Comments by Fergus Duniho
The relationship between hexagonal and cubic grids runs deeper, and in that sense hex bishops do correspond to unicorns. See e.g. https://www.redblobgames.com/grids/hexagons/#coordinates-cube
The article you linked to describes how hexes on a hexagonal grid could be described with three coordinates, because the six sides of a hexagon give it three axes. It's because of this that there is a choice when it comes to which two axes will be used to represent spaces for a particular hexagonal game, and different games have represented coordinates differently. Anyway, this article tries to make a correspondence between hexagonal grids and 3D grids, because each can be described with three coordinates. That's interesting as far as it goes, but the question remains whether there will be any correspondence between how pieces move on a 3D board and how they move on a hexagonal board. Since the article is simply about geometry and not about Chess variants, it does not cover that.
On a 3D cubic board, a Unicorn should move an equal distance in all three dimensions. So, if it starts out at (0,0,0), it could move to spaces with coordinates like (1,1,1), (2,2,2), (-1,1,-1), and so on. Looking at the diagram for axial coordinates on a hexagonal grid, the spaces that a Bishop could move to from (0,0,0) are all at a distance of 1 in two dimensions and a distance of 2 in the other dimension. These distances do not match how a Unicorn moves on a 3D board, and in fact, the grid does not contain a single hex with coordinates matching any space a Unicorn could move to from (0,0,0) on a 3D board.
I will also note that the spaces a hexagonal Bishop could reach from (0,0,0) do not match those that a 3D Bishop could reach. A 3D Bishop would be able to move to a space that is the same distance away in two dimensions but a distance of zero in the other dimension. Looking at the Axial diagram, these are on the orthogonal rows passing through (0,0,0). In other words, the way a Rook moves on a hexagonal board corresponds with the way a Bishop moves on a 3D board, at least in terms of how the coordinates change.
On a 3D board, a Knight move should adjust one coordinate by 1, one by 2, and leave the third the same. Looking at the coordinates on the Axial grid, there are no coordinates in that relation to (0,0,0). Every coordinate with no change in one coordinate has the other two at equal distances from (0,0,0). These all match the spaces a hexagonal Rook could move to.
So, in terms of Chess piece movements, I do not see much correspondence between hexagonal boards and 3D cubic boards. The changes in coordinates that Chess pieces are capable of will not be the same on each board. And even with three coordinates, the hexagonal board is missing many spaces from a 3D board and does not seem to support the sorts of coordinate changes that Rooks, Knights, and Unicorns are capable of in 3D Chess. While it supports the sort of coordinate changes Bishops are capable of, these belong to the hexagonal Rook, which can reach every space on the board.
Given all this, I do not see any way to support the idea that hexagonal Bishops correspond to 3D Unicorns. My takeaway from this is that piece movement should be defined in terms of geometric relations and not in terms of numeric coordinate changes. By defining it in terms of geometric relations, I can use the same definitions for Chess pieces on square, cubic, and hexagonal boards, and the pieces will move appropriately on each board. But if I try to define them in terms of coordinate changes, six-sided polygons (hexes) and six-sided polyhedrons (cubes) will not work the same way.
In 3d geometry one distinguishes body-diagonals and face diagonals.
Where is this terminology from? I am not familiar with it and have not found it anywhere I looked.
BTW, isn't it possible to cut through a cubic lattice so that you get hexagons (in a plane perpendicular to a body diagonal)?
I do not understand your terminology well enough to understand your question.
The Unicorn was invented for 3D games, and I disagree with the claim that the diagonals on a hexagonal board are the Unicorn's diagonals rather than the Bishop's. In various hexagonal adaptations of Chess or Shogi, the Bishop has been understood to move along lines of spaces that are the same color.
His reasoning for this begins with how he defined standard diagonals:
Standard diagonal (SD) directions, the diagonal of the standard square-cell geometry formed by simultaneously moving equal numbers of orthogonal steps at right angles have steps of length root-2.
I would disagree with this, defining diagonals instead as lines formed by passing through the opposite corners of spaces. This fits with the etymology of diagonal, for dia- means through, and -gonal refers to the corner. Similarly, a polygon literally means many-angled. If the shapes are not squares, then diagonal movement will not reach the same spaces as you could reach by making an equal number of moves in two directions at right angles to each other. It is merely a coincidence that it works this way on a square board, but it is not the definition of diagonal.*
The problem comes in when we start thinking about how diagonals should work on 3D boards. Since the Bishop is a colorbound piece on the 2D board, the decision was made to make it a colorbound piece on the 3D board too. But to do this, it was given a move that does not pass through the corners of a cubic space. Thinking of each space in a 3D game as a cube, the level-to-level movement of a Bishop in 3D Chess goes from one cube edge to the edge that is opposite in all three dimensions. However, this can be thought of as diagonal movement if we think of each vertical slice of a 3D board as a 2D board of squares. In fact, this is not so different from how we think of movement on the same level. We easily see the usual Bishop's moves as diagonal on one level of a 3D board, because we view it as a 2D board of squares. But if we thought of each space as a cube, the movement would no longer appear diagonal. This is easier to see, because of the way 3D boards are constructed. But mathematically, the board can be divided into vertical slices just as easily as it can be divided into horizontal slices. (Note that these vertical slices come in same-rank varieties and same-file vartities.) If we view the Bishop's movement as 2D diagonal movement through a single vertical or horizontal slice of the board, it can still be understood as diagonal in a 3D game.
Therefore, I would distinguish between 2D diagonals and 3D diagonals, and not between standard and nonstandard diagonals. Bishops move along 2D diagonals, even on 3D boards, and Unicorns move along 3D diagonals, which just don't exist on 2D boards, including 2D hexagonal boards. So it is wrong to say that the diagonal movement on a hexagonal board is how a Unicorn rather than a Bishop would move. The 3D Unicorn simply has no legal moves on a 2D board of any kind.
* As an afterthought, his definition of Standard diagonal could be tweaked for spaces of other shapes than square. On a hexagonal board, you can reach spaces along the diagonal though pairs of alternating orthogonal moves that are 60 degrees apart from each other. This works, because each angle of a hexagonal space is 60 degrees, not 90 degrees like it is for a square.
On Designing Good Chess Variants. Design goals and design principles for creating Chess variants.[All Comments] [Add Comment or Rating]Why are all the comments bolded on this page?
There was an unclosed B tag near the end of the page. I have now removed it.
Aside from that, I'm wondering if there's any disadvantage to rotational symmetry compared with mirror symmetry. Does it encourage symmetric positions more? It seems less popular for some reason.
Rotational symmetry is less popular, because Chess has mirror symmetry, and most Chess variants copy this feature of Chess. This mass copying of Chess should not imply that mirror symmetry is inherently better. Shogi has rotational symmetry and doesn't seem to be the worse for it.
One difference between Chess and Shogi is that Chess has a Queen beside the King. With mirror symmetry, Queens start in the same file, and moving a Pawn from the King's file has less chance of exposing the King to check. In Shogi, Kings both start in the center file, and rotational symmetry mainly affects the positions of Rook and Bishop, which are toward the sides. The rotational symmetry allows an early exchange of Bishops and keeps Rooks from engaging with each other until later in the game. So I would say that whether mirror or rotational symmetry is better will depend upon what pieces are in the game, where they are placed, and the dimensions of the game. So I don't think there is any hard and fast rule that one is always better in any Chess variant.
Piececlopedia. 
An encyclopedic reference on Chess variant pieces.[All Comments] [Add Comment or Rating]I don't think the adjective supernumerary makes much sense for pieces that affect other pieces. The latter doesn't have anything to do with a number. In the context of chess engines I always use the term 'supernumerary pieces' to refer to the situation where you have more pieces of a certain type than were initially present, e.g. 2 Queens or 3 Knights.
You are using the word in a very literal sense, as it literally means "exceeding the standard or prescribed number." In that sense, though, supernumerary applies to the number of pieces and does not describe particular types of pieces outside of the context of a particular game.
But I borrowed the use of this word from Anthony Dickins, who uses it in a more figurative way in A Guide to Fairy Chess. After covering Leapers, Riders, and Hoppers, he has a section called "SUPERNUMERARY PIECES". He says underneath the heading:
We now come to a group of pieces that are not derived from normal chessmen but have movements or powers that are entirely new. We shall call these 'Supernumerary' pieces.
He then proceeds to describe the Imitator and Joker, which are both mimics, the Diplomat, which offers safe harbor to adjacent pieces but cannot move or be captured, and the Pyramid, which also can't be moved or captured.
He then mentions combined pieces, snipers, and hunters, which I would say are not entirely new in the same way that the first four pieces he mentioned were. This is followed by a section on special powers, which covers capturing, imitating, imitative, moving,neutrality, protean, reflecting, royalty, and invisibility. This ends his section on supernumerary pieces, as the next section has a higher level heading.
I am taking the general idea behind what Dickins meant by supernumerary but applying it more strictly to pieces that do not clearly fall into the Leaper, Rider, Hopper, or Hybrid categories. It is a catch-all for pieces with powers other than fixed powers of movement and the ability to capture by displacement.
The page doesn't really stress that how a piece moves and what it can do otherwise are really independent properties, which could be combined arbitrarily. E.g. you can have immobilizing Knights or Gold Generals. I would refer to this as 'additional powers' (as opposed to 'power of movement').
I had been considering the term "Superpowered Pieces" but felt that this term would suggest that these pieces are more powerful than other pieces, whereas "Supernumerary Pieces" would be neutral on this matter. For example, non-displacement capture usually comes at the expense of not being capable of displacement capture, and switching one for the other doesn't necessarily make a piece more powerful. I also prefer the compactness of putting an adjective before pieces over the longer "Pieces with Additional Powers". If I did go with a longer phrase, "Pieces with Special Powers" might work better, as it has fewer syllables, and the powers might be replacements for rather than additions to the usual piece powers. I'm also considering something like "Novel Pieces" or "Unconventional Pieces" for what I'm presently calling supernumerary.
I think Windmill should fall into the hopper category since it requires a hurdle to move.
Well, it doesn't hop over anything. So it's not a hopper. But because it does require a piece to go around, it is also different from a regular rider. Maybe it could be classified as an Orbital Rider, because it requires another piece to orbit around. The question that remains is whether this category belongs under Riders or under Hybrids. It has one element of hopping, which is requiring another piece to move, though it works differently than hopping does.
I suppose what it has in common with hoppers is that hoppers and orbital riders could both fall under a larger category of pieces that are dependent on the proximity of another piece to move. But without lots of other piece falling into this category, it may be best to keep Hoppers as a top-level category.
I have restored the Piececlopedia as an indexed HTML page. For those who want it, the database link we had been using for the Piececlopedia is at the top of the page. This new Piececlopedia index differs from the old one by being arranged taxonomically. While it takes some categories from the Taxonomy page, it also departs from it in some points.
A Taxonomy. Categorizing several types of pieces.[All Comments] [Add Comment or Rating]'Double' and 'triple' are more precise specifications of 'compound'. So I would just say 'double leaper' and 'triple leaper'.
As pointed out, a double leaper could be a piece that leaps twice, and a triple leaper could be a piece that leaps thrice. Sticking with the word compound makes what I mean less ambiguous.
'Hybrid' wouod be a useful designation for compounds of different move types, like leaper + rider or leaper + hopper.
One of the main contexts in which the word compound comes up is in speaking of the other compounds you could have of Chess pieces besides the Queen. If you called the Queen a compound but referred to the Knight-Bishop and Knight-Rook compounds as hybrids, that would confuse matters. So I am not in favor of using the word hybrid for a type of compound. It might be better used to describe a piece that combines the moves of two different pieces in a hybridized manner, such as the Chinese Chess Knight, which first moves as a wazir, then as a ferz, or bent riders, which start with a leap in one direction, followed by riding in a different direction.
As an afterthought, I could go along with speaking of hybrid compounds, which is different than substituting the word hybrid for compound. After Leapers, Riders, and Hoppers, I could include a new section on Hybrids, which includes hybrid compounds and other kinds of hybridized pieces.
Simple compound is for a piece with only one move type.
Since I contrast simple pieces with compound pieces, I would not refer to simple compounds.
Hybrid is for a piece capable of 2 different types of move.
Instead of using a different word than compound, I will just distinguish the kind of compound I'm speaking of.
I just think splitting Leaper Compounds into Double-Pattern & Triple-Pattern is neat.
I think it is less confusing to use the same term when the meaning doesn't change. I have now broken "Leaper Compounds" into "Double Leaper Compounds" and "Triple Leaper Compounds."
Darter is a term for a blockable leaper
Lame leaper is the more established term, and I don't understand the significance of calling them darters.
Leaper-Rider is a type of piece that makes a leap & may continue riding
Ralph Betza has already given us the term bent rider, and I want to favor established terminology over the use of new terms.
An X/Y-sniper travels like piece X & captures like piece Y, so Murray Lion is an Alibaba/Pasha Sniper.
I was thinking of rifle pieces, since rifles are usually used for sniping. Since a Pasha is a compound of a Man and an Alibaba, I don't think it properly counts as an X/Y-sniper. The examples Anthony Dickins gives for Snipers in A Guide to Fairy Chess have no overlapping capturing and non-capturing moves. In this case, though, one is a subset of the other, and if you allow that, all divergent pieces may be counted as Snipers. Since divergent is the more common term, and the Murray Lion has overlapping capturing and non-capturing moves, I do not favor calling it a Sniper.
Argentinian Pieces aren't of Argentinian origin either.
Since I don't think either one of us have been using that as a category, it's not a point in favor of placing Leo and Vao in a Chinese piece category.
Modern Chess. Variant on a 9 by 9 board with piece that combines bishop and knight moves. (9x9, Cells: 81) [All Comments] [Add Comment or Rating]Strange! Despite using the '?nocache=true' suffix and flushing my browser cache the updated version of betzaNew.js refuses to appear in my browser.
I have now purged it from Cloudflare's cache.
A Taxonomy. Categorizing several types of pieces.[All Comments] [Add Comment or Rating]Why would you make a distinction between hybrids and compounds? What is that distinction in your mind? Why refer to leaper compounds as double and triple pattern instead of compounds? Why would you call the Mao and Moa Darters? What does that term mean for you? Why list bent riders under the heading of Leaper? Was that a mistake?
The Murray Lion is not a sniper. It captures by displacement. While based on the Cannon from Chinese Chess, the Leo and the Vao are not of Chinese origin.
Reflectors, Multi-Path, and Doubly Bent might be useful categories.
Xhess. Decimal variant with Nightriders and Cannons. (10x10, Cells: 100) [All Comments] [Add Comment or Rating]Variants tend to [be] best when varied from traditional chess (IMO).
The use of Horsemen instead of Pawns, as well as the use of Nightriders and Cannons already makes this very different from Chess. In general, the best Chess variants will maintain a balance between being like Chess and being different from it. They should be different enough to offer something of interest that Chess does not, but they should also be similar enough to not lose what makes Chess a worthwhile game. So more differences from Chess wouldn't necessarily make it an even better game.
Here I'd opt to change the knights to camels for a bit more reach on a 10x10 board.
There are 10x10 Chess variants with Camels, such as Devingt Chess or Cardinal Super Chess. There are also other large variants with Camels, some of them being listed on the Piece:Camel tag page. While Camels do move further than the Knight, they are colorbound, which means they cannot actually reach as many spaces as a Knight.
The suggestion -- if you are adding the cannon -- sort of sister to the rook -- why not the Vao as a sister to the bishop. It would create some added symmetry to the game.
My own games Eurasian Chess and Gross Chess do this.
A Taxonomy. Categorizing several types of pieces.[All Comments] [Add Comment or Rating]I mentioned in my earlier comment that I already had something like this page in mind. What I actually had in mind was a taxonomic index of the pieces in the Piececlopedia, which would be helpful for getting a better idea of what is missing and could be added. I have made an initial draft of it at /piececlopedia.dir/pindex.html, which I plan to eventually make into the new index page for the Piececlopedia directory. I have sometimes used the terminology from this page, but I have also added new terminology for types of pieces not covered here. Please take a look and let me know if anything is wrongly classified or might be better classified.
Modern Chess. Variant on a 9 by 9 board with piece that combines bishop and knight moves. (9x9, Cells: 81) [All Comments] [Add Comment or Rating]How about placing a dummy piece on an invisible square? The swapping move could be legal only if the piece is still on the board, and it could remove the piece when it is completed.
I have deleted his spam, marked him as a spammer, and changed his password to some random text I just forgot.
Modern Chess. Variant on a 9 by 9 board with piece that combines bishop and knight moves. (9x9, Cells: 81) [All Comments] [Add Comment or Rating]Could the color-changing move also include a conversion of the other Bishop to a regular Bishop?
I improved the diagram and the text describing it today, but it could use an Interactive Diagram. Would someone like to provide one?
Featured Chess Variants. Chess Variants Featured in our Page Headers.[All Comments] [Add Comment or Rating]Add a second for Modern Chess from me
I did and made it the featured variant for January, 2025.
Ajax chess has rule enforcing play and game records.
Okay, I found the link to the preset scrolling down its rules page. It is not listed on its own separate page, which stops it from being indexed as a Game Courier preset. I have updated the columns for this game.
Duck chess can be played at chess.com if that counts.
It would be preferable if it could be played here. But if it had no competition from nominations better supported on this site, that could count.
Instead of listening to it all, I just skipped ahead to multiple points to get a sense of what it was. Everything I heard was instrumental. It mostly sounded electronic, and I suspect the more acoustic sounding instruments might have been synthesized. I didn't recognize any of it. It seemed like it might be the work of one artist. Is this your work, or were you putting together music you like by different artists?
A Taxonomy. Categorizing several types of pieces.[All Comments] [Add Comment or Rating]I was thinking of doing something along the lines of this page when I came across it on an early CD-ROM of this site. One detail I found surprising is the claim that "compound" was coined by John William Brown in Meta-Chess. Checking out earlier sources, I found V. R. Parton using the word. So I have removed the claim that it was coined in Meta-Chess.
Dzura. Members-Only Shogi-inspired, tactics-oriented game on a hexagonal board. (9x9, Cells: 61) [All Comments] [Add Comment or Rating]Since this comment is for a page that has not been published yet, you must be signed in to read it.
Xiangqi: Chinese Chess. Links and rules for Xiangqi (Chinese Chess). (9x10, Cells: 90) (Recognized!)[All Comments] [Add Comment or Rating]Could I get SVGs made of the elephantrev2 and envoy images in the Alfaerie set that are used on this page for the Elephant and Guard? These would allow me to use better quality images for the western pieces.
Next Generation Chess is a rather generic name along the lines of New Chess, Chess 2, or 21st Century Chess. While the name suggests something different from Chess, it doesn't indicate the nature or extent of this difference, and it doesn't distinguish it from other Chess variants, which are all in their own ways different from Chess.
Asymmetric Chess. Chess with alternative units but classical types and mechanics. (8x8, Cells: 64) [All Comments] [Add Comment or Rating]I have deleted the comment, which tried to link to a casino site.
25 comments displayed
Permalink to the exact comments currently displayed.
So far, I have not addressed his math, which I will now check here:
Here is what he wrote on the page:
I now understand that by root-2 and root-3 he means the square roots of 2 and 3. For square and cubic boards, these values check out. The diagonal distance between the midpoints of two diagonally adjacent squares is the same length as the diagonal of the square itself. This is the hypotenuse of a right triangle with two of the sides. Taking each side to have a length of 1, the Pythagorean theorem gives it a length equal to the square root of 2. Likewise, the distance between the midpoint of two cubes sharing a single corner is the same as the distance between the two opposite corners of a cube. When I looked this up, I learned that it is equal to the square root of 3 times the length of one side. Giving the side a length of 1, this diagonal has a length of root-3.
Let's now consider the hexagon. When the side is equal to 1, the distance between two opposite corners will be 2, as a hexagon can be divided into six equilateral triangles sharing a common point. See How many equilateral triangles are there in a regular hexagon?. However, since there is a gap between two diagonally adjacent hexagons, this has to be added in. With a side length of 1, this gap is 1, making the total distance a Bishop travels between two diagonally adjacent hexagons as 3. Well, 3 is not its square root. So this doesn't give us the same value.
But when we start with a side length of 1, the distance a Rook moves between two adjacent squares is not 1, which is already a problem. To calculate this distance, I will refer to this Regular Hexagon Cheatsheet:
The first thing I'll do is calculate the distance between two opposite sides when a side has a length of 1. For this, we want to look at the Inradius formula for Ri. With a side length of 1, a is 1, and Ri is half the square root of 3. Since Ri is half the length we're looking for, the distance between two opposite sides is root-3, which is 1.732050807568877. So, with a side length of 1, a Rook in hexagonal Chess travels the same distance as a Unicorn in 3D Chess.
To better harmonize the distances that pieces travel on hexagonal and 3D boards, I figured it would be best to give the Rook a distance of 1 on a hexagonal board. This allows Rooks to start out traveling the same distance on each board. Since the distance between two opposite corners is twice the length of one side, I need to figure out the length of one side when the distance between opposite sides is 1. I can do this with a ratio, since I already know three of the values and can solve for the fourth. So, x is to a length of 1 as 1 is to the square root of 3. As an equation, it looks like this:
x/1 = 1/root-3
So, x = 1/root-3
Since we want three times that, we get 3 divided by its square root, which is its square root. So the math checks out.
While the math does check out, I consider this a reductio ad absurdum against his idea of defining piece movement in terms of distance. A piece that moves in all three dimensions, as a Unicorn does, cannot have any legal moves on a 2D board. Although the hexagonal Bishop moves the same distance as a 3D Unicorn, the distance a piece moves is not the same as the manner in which it moves. I conceive of the Bishop as going through 2D diagonals of polygons and of the Unicorn as going through the 3D diagonals of polyhedrons. If you change the shapes of the spaces, you are going to get different measurements, but the idea behind 2D diagonal movement and 3D diagonal movement will remain the same. In circular or spherical Chess, the measurements will even vary between different spaces on the same board, but the idea of 2D diagonal movement will still make sense when it is understood as going through the corners of spaces.