Comments by gwalla

Because of the "inside-out rook" problem, I started trying to define equivalence for pieces in terms of what conditions (relative piece positions) a move can be performed under, until I realized that this was a much better way of defining moves in the first place. the inside-out rook demonstrates that the sequence of steps is entirely irrelevant, only which squares relative to the starting square can it be blocked on and how.

So at this point, I changed my approach. Instead of a path being a sequence of steps, it becomes a destination vector and a set of "tests", where a test is a pair of a vector and a state that must be true of the square that vector away from the starting square for the move to be legal.

A welcome byproduct of defining everything in terms of vectors from the starting square, instead of vectors from the previous step in a sequence, is that "exotic" board topologies become much easier to deal with. The sequence method was fine for hex boards (it's just a different set of basis vectors), but trying to define, say, a bishop on a moebius strip board was weird (try it!). But if all vectors are treated as having the same origin, there's nothing special about it.

Nice, right? Oops, forgot about divergent captures. Okay, instead of just a destination vector, we have a set of results, a set of "things that happen when the move is taken", which can include a destination and/or capture vectors. The Chu Shogi Lion's igui capture can be expressed as a move with a capture vector but no destination vector, since the Lion doesn't move; a null-move like the Lion's jitto is a move where the result set is the empty set ∅. Promotions? Another kind of member of a result set.

There's still a problem with promotion though. We need to know when it's an option. All moves are relative, but pawn promotion happens at a certain absolute position. Let's say that there is a "promotion zone" state for squares, and any pawn step onto a square with that state is a move with a promotion result. En passant can be defined in terms of a capture move to a "just stepped over by a pawn's double move" state. Of course hte problem here is that "state" is very vague. A promotion zone is permanent, but the en passant square only exists for one turn, and there's no way of specifying that sort of thing within this system yet.

Another important thing has been left out: black and white, or more generally sides. At this point I'd been mostly ignoring them and was thinking in terms of just e.g. "knight" rather than "black knight" and "white knight". I was more or less treating captures as explicitly affecting the opposite side. But that's not very flexible (it disallows, say, a hopper that can only use friendly hurdles, or a piece that can capture friendly pieces). Abstracting away the actual human players (who may not even exist, e.g. in a chess problem), what is the distinction between the sides? Alternating turns! Counting from the beginning of a game, white pieces move on odd-numbered plies and black on even-numbered plies (and neutral on all). So we should include, as part of a move's condition (set of tests), which plies it may be performed on, in most cases some number modulo the number of sides. Together with obligate promotion, we can even define oddities like Petkovian half-neutrals, which change sides when moved. Now that we've introduced a time dimension, we can even include that in our test vectors, allowing us to define en passant more concretely in terms of the previous turn's position instead of hand-waving an "en passant square" state.

I'll get into royal status, mimics, what a "state" really is, rethinking captures, and generalizing to almost everything later, but it's getting late and I have work tomorrow.

For a while now I've been playing with the idea of defining chess pieces in strictly mathematical terms. Originally this was meant as a more flexible/descriptive alternative for Betza Funny Notation, perhaps as a way of specifying pieces for chess-variant-playing AIs, but there is also the possibility of defining operations and functions over them, and maybe even proving some things algebraically.

My early attempts were to define pieces as sets of possible paths, where paths are sequences of steps, and steps are vectors; notation would use the Kleene star for unlimited equal steps. So a wazir, for example, would be {(0,1),(1,0),(0,-1),(-1,0)}, and a rook would be {(0,1)(0,1)*,(1,0)(1,0)*,(0,-1)(0,-1)*,(-1,0)(-1,0)*}. Since most pieces are symmetrical it would make sense to parameterize this as a kind of shorthand, but since some pieces aren't symmetrical it shouldn't be assumed (in the way that "a 1,2 leaper" usually implies the (1,-2), (2,1), etc. leaps). This works for the usual steppers, leapers, and riders, and even bent riders; it's basically how people usually think of chess moves, just in more formal terms.

But this starts to get awkward where pawns are involved (and a way of describing chess pieces that has trouble with pawns is seriously flawed!). First off, we need to separate passive moves from captures. That's not too bad; we can even do so by introducing a "capture step" distinct from a passive step, which must be to a square occupied by an opposing piece, and this lets us do fun things like define the Chu Shogi Lion and the Locust. Then there's the initial double move, and promotions. We can introduce a new kind of "step" that is a promotion to another piece type to deal with these: the initial double-step can be implemented by defining a "starting pawn" that has it, where every move ends with a "promotion" to a "basic pawn" piece that doesn't. The promotion step is also strange, because it always happens at the end, but if you define operations that derive pieces from other pieces (like a "double move" operation that derives a Hook Mover from a Rook) in terms of sequence concatenation, you can end up with a promotion in the middle, or even more than one. So we've unfortunately introduced a distinction between "well-formed" and "ill-formed" pieces, and a need for some sort of normalization.

Introducing hoppers also complicates things. We have to introduce another step, one that requires a hurdle like the capture step but doesn't capture it. But now we've introduced the possibility of moves that "collapse": after all, what is the difference between a (1,2) knight move, and a pair of a regular mao-move and a mao-move over an obligatory hurdle?  Or between the latter and a pair of a regular moa-move and a moa-move over an obligatory hurdle? Equivalence is a mess.

I never found a satisfying way of dealing with en passant, or castling. Defining a piece in terms of the path is travels doesn't lend itself to a move where two pieces are repositioned.

Finally I realized that this approach fundamentally allowed for pathological alternate definitions of pieces. For example, take the rook. One or more equal orthogonal steps, right? Now let's define an "inside-out rook": a piece that leaps orthogonally directly to the square right next to its final destination, then slides by orthogonal steps until it reaches its starting square, then finally leaps directly to its final destination. This piece behaves exactly like the basic rook. It's entirely equivalent: any square it can reach, a standard rook can also reach under the same conditions, and vice versa. Yet it is considered a different piece. While the example is obviously contrived, similarly weird things could be produced by operations that concatenate or interpolate paths.

Looks like I misremembered. I thought that the Unicode xiangqi characters had been added but apparently not.

The Westernized xiangqi and shogi characters are with the other variant pieces in the Private Use Area. They have names like BLACK CHESS MANDARIN and WHITE CHESS SILVER GENERAL. 

The "appendices" to Nishiki-teki are just other (non-free) fonts by the same designer. The one with variant pieces is obsolete because those pieces are all in Nishiki-teki proper now (you can find them in the PDF listing Nishiki-teki's Private Use Area). The basic xiangqi pieces are also in Nishiki-teki at their correct Unicode code points; Westernized symbols are in the PUA, as are Westernized Shogi pieces, though the dedicated fonts also have extras for Xiangqi and Shogi variants.

The current text kind of makes it sound like you need the appendices for variant pieces, but you don't. They're in there already.

I'd like to nominate Nishiki-Teki. It's a cartoony "pop" font that takes a manga-styled approach to the detailed "traditional style" (kings and queens with decoration, rooks with masonry, knights with bases, realistic pawns) chess symbols. It also has variant pieces in its very extensive Private Use Area, mostly following Quivira's lead but with a few others from Unicode proposals, from U+FC2E0 to U+FC39E. The site is bilingual in Japanese and English.

The last one does on my computer:

So I was wrong, the font name is Noto Sans Symbols, not Noto Sans Symbol. The other two match DejaVu Sans on my computer, which is probably the first fallback in my font stack.

Fedora (a fairly old version thereof, since I'm not very dutiful about upgrading).

I wonder if Mint messed something up in their distribution? It doesn't make sense for them to deliberately take them out, and it's not like they're new additions to Unicode (they've been included since Unicode 1.0).

The version I have was provided by my Linux distribution. Google's own link is here.

The "unknown font" appears to be Noto Sans Symbol, part of the Noto font family provided by Google that comes standard on Android.

Well, that's partly why I asked. The current draft is limited by my research skills. Information on the symbols used by fairy chess composers was relatively easy to come by, but I had a much harder time finding info on symbols used by variant creators, so I only included the knighted compounds since I knew they were used in several relatively popular variants, and the shatranj pieces because I had an unambiguous example of their use in notation in a published document. That doesn't mean a case can't be made for other pieces, it just means that I don't have enough data to make a case. So I'd like to involve people with more expertise on that front.

There are also open questions about several pieces. For example, should a camel piece be considered unified with the existing DROMEDARY or BACTRIAN CAMEL characters? Since some games call a vao an "archer" or "bowman", should an archer and a diagonal cannon be considered variations on the same character? These aren't questions I can answer on my own, they would depend on consensus.

But I never expected you to replace images on the website or in Game Courier with Unicode characters, particularly in board diagrams. That's not really what this is about. This is so that people can use figurine notation for chess variants in plain text contexts like email, and more conveniently in styled text contexts like Word documents.