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# Comments by Tony Paletta

Bishop:Bishopper :: Rook:Rookhopper

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.

[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'.

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'.

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.

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.

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.

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.

Mark, Latin Squares are typically used when experimental plans involve 'treatment ordering' or 'incomplete block' (nesting of subjects under some combination of treatments) designs where there is a possibility of correlation between treatment and assignment. The 'orthogonal' is the sense of 'uncorrelated' (== zero cosine == 'right angle'), meaning that there no overall correlation or covariance is introduced between treatment and assignment (which would otherwise 'confound' a treatment effect, making it indistinguishable from the assignment or ordering effect). (Just a rough sketch from memory; if this sounds like Greek to you, rest assured that the things called Greco-Latin Squares serve the same 'orthogonal' master).

Fergus, 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.)

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.

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.

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.

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.

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).

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 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.

There is nothing especially simple, elemental, basic or natural about the rules for movement proposed by Glinski in developing a chess variant played on a hexagonal board -- in fact, the Glinski-B interpretation is considered something of a kluge by some CV designers (others view it more favorably). Other interpretations are not 'exotic' -- they are simply other, currently less well-explored, possibilities for defining movement. As I see it, Glinski's 'Hexagonal Chess' is fully equivalent to a game with half-Bishop + half-NRider (g-Bishops), Rook + half-Bishop (g-Rooks), half-N + half-Zebra + half-Camel (g-Knights), Rook + Bishop + half-NRider (g-Queens), King + half-Knight (g-Kings), and Berolina-type pawns (g-Pawns) played on a portion of an diagonally-oriented 11x11 chessboard. (Oddly enough, it's still an interesting game). I personally feel it might be better to embrace this equivalence (and others like it) rather than insist on somewhat arbitrary distinctions between hexagonal- and square-tiled playing fields.

Charles, Thought experiment and philosophical question. I select a an 8x8 playing area of regular hexagons resting on their points and arranged in the overall shape of a rhombus (let's say the major diagonal is NE-SW). The leftmost line of hexes can be called the 'a-file', the second (parallel) line from White's left can be called the 'b-file', etc.; the line of hexes closest to White can be called the '1st rank', the next parallel line the '2nd rank', etc.. A certain type of movement can be described as 'Wazir-like': one space along a file or a rank. Pursuing this analogy, we can generate analogs for all the units of standard chess and adopt analogous rules. A Staunton chess set is arranged so the chessmen or on the spaces with the same labels as in standard chess (White Staunton Rook on a1). Play begins: 1 e2-e4 e7-e5; 2 g1-f3 b8-c6; 3 f1-b5 a7-a6; 4 b5-a4 g8-f6; 5 Castles ... My question: What would be an appropriate name for the unit White moved first?

Ravioli Chess seems like a fundamentally interesting idea. A somewhat similar family of CVs can be developed if we assume that: (a) The playing field is one or more plane surfaces - folded, rolled or layered onto itself. (b) Specific, regularly placed pairs of locations on the playing field are points of contact. (c) Pieces occupying a point of contact may move from either their physical or their virtual location (the location in the pair). (d) Friendly units can virtually coexist but cannot physically occupy the same location. Opposing units cannot coexist and moving to one location of a location pair captures any opposing units on either location. (e) Piece move along (or around) any plane region, provided that the path from the starting location is open. Some examples: Toaster Pastry Chess: The Q-side and K-side are notionally two layers, with the edges of each side (perimeters a1-a8-d8-d1 and e1-e8-h8-h1) the points of contact. Units on an edge of either half are also virtual occupants of the space four spaces away, along the rank, in the other half-board and may move within either half-board containing the physical or the virtual starting space. Pierogi (Calzone?) Chess: The board is notionally rolled so that the a- and h-files are aligned vertically, with the surface making contact along the a-h files and each end rank. Units on the edge (a1-a8-h8-h1) may move as in standard chess from either their physical location or from the horizontal mirror image edge space. Taco (Omelet?) Chess: The board is notionally rolled as in Pierogi Chess, but the contact is accomplished by the units -- each unit is effectively a domino occupying both its physical location and its horizontal mirror image (virtual) location. From either location, units move as in a standard chess plane. Turnover Chess: The board is turned 45 degrees as in a diagonal form of chess, and also notionally rolled so that the side corners (e.g., a8-h1) align vertically. Pieces on the edge (a1-a8-h8-h1) move from their physical location or from the (virtual) horizontal mirror image edge location. Pawns are Berolina and a standard array might be adopted from 'Diagonal Chess' (L.A.Lewis) or 'Diamond Chess' (A.K. Porterfield Rynd) -- see Pritchard's ECV. Of course, these CVs will not suit everyone's taste. I will (wisely, I think) omit the details of possibilities such as 'Burrito Chess' or 'Cannoli Chess'.

Balancing a game with different pieces AND possibly different objectives is a tricky business. Since Magna Carta Chess started as a particular proposal with historical theme constraints, it should be finalized as such. I would not consider subsequent efforts seeking a similar or maybe somewhat dissimilar CV to be spinoffs.

(Partial reply to your post. You mentioned several topics.) I assume you mean a Dabbaba-analog on a hex board, using the Glinski-based analogy to FIDE chess. Mapping the hex board onto a standard chessboard (where the 91 Glinski hex board >> diagonal oriented 11x11 chessboard with 15 squares cut from each of the two side corners, for example) helps clarify the situation: the h-Dabbaba is more strictly equivalent to a true Dabbaba plus half an Alfil and, since the Alfil adds nothing to the Dabbaba's possible squares, the net effect is (not so surprisingly) the same type of boundedness as the true Dabbaba. One type of 'half-bound' piece on the hex board would be a piece moving in any single h-Rooks direction (e.g., N-S) and in the perpendicular h-Bishop direction -- its a FIDE Bishop. A conventional Camel is also possible, as are any other half-bound pieces mapped from a square-tiled chessboard.

I've done some work recently on a family of chess variants that generalize basic R-B-N piece movements and I more-or-less 'found' a generalization in the Mao-Moa family. I was wondering if anyone had run into the 'new' piece in other games and could provide me with any CVs that use it. The piece of interest is similar to a 'Bison' ([3,2] leaper + [3,1] leaper). Instead of leaping directly to its destination, it must first pass over a vacant space a Knight's-leap away, then continue one space orthogonally (in the N-leaps long direction) or diagonally (in the N-leaps combination of directions) to its destination. The context I'm using it in is 'Octoid Chess'. Instead of the 4-4-8 basic directions of the 'tetroid' R-B-N, I adopt 8-8-16 directions. On an 8x8 board R moves one ore two spaces like a standard chess Queen, B is one or two leaps like a Nightrider, the 'new' piece has the 16 directions, one step like the Octoid B, then like the Octoid R (Ks/Qs/Ps as in standard chess, castling also standard).

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