Bishop's movement is both diagonal and uniformly
biaxial. This convergence is a coincidence caused by the fit between the
geometry and the coordinate system of the chessboard.

Let's now examine the divergence of these two approaches. A 2D hexagonal
board has two axes. In Glinkski's Hexagonal Chess, the two axes are
vertical and horizontal, as in Chess, but the horizontal axis corresponds
with diagonal, rather than lateral, lines of spaces. In the generalized
approach to hexagonal coordinates used by Game Courier, both axes describe
lateral lines of spaces, but they intersect at 60 and 120 degree angles
instead of at right angles. Whichever method of coordinates you use for a
hexagonal board, the geometrical approach and the mathematical approach no
longer converge. In Glinski's Hexagonal Chess, for example, the Bishop
sometimes moves uniaxially, and the Rook has only one line of uniaxial
movement. Using the other method, the Bishop always moves biaxially,
through not always uniformly so, while the Rook has only two lines of
uniaxial movement, and its movement across the other line is biaxial. So,
for a hexagonal board, the mathematical approach breaks down, and only the
geometrical approach is useful.

For a 3D board, the mathematical approach is useful and commonly used. In
addition to uniaxial and biaxial movement, it introduces triaxial
movement, which is movement that changes the place of a piece on all three
axes of a 3D board. Although the mathematical approach is useful for 3D
boards, the geometrical method can also be used. Diagonal movement goes
through opposite corners of a cubic space; lateral movement goes through
opposite faces; and edgewise movement goes through opposite edges.
Although both approaches can be used for a 3D board, they no longer
converge. Although lateral movement remains uniaxial, diagonal movement is
no longer biaxial. Instead, it is triaxial. In Raumschach, a well-known 3D
variant, the Bishop of Chess has been replaced by two pieces, one still
called a Bishop and the other called a Unicorn. The Raumschach Bishop
moves biaxially but not diagonally; the Unicorn moves diagonally but not
biaxially. No piece in Raumschach can move both diagonally and biaxially
at the same time.

Although either approach can be used for 3D Chess, only the mathematical
approach is really useful for 4D and higher dimensional games. The
geometrical approach is useful for both 2D and 3D games, because we can
easily visualize 2D and 3D geometrical relations. But it is much more
difficult, if not impossible, to visualize 4D relations. On a 4D tesseract
board, each space would be a tesseract, but who can visualize a tesseract?
I can't. But the mathematical approach doesn't require visualization of
multi-dimensional shapes, and it is easily adapted to endlessly multiple
dimensions. So, for a 4D game, we would just add tetraxial movement, then
pentaxial for 5D, then hexaxial for 6D, etc. In trying to play such games,
we would be pushing our own limitations, but we would not be pushing any
limitations of the mathematical model for describing piece movement. It
could adequately describe movement on boards of any number of dimensions.

One practical use of the mathematical approach is for describing the
movement of pieces to a computer. Computers have no understanding of
geometry and can do geometrical calculations only by having the geometry
reduced to mathematics. In creating ZRFs for Zillions of Games, for
example, we define directions in terms of the changes in coordinates. The
computer has no understanding of the spaces as squares, cubes, or
hexagons. All it knows are coordinates and how directions of movement
change coordinates. Despite our inability to describe the movement of
pieces to a computer using the geometrical method, it remains a perfectly
valid method for describing movement, and it is well-suited for human
understanding of piece movement.

Let me now turn to the word triagonal, which started this line of thought.
This word creates confusion, because it tries to conflate two different
methods of describing piece movement, the geometrical and mathematical.
Since it is used to describe the triaxial movement of the Unicorn, and
given that it shares 'agonal' with diagonal and 'di' is sometimes a
root meaning two while 'tri' means three, it misleadingly suggests a
contrast with diagonal. In actuality, there is no contrast between the
meaning of triagonal and diagonal. Triagonal has one of two meanings. It
either describes movement that is both triaxial and diagonal, or it
describes any triaxial movement. We can safely assume that it is not
synonomous with diagonal; otherwise, there would have been no use for this
neologism. If it describes movement that is both triaxial and diagonal, it
no more contrasts with diagonal than integer contrasts with real, for all
integers are real numbers. If triagonal is synonymous with triaxial, then
there is no more contrast between triagonal and diagonal than there is
between even and prime. A number can be both even and prime, as 2 is, or
neither, as 4 is, or odd and prime, as 1 is, or odd and not prime, as 9
is. Likewise, a line of movement can both triaxial and diagonal, neither,
or one and not the other. 

No matter which definition of triagonal we go with, it sows confusion. In
both cases, it suggests a contrast that does not exist. While the meaning
of diagonal can be found in its roots, which are 'dia' for through and
'gonal' for angles, the meaning of triagonal cannot be found in its
roots. Going by the roots of the word, all it should mean is triangular,
which does not describe a kind of movement. If triagonal describes
movement that is both triaxial and diagonal, it's a term that does not
clearly belong to either method of describing piece movement, and its use
is limited to games where triaxial movement and diagonal movement
converge. If it is synonomous with triaxial, we would avoid confusion by
abandoning the word in favor of the more accurate triaxial, whose meaning
actually is contained in its roots. Triaxial has the added advantage of
fitting into a group of terms that progressively describe movement along
increasing numbers of axes. If we followed the model of triagonal, we
might say that a 6D game has hexagonal movement, and that would be
terribly confusing. In short, using the word triagonal invites confusion,
and Gilman's description of the hexagonal Bishop's movement as triagonal
but not diagonal is evidence of this confusion. By clearly distinguishing
between the two alternate methods for describing piece movement, we can
avoid further confusion.

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).

V.R.Parton used the term 'vertexal' to describe the 3D movement which
involved the change of three axes in a cubic field.  This would be
etymologically comparable to 'corner-wise'.  Both both of these terms
would change meaning when applied to a 2D structure.

And, like this article, this discussion has highlighted the differences
in
titles and descriptives.  Between what is common, what is proper and what
is ideal.  Could it not best be said that a word is only as good as what
it means to describe? ;-)

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.

My two cents' worth is that 'orthogonal' (as used in game rules) and
'triagonal' are 'terms of art,' useful in descriptions of game rules
and hardly anywhere else, and therefore known to people interested in
games but not to most others (including lexicographers). I'm mildly
interested to learn from the discussion here that their derivations are
probably based on confusions, but this doesn't diminish them in my
regard. Lots of good words were originally coined ineptly. Any attempt to
replace 'orthogonal' with 'lateral,' or 'triagonal' with
'vertexal,' or with any other new coinages, is more likely to create
confusion than remove it.

Tony P.,

I have not distorted any word. With respect to orthogonal, I am simply
suggesting a sense that works with our current usage. I don't have to
redefine orthogonal to allow for three orthogonal paths to intersect on a
hexagonal board, because current usage of the word already allows for
this.

The term point-paths does not work for me, because I play Chinese Chess,
which is played on points, and these points are connected by orthogonal
lines. Calling something a point-path doesn't tell me whether the lines
of movement going through the points are orthogonal or diagonal. Diagonal
is already a common English word that perfectly describes the movement I
think you are describing with point-paths.

Mark,

My objection to triagonal is not that it is based on confusion, but that
it invites confusion. The fact that it invites confusion does diminish it
in my regard. My main point is that there are two alternate methods of
describing piece movement, and triagonal does not fit neatly with either
method. Instead of using this confusing term, we should use terms that
clearly identify one method or the other.

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.

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.

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.