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Wow. I just read all the comments for this section, and my eyes and brain are very tired. I would like to say that the terms orthogonal, diagonal, and even hippogonal work just fine for me, whether it's on a square board or a hex board. When you try too hard to read into things, they become less fun, in my opinion. On a lighter note, here's some suggestions for chess movement directions, which I'm sure are linguistically and mathematically incorrect enough to really annoy some people: Vertizontal, Hedrogonal, Hexaxial, Hipporadial, Horizagonal, Paraheximal, Gonalogonal, Trigonagonal, Hexohippozontal, Radiogonal, Orthozontal, Dihedraxial, Hexozontal, Verteximal, Zontifical, Edge-ognal, Point-edgimal, Side-o-zontal, Polyhedronomical, Rookogonal, Bishagonal, Knightaxial, Geogonal, Dihippaxial, Extragonal, and of course, Tri-di-hippo-horizo-vertico-radio-axio-hexo-agonalogonalogonal-wise I'm sure these are all thoroughly useless but if you like them you can have them.

Hippogonal seems a non sequitur to me. Better to describe oblique (non-radial) directions by their coordinates: thus the Knight, Camel, and Zebra move along the 2:1, 3:1, and 3:2 obliques respectively. Many others on the square board are listed in From Ungulates Outwards, which also gives the formula for duality of moves at 45° to each other. Here are some reasons to avoid square-board names for hex-specific pieces. Matching is more 'obvious' to some than to others. In the all-radial Wellisch Hex Chess, the diagonal piece (which is short-range) is called a Knight! Many of my piece names use duality, as this is linked to binding. If both coordinates are odd, as in the Camel, binding is the same as for the Bishop - inspiring the names Zemel, Gimel, Namel for leapers in other such directions. Duality on a hex board is entirely different. Extrapolating from the Glinsky/McCooey use of 'Knight' would give an unbound 'Camel' but a 'Giraffe' bound to a third of the board. It is unwise to ignore the cubic board when dealing with the hex one. Consider a hex board with cells designated ia1 to ph8. Now consider just ia1, ja2, jb1, ka3, kb2, kc1, la4, lb3, lc2, ld1 &c.. The diagonals between those cells are exactly like the orthogonals on a hex board. This is like the link between just the white squares of a FIDE board and a board of 7 rows of 2/4/6/8/6/4/2 squares. However the latter transforms orthogonals to diagonals as well as vv, whereas the cubic-hex transformation LOSES the original orthogonal. This means that the hex board does not have the standard diagonal. The diagonal that it does have transforms from the 2:1:1 oblique, which may explain why Wellisch considered that direction oblique itself. Now consider a hex-prism board of 15 8-cell files in triangular formation a bc def ghij klmno (used for a variant that I will soon be submitting). The line k1-l2-m3-n4-o5 is plainly a square-board diagonal but, by rotation, so is a1-c2-f3-j4-o5. The logical move for a Bishop on this board would be all such diagonals. With such a range of moves a Bishop could actually reach the whole board (note j4 and n4 both being reachable from o5), as could a Camel or Zemel similarly defined by the three planes of square boards. There is no need or reason to include a same-rank (i.e. same-hex-board) move; after all, the square-board Bishop has none.

<P>I have recently been attempting to adapt as many chess pieces as possible to a succinct description applicable to any board, particularly square and hexagonal. In doing so, I have liberally used the terms orthogonal, diagonal, radial, and hippogonal, defined as follows (before reading this discussion):</p>
<ul>
<li>Adjacent: Sharing either a side or a vertex.
<li>Orthogonal: Directions in which spaces share a side.
<li>Diagonal: Directions in which spaces share two orthogonally adjacent spaces, but are not themselves orthogonally adjacent.
<li>Hippogonal: A jump one step orthogonally, followed by one step diagonally onward.
<li>Radial: Includes orthogonal and diagonal directions, but not hippogonal.
</ul>
<p>The odd part is that there are no diagonally adjacent cells on a hexagonal board under these definitions, and no cells diagonal to the center circle of <a href='http://www.chessvariants.com/43.dir/diplomat/diplomat-chess.html'>Diplomat Chess</a>, but it provides a framework for using as crazy a board as you desire, while remaining true to the way those terms have most often been used.</p>
<P>I have not been including specifically three-dimensional pieces (so most of the pieces described here are out), so I have not included triagonal directions. That term to me describes the [1,1,1] direction (uniformly triaxial), without any sense of jump length. Thus, spaces in a triagonal direction should each have three orthogonally adjacent cells that are one step diagonally from the next and previous.</p>
<P>I have recently run across Chatelaine in literature, exactly as Gilman describes, so I would not be uncomfortable with using it as a chess piece. However, the piece described has been used elsewhere, particularly in Shogi variants, so I might not include it in my collection.</p>

Thanks to everyone for the info on Berse. Were there a 'rating other comments' list I would select excellent on it! On reflection I have decided against the change. It does not fit in with my rule of all radially-enhanced Rooks (as distinct from the Marshal which is obliquely-enhanced) being female titles. The canine/feline connotations would particularly conflict with this, by offending both old chivalric sensibilities and modern feminist ones! Finally I have been considering extensions of the Crab for a future articles; the name I think most logical for the 3:2 version of the Barc is Berz, which is just different enough from Ferz but not from Berse. Two further comments while I am at it. Firstly HOSTESS in square brackets is based on a pre-publication layout and I have submitted a correction to delete it. The correct reference to this name occurs later. Secondly in Britain it is the other meaning of ounce that is former!

I have read and enjoyed the book _The Chess Artist_ by J. C. Hallman. He describes his friendship with a Mongolian woman who is a Grand Master. In one of their conversations she seems to deny that 'Bers' or 'Berse' means anything in Mongolian--except 'chess queen' of course. It is the word 'Fers' adjusted for the fact that Mongolian has no f. It somewhat resembles the words 'Merzé' (mastiff) and 'Bars' (snow-leopard, formerly 'ounce'), both of which have been taken as guides in carving the pieces.

Berse could possibly have its initial roots in bersit, meaning to burst. This would explain the term bersim for a flower.

Berse appears to be a form of bersim, which is a flower.

According to Cazaux's book on chess variants (in french) it is a species of cat, latin name Panthera unica, french once. Don't know german or english names. --JKn

As Chatelaine seems an unpopular name I am considering substituting the more established name Berse, from Mongolian Chess, but would like to know what it means first. Does anyone know the literal meaning?

**Fergus Duniho**wrote on Mon, Dec 15, 2003 03:16 AM UTC:

<P>TonyP, you write:</P>
<BLOCKQUOTE>'To try and wrap up my end of this discussion of 'angles dashing from a
hex in a plane'. There exists a usage convention (tradition with a group
of supporters) for using 'orthogonal' and 'diagonal' to describe some
possibly paths on a hex grid. The usage (1) isn't especially apt, since
it conflicts in some important ways with the usual meaning of orthogonal
and diagonal in both chess and mathematics (especially plane geometry)'</BLOCKQUOTE>
<P>I just did a google search on the terms 'chess orthogonal', and page after page was on Chess variants rather than on Chess. I then went to the FIDE rules at the FIDE website and looked at its description of how Rooks moved. It said that Rooks moved along ranks and files, and it made no mention of the word orthogonal. I then searched for the word 'orthogonal' on the page, and the search turned up nothing. As I mentioned earlier, I did not learn the word orthogonal when I learned Chess. Instead, I learned that Rooks move straight, and I didn't learn the word orthogonal until I studied Chess variants. Based on all this, I surmise that the word 'orthogonal' is not commonly used for describing the rules of Chess, and, contrary to what you say, there is no usual meaning of orthogonal in Chess. The word, insofar as it is used in a Chess context, seems to be primarily used in Chess variant contexts.</P>
<P>As for the usage of the word in mathematics, I don't see the conflict. Like the statistical usage of orthogonal, which is based on the mathematical usage but not identical with it, the Chess variant usage of orthogonal is also based on the mathematical usage but not identical to it. And this is to be expected. Mathematics is a different field than Chess variants, and it has different concerns.</P>
<BLOCKQUOTE>and (2) suggests a 'rightness' (based on the analogy to standard chess) that is misguided, a frequent source of confusion, and somewhat stifling for
developing other approaches to hex chess. I therefore feel its a usage
ripe for replacement.</BLOCKQUOTE>
<P>I disagree with all of this.</P>

Fergus, Both the Bishop and the Rook do indeed have orthogonal lines of movement. I touched on this this in a 12-13(?) comment directed to Charles concerning why Rooks, and not Bishops, are usually described as are orthogonal movers; basically, my answer was that its a convention -- meaning a tradition -- and a bow to common usage; since Bishops are described as diagonal movers it seems relatively harmless to describe Rooks as orthogonal movers. In fact Solomon Golomb (who developed Cheskers, Pentominoes and was a leading light in recreational math), in a write-up on Cheskers, once described Bishops as Rooks on the 32-space board formed by one color of the chessboard, and Camels (Cooks in 'Cheskers') as Knights on the same board. I certainly don't find it a problem to think of Bishops as orthogonal movers, and I think any rule that uniquely identifies Rooks and not Bishops with 'the possible set of orthogonal movement patterns' would be somewhat deficient, since they are simply rotations. [Aside: I have used the 'Cheskers' game as an inspiration for a very odd game called 'Dichotomy Chess' (modest - goal variant), where I also tacked on a Dabbaba-rider + Ferz (B+K on 32!)]. My comment about 'straight lines'? It illustrates a construction guideline that does give rise to straight lines in one context (planes) and arcs in other (spheres), even though we might have been trying for 'meaning the same thing' and used a rule that is used to produce straight lines in planes. I certainly don't consider straight lines and arcs the same thing -- and I don't feel a need to call them both straight lines, or both arcs. They are simply analogous with respect to the rule of construction, but do not fully represent the same meaning. Walking the 'straight-lines' over to the orthogonal discussion: a rule that does produce paths of orthogonal movement on a square-grid and can be applied to produce paths on a hex-grid does not replicate orthogonal movement on the hex-grid -- it produces sets of movement paths through a point that are orthogonal on square-grids boards, but not on hex-grids. Analogous with respect to the rule of construction (and even using the word right angle -- so it must be legit?) if we apply the rule to square- and hex-grids, but producing results not reflecting the same type of thing. On a hex-grid, the simplest orthogonal movement pattern involves an 'edge-path' and a 'point-path' (e.g., vertically and horizontally on the Glinski board). A while ago (few weeks), I indicated to Charles G. that this is a mapping of a standard Bishop (e.g., from a chessboard rotated 45 degrees) that was 'halfbound' as opposed to the 'thirdbound' pattern of g-Bishops. To try and wrap up my end of this discussion of 'angles dashing from a hex in a plane'. There exists a usage convention (tradition with a group of supporters) for using 'orthogonal' and 'diagonal' to describe some possibly paths on a hex grid. The usage (1) isn't especially apt, since it conflicts in some important ways with the usual meaning of orthogonal and diagonal in both chess and mathematics (especially plane geometry) and (2) suggests a 'rightness' (based on the analogy to standard chess) that is misguided, a frequent source of confusion, and somewhat stifling for developing other approaches to hex chess. I therefore feel its a usage ripe for replacement.

<i><blockquote>
Interestingly, in his earlier (more informal, mass market) 'Brain
Games', (Penguin Books, 1982) Pritchard used 'files' and 'lines' in
describing the paths in Glinski's 'Hexagonal Chess', rather than
'orthogonals' and 'diagonals'.
</blockquote></i>
<p>
Interesting. I will note that the Prichard quote I made earlier was from the quick description of FIDE Chess in his introduction to <u>The Encyclopedia of Chess Variants</u>, so apparently he was comfortable using the term that way on a rectangular board. However, I suspect he wasn't entirely happy with <strong>any</strong> terminology for Hexagonal Chess, as in the ECV he avoided using any terms at all for hexagonal Rook movement, relying on diagrams instead.

**Fergus Duniho**wrote on Sun, Dec 14, 2003 06:45 PM UTC:

Tony P., I already responded briefly to this quotation of yours, but now I will respond in more detail: '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).' There are certain problems with using orthogonal to describe lines of movement that are orthogonal to each other. First, it does not describe a quality that belongs to any line of movement. Rather, it describes a quality of the relation between two different lines of movement. Second, it does not distinguish how a Rook moves from how a Bishop moves. On a standard chessboard, the Bishop's lines of movement are also orthogonal to each other, for they too are at right angles to each other. But if we take an orthogonal line of movement to be a line of movement that intersects the boundaries of a space at right angles, it describes a quality of the movement itself, and it distinguishes how a Rook moves from how a Bishop moves. It also fits well with the very first definition given in Webster's: 'intersecting or lying at right angles.' It is movement along a line that intersects the boundaries of spaces at right angles. Now, if we combine these two senses of orthogonal and call it the FULL MEANING of orthogonal, as you seem to suggest we should do, we have really just conflated two independent ideas.

**Fergus Duniho**wrote on Sun, Dec 14, 2003 06:26 PM UTC:

Charles, We're in agreement on the meaning of orthogonal, but we're not in agreement on the words standard and nonstandard. You say that nonstandard means 'different from that used in the standard games'. I disagree. A standard is a rule or principle that establishes how things should be. For example, there is a standard that diagonal lines of movement pass through spaces at their centers and corners. On hexboards it results in different numbers of diagonals at different angles than it does for square boards, but it's the same standard. All that's different is the application. We may call a hexboard a nonstandard application, but it's still the application of the standard concerning what diagonal lines of movement are. The inventor of the oldest hex variant may have been ignorant of this standard, but his ignorance is not an adequate argument against it.

**Fergus Duniho**wrote on Sun, Dec 14, 2003 06:10 PM UTC:

Tony P. You write: 'The fact that I can use 'the shortest possible distance between two points on the surface' to connect points on both planes and spheres does not tell me that it is appropriate to refer to both types of constructions as 'straight lines'.' Again, your comments are lacking sufficient context for me to know what you're talking about. Although this comment was addressed to me, it does not seem to pertain to anything I have said.

[Sorry I accidently posted my last comment under a 'Fergus' thread (Game Courier), rather than the 'Constitutional Characters' thread.] Fergus, The fact that I can use 'the shortest possible distance between two points on the surface' to connect points on both planes and spheres does not tell me that it is appropriate to refer to both types of constructions as 'straight lines'. [and now, new comment] Peter, Interestingly, in his earlier (more informal, mass market) 'Brain Games', (Penguin Books, 1982) Pritchard used 'files' and 'lines' in describing the paths in Glinski's 'Hexagonal Chess', rather than 'orthogonals' and 'diagonals'.

People are still how there can be three orthogonal directions in a plane. Because they are orthogonal to different things. Orthogonal is shorthand for orthogonal to the cell boundar[y/ies] crossed. This is true of the Rook move between square, hex, cubic, and hex-prism cells, and not of Bishop moves. Qualifications of diagonal clearly need not be used in a page referring only to square boards or only to hex ones. Where they are needed standard is shorthand for 'used in games that are or have been standard in their part of the world' such as FIDE Chess, Chaturanga, Xiang Qi, and Shogi. All these use either phyusical square cells or, as in Xiang Qi, something functionally equivalent, and all use the same diagonal in terms of move length and colourbinding (again even if squares are not physically coloured differently). Nonstandard means 'different from that used in the standard games' in terms of move length and colourbinding. Does anyone expect a hex or 3d game to ever become a regional standard? Hybrid means combining orthogonal moves at different angles to each other between hex-prism cells. Regarding '3 at 90º or 2 at 60º' you must remember to read it as following 'equal distances in...:' Only 2 hex orthogonal moves can be mutually at 60º as a third at 60º to one would be at 120º to the other.That the hex diagonal is not self-evidently a Bishop direction is demonstrated by the oldest hex variant (Wellisch, 1912), in which there is no Bishop and Knights move one square hex diagonal (Viceroys in my terminology - the Queen is my Vicereine and the King a Xiang Qi General). Noting that the faces of the tetrahedron in Tetrahedral Chess can be seen as hex boards, it becomes apparent that the same kind of Knight is used (invented independently, no doubt) there!

**Fergus Duniho**wrote on Sun, Dec 14, 2003 03:13 AM UTC:

Tony P., You write: '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,' That's fine. I have never imagined that this was your argument, though I do appreciate you giving the clarification. '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).' I disagree. Orthogonal and diagonal have the same meaning on a hexboard as they do on a square board. I see no need for extraneous terms that already do the job of established terms.

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.

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.

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 Duniho**wrote on Sat, Dec 13, 2003 10:27 PM UTC:

Some anonymous person (Charles Gilman?) writes: '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;' Do you mean 3 at 60º? A hexagonal board has diagonals along 3 axes. '2 at 60º AND another at 90º to both = hybrid diagonal?' What kind of board has that? 'Surely everyone can agree that Hex boards 'have a nonstandard diagonal but no standard diagonal'.' No, I don't accept this as a valid distinction. All that's nonstandard is the board. Given the standard for what diagonal means, diagonals on a hexboard are as standard as diagonals on a square board. Of course, they may be less familiar to those who only know the standard board, but unfamiliar doesn't mean nonstandard. 'First mentions could be clarified in more detail, e.g. (colloquially called triagonal).' I'm all for describing movement in detail for nonstandard boards. Even though I will maintain that the hexagonal Bishop moves diagonally and that the hexagonal Rook moves orthogonally, I would not say so little in a description of a game and leave it to the readers to figure out. See my description of Hex Shogi as an example.

**Fergus Duniho**wrote on Sat, Dec 13, 2003 10:12 PM UTC:

Wonderful. Parlett uses orthogonal in the same sense I finally arrived at independently. Let me add that Anthony Dickens also uses the term orthogonal in A Guide to Fairy Chess (1969), though he doesn't explain why it's a good term for what it describes.

**Fergus Duniho**wrote on Sat, Dec 13, 2003 10:03 PM UTC:

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.

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