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Constitutional Characters. A systematic set of names for Major and Minor pieces.[All Comments] [Add Comment or Rating]
L. Lynn Smith wrote on Thu, Dec 11, 2003 06:28 AM UTC:Charles Gilman wrote on Thu, Dec 11, 2003 09:35 AM UTC: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.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.
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.
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
You make a good reductio ad absurdum argument against compressing bi-diagonal.
I actually think it falls into the 'utinam logica falsa tuam philosophiam totam suffodiant' category. ;-)
I don't read Latin well enough to know what you're saying.
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.
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.
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.
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.)
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.
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).
<P>When I first learned to play Chess as a child, I learned that the Rook
moves straight. I did not know the word orthogonal until I began studying
Chess variants in more recent years. Because of the definition of straight
that I learned in geometry class, straight seemed like an inadequate term
for how the Rook moves. After all, the Bishop also moves in a straight
line. But the word straight has senses besides the one used in geometry,
and there is one common and everyday sense of straight that adequately
describes how a Rook moves even on a hexagonal board. Let me now quote
from Webster's: 'lying along or holding to a direct or proper course or
method.' And let me continue with some related definitions: 'not
deviating from an indicated pattern' and 'exhibiting no deviation from
what is established or accepted as usual, normal, or proper.' Suppose I
live on a curved road, and we are on the road, headed to where I live. And
I say to you, 'I live straight down the road.' Would you think me mad
because I don't live on a straight road? Would you drive off the road in
order to go in a straight line? Or would you understand that you will find
my house by continuing down the road? In the same sense that I used
straight here, the hexagonal Rook moves straight, and the hexagonal Bishop
does not. The geometry of the board defines certain natural paths, and
these are what the Rook moves along. In contrast, the Bishop moves along
paths that cut across the natural paths of the board. As it happens, the
roots of orthogonal allow an interpretation of orthogonal that is
synonomous with this sense of straight. So either word may do for
describing how a Rook moves.</P>
<P>Now let me amend what I was saying about diagonal last night. In <A
HREF='http://www.chessvariants.com/misc.dir/coreglossary.html'>A
Glossary of Basic Chess Variant Terms</a>, John William Brown provides the
term 'radial move,' which he defines as a move that is either diagonal
or orthogonal. In looking up radial in the dictionary, I don't find any
mention of diagonal or orthogonal directions, but I do find that it can
describe lines originating from a common center. So, the idea behind this
technical sense of radial is that diagonal and orthogonal lines of
movement converge at a common center. So let's now apply this concept to
movement along a Chess board. A radial line of movement would be one that
passes through the center of every space it connects. This distinguishes
it from an angular line of movement, which doesn't always pass through
the center of connected spaces.</P>
<P>Now, as Brown was defining the term, it includes both diagonal and
orthogonal movement. It is now simple to distinguish between these. An
orthogonal line of movement is a radial line of movement that never passes
through corners. A diagonal line of movement is a radial line of movement
that does pass through corners.</P>
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.
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., 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,
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.Anonymous wrote on Sat, Dec 13, 2003 08:01 AM UTC:After much thought I have drawn some conclusions. Straight need not be orthogonal or even radial. The difference betwen Nightriders and Roses is that Nightriders move straight. Conversely you could have a hex piece making successive Wazir moves with 60º turns in between - orthogonal but not straight. The current CV usage of orthogonal is valid, as such a move connects cells by passing through the middle of cell boundaries at right angles. Whether such moves are at right angles to EACH OTHER is irrelevant as there are right angles between FIDE diagonal moves and between some oblique (e.g. Knight) moves. Non-orthogonal radial moves are all IN SOME SENSE diagonal, as they connect cells by passing BETWEEN cell boundaries at acute angles. Triagonal is of degenerate etymology and none of you like my extension of it to Hex variants, and those objections automatically extend to tetragonal. The confusion from these terms is evident from some comments mentioning 4d games, which I did not consider. However, some diagonals have longer shortest moves than others and I still wish to distinguish between them on that basis. How about equal distances in...: 2 orthogonal directions at 90º to each other = standard diagonal; 3 at 90º or 2 at 60º = nonstandard diagonal; 2 at 60º AND another at 90º to both = hybrid diagonal? Surely everyone can agree that Hex boards 'have a nonstandard diagonal but no standard diagonal'. First mentions could be clarified in more detail, e.g. (colloquially called triagonal).
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.
The use of orthogonal to indicate 'rook-wise' movement is not restricted to the CVP. Pritchard uses the term in <u>The Encyclopedia of Chess Variants</u> ('<i>Rook</i> as Queen, but orthogonally only'), and Parlett uses the term in <u>The Oxford History of Board Games</u>. Parlett's definition is kind of interesting:
<p><blockquote>
<i>Orthogonal</i> describes a move in which a piece, travelling in a straight line through the centers of two or more cells, crosses each boundary at right angles to it. (The word derives from roots meaning 'right angle'.)
</blockquote>
<p>
Now Parlett might have been influenced by the CVP -- he gives us as a source -- Pritchard is and has been an influence <strong>on</strong> us. Where Pritchard's usage might have came from, I don't know.
<p>
A Google search on 'Orthogonal Movement' brings many interesting things to light, including usages from other fields that seem to echo our use of the term. It also seems to be widely spread through the wargaming field, but that might be Parlett's fault.
<p><hr>
Parlett also coins the term <i>Hippogonal</i> for a Knight's direction of movement. Now, this means 'horse-angled', which I think is a kind of nice usage.
H. J. R. Murray used the word 'orthogonal' in the sense that we modern variantists do in 'A History of Board-Games Other Than Chess' (1952) in his description of Tablut on page 63. In his 'A History of Chess' (1917), I was unable to find the use of the word 'orthogonal'. He instead uses the phrase 'horizontal or vertical'.
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