Comments by FergusDuniho

I have two ideas for how the tournament can be run. One is to do it as a
set of concurrent sub-tournaments. Each sub-tournament would focus on a
single game, and it would consist of a series of elimination rounds to
determine a champion. The champion of the tournament as a whole would be
whoever is champion of the most sub-tournaments. Prizes would be awarded
for championship in each sub-tournament, and a bonus prize would be given
for winning the whole tournament. By this method, winners would play more
games than losers.

The other idea is for everyone to play the same number of games. Points
would be given for each win or draw, and the whoever scored the most
points would win the tournament. This is how the last multivariant
tournament was done.

With each method, appropriate methods of tie-breaking would be employed if
needed. Let me know your thoughts on which method you prefer and what
reasons you have for favoring one or the other.

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.

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.

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.

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.

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.

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.

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.

Tony P.,

You wrote: '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.'

Okay, here is why your comments puzzled me. You are entirely mistaken in
your assumption that my comments on equations had anything at all to do
with your comments on the statistical use of orthogonal. In fact, I have
said nothing on the subject of the statistical use of orthogonal since my
one-time mention of it. My comments on equations between sets of
coordinates was on an entirely unrelated thread, in which I was simply
discussing the two different methods for describing piece movement.