the dimensions and the pieces

Having a general template for ND spaces on how the pieces move is needed for the evolution of these games.

As well as some pieces are tricky in 3D chess in relation to 2D chess, 4D chess or 5D chess is challenging in relation to 3D chess

1. The dimensions

1D chess is a simple row of squares where only pieces like the rook, the dababba, pawns or grasshopper can move on. The only reason for the use of squares instead of lines is because there is need to 2D space for showing the pieces.

This has been submitted as a WoF 2 player game, this year.

2D chess is classical chess, that we all know, where pieces can move in two dimensions, with ability to combine directions, like the knight, bishop, king (instead of wazir) or queen (instead of rook).

3D-GrandChess has the same figures as conventional chess, but the pieces move differently, with the ability to move in the 3rd dimension, even if there cannot be made the combination of 3 dimensions in a single move. instead, they choose a 2D plane for each move (or a 1D line for the rook)

[Note: i will be using Japanese letters for the 4th dimen] The fourth dimension can refer to time, where pieces can move to the past, being unable to move for the rest of the game, and disappearing from next boards, or be treated in the fifth dimension as in a parallel universe. a fourth dimension can also be static, like our 3rd dimension, without referring to time, by considering an expanded cube, and treating the moves through dimensions as with a 2D chessboard compared to a 3D chess-framework. Or, by creating custom pieces with specific movements on the four dimensions. this system can go further and create squares of cubes, cubes of cubes, rows of cubecubes (7D), etc. until ∞D

2. Comparing dimensions - the directions

a piece can have a force of movement in a certain number of dimensions - the directions. a reduced number of dimensions may leave open the question on how a piece would move with more dimensions, because multiple options lead to the same in 2D, or sometimes, 3D.

Establishing the steps on each direction of a piece is important, or confusions may happen. The knight, for example can be thought of two ways:

1 - it moves 2 squares in one direction and then one square in another

2 - it moves in a diagonal step and then one orthogonal step respecting the diagonal direction (this is wrong)

both are fine in 3D and 2D, but if we go 4D, what does an orthogonal step mean? that means it keeps ONE 3d world.

1: あαa1 - うβa1 is possible

2: あαa1 - うβa1 is not possible

Time traveling method would be similar.

so defining how a piece moves in general should be made using directions.

3. Pieces movement

The rook moves any number of squares in one direction. [one square for wazir]

The bishop moves any number of squares, the same in two directions and never moving on a third one [one square for ferz]

The king combines wazir and ferz.

The knight moves 2 squares in one direction and then one square in another

The queen combines the rook and the bishop

The pawn moves in a 1D row and captures like a forward ferz

It is important to note that more dimensions give more opportunities to make more pieces, that move, by force, in 3D space.

Perhaps as a solution to the 4th dimensional problem it might be more convenient to think of the "time" axis as an extra spatial dimension. In general relativity, "time" isn't a special dimensional component of the universe that is seperate from space, it is all "space-time".

You would solve this problem by having 8 "3-D" boards next to each other that would represent a fourth axis of freedom. You can use Japanese letters if you like, but forgive me if I use hebrew (oi vey ):

Let's use the letter t to represent the fourth axis (since there is a kind of relation to time if you like...)

We can now represent each "Tesseract" using 4 distinct coordinates representing its position within the 4D matrix: (x, y, z, t) - note that we would only see a cube within the larger cube (but then that's kinda the point).

From there if we define a 4D-GrandChess as similar in that each piece can only move relative to some 2D plane, then you would probably end up with a "3D board area" full of pieces for White, a "3D board area" full of pawns for White, a "3D board area" full of pieces for Black, a "3D board area" full of pawns for Black and 4 empty "3D board areas". This doesn't incorperate any classical notions of time at all but merely another axis of freedom that can be exploited.

After Hebrew we should use Phonecian then Cuneiform :0 (joke joke)

Tempted to make a club for math and physics nerds...

Here's a Hexadecimal number system I invented (a=0, b=1, c =2 ...):

Notice how within the digit symbol itself, each number is represented using a binary composite of the "arms" to represent its digit :v

Actually I did not used Hebrew because I did not found the symbols lol

Discussions on 4 dimensions or more can be exciting for fans of multidimensional geometries, as we can be; but I'm afraid they are unlikely to appeal to the masses of potential players who want to enjoy exercising the arts of strategy and tactics; challenging other players to exciting competitions of intelligence.

We humans are physically made for the 3 dimensions. There are two small organs inside our ears, called labyrinths, which our brain uses to orient itself in space; they're made up of three tiny loop-shaped tubes, arranged according to the perpendicular faces of a cube. Each pair of tubes is practically equivalent to a Cartesian axis.

We are also made to live on a plane, because of gravity; although we actually live on a sphere; but let's say the plane tangent to the sphere at the point where we are. This is why many people are confused if we show them a picture turned upside down.

This is not the case for computers and intelligent machines. For them, adding a dimension is tantamount to multiplying positions, movements and related reasoning: they do not feel any confusion "in orienting themselves". Humans, on the other hand, would have to expend all their brain power just to identify the 'position' of a single piece and the 'positions' in which it can be moved.

Just imagine a 4D-GrandChess made of 8 3D-cubes as you proposed. You have 4096 squares/cubes/hypercubes and 1024 pieces for each player.

First question: how do we move? One piece at a time as in chess? Can we imagine how slow such a game would be? In large armies many units move simultaneously to perform different tasks. Maybe we should allow multiple moves at a time; this would be a very different kind of game, perhaps unattractive to the many millions of chess fans.

Then let's consider one single piece, a very simple one, a white pawn positioned at its starting place, at the second cube in ζf7-2. How do we define advancement, which is important for pawns? Maybe as advancing into the next cubes; and promotion when they reach cube 8. So I can move it into ζf7-3 or ζf7-4 and we will see this pawn all alone in cube 3 or 4, totally empty of other pieces.

After ζf7-4, on next move maybe I can move it on ζf7-5 or capture some opponent's pieces, where? In ζf6-5, ζf8-5, ζe7-5, ζg7-5, εf7-5, ηf7-5. Can you visually imagine these positions in the fifth cube? They are the 6 small cubes adjacent to the 6 faces of ζf7-5; but this set of positions hardly conveys the idea of "advancement" to us humans.

I defy anyone to imagine the range of positions that can be reached by a bishop or queen when one of the dimensions involved in the move is the transition to another cube. There's no way to make this intuitive for us humans, or at least for the majority of possible players.

One of the great attractions of chess is that much of the reasoning is done intuitively. I think 3D playing is much more challenging; but where the moves are always made on a plane, much of the reasoning can be made intuitively; thus attracting many intelligent players. A variant needs to be played by many people for it to become popular; otherwise it remains a theoretical variant.

My dream is that 3D-GrandChess could become so popular that some industry finds it worthwhile to produce the physical frameworks to play it. Or some good programmer builds the computer platform or an app for online playing, and we have our own international federation.

To achieve this, the game has to be attractive to humans; many humans; fans of chess, rational games, geometry, or similar.

Who said it needed to be 8x8x8x8?

4D chess can be 4x4x4x4, we can use different pawns, with different capturing. For me, advancement means keeping in the same cube. So they promote by reaching 8th rank inside a cube. For capture, they move one step in two directions. I also really love the idea of making 4D into time.

Defining the advancement of pieces such as pawns would just extend the logic of how they move on the 2D plane according to the system you defined for chess pieces within the 3D-GrandChess space. I think maybe I got it wrong to begin with. Perhaps rather than having an entire 3D "Block" filled with pawns and piece, we should instead keep the same proportions through each 3D block - sorry, just realised this, otherwise we would be changing the orientation of the axes (my mistake). Travelling "forward" would still be on the y axis, and we would extend our 2D planar movement but incorperate it with 4 Dimensions, simple. Using the notation I defined in the thread on pieces and (x, y, z, t) to define the dimensions) we could define a pawns movement as such:

"Pawn (without en passent and first move allowing for 2 square movement)"

mF(0, 1, 0, 0) & cF(1, 1, 0, 0) & cF(0, 1, 1, 0) & cF(0, 1, 0, 1)

-

How do we move? One piece at a time is possible, but I've been pushing lightly to promote awareness of synchroneous movements. Personally, I prefer my own rules of Synchroneous Chess that I invented independently from the true inventor of Synchroneous Chess (The games are very similar with only a few subtle differences) - the point is that synchroneous rules would effectively half the time because both players would be moving at the same time. Other options could of course include an agreement of both players to move up all their pawns by one or two ranks (as I have suggested for collosal chess) - another, and my personal favourite, is to negate the rules of castling and double pawn movements in favour of each player arranging their pieces strategically how they like before the game begins - just like in classical war, each sides Generals decide their battle formations prior without detailing their strategy to the enemy.

-

I agree, it is not intuitive to the majority of neurotypical persons. I am lucky in that I have had frequent exposure to things like 4Dimensional geography due to my interest in Gravitation and Relativity (also, as a Maths and Theoretical Phsyics student at University, everything you were saying makes perfect sense to me, so it is nice to meet a fellow enthusiast) - as you may remember me saying before, I believed even 3 Dimensional Chess would find its core base and interest amongst Mathematicians, Physicists, Computer Scientists and I add to that list Game Developers.

-

Yes, Federation for variants would be magnificent. Personally I'm a little tentative about this though because, like yourself I imagine, I believe that Chess is the inheritance of Humanity - even if there exists an organisation to promote, facilitate world tournaments and assist in the definitions of rules, there cannot really ever be an "official" Chess representative because we all have as much claim to it as everyone else. I firmly believe this and as shocking as it may sound, I do not recognise any organisation or entity as an authority over Chess - not even FIDE (actually, FIDE is well known and documented to be broiled in scandle and corrupt practises so something else to watch out for).

I quite like the idea of a cooperative or joint council of major chess organisations agreeing to cooperate jointly for the greater interest of Chess as a whole (including variants) because I don't personally believe that FIDE is adequate to the role they have assumed for themselves. This is something we can debate extensively though.

-

@TheUltraTrap, Yep. The great thing about our game and the creation of variants is we are not confined to any particular definition anyone defines and we have practically infinite possible ways of playing the game including with smaller parameters and boards (and you are right about promoting within the same "cube" - I made a bit of an error earlier with my suggestion of piece placement suggesting that forwards was within the 4th spatial dimension opposed to what it was previously).

As for Time, God knows lol. There are so many sci fi definitions of time, do you propose a multiverse system or something more like Back to the Future?

-

Final note: I have been working on an extreme variant that utilises custom boards to affect movement where players can grow their armies a bit like "Age of Empires" - in this game I am developing I considered that all players would "move at the same time" but all pieces can have at least one movement/action within each time "frame" - perhaps this might also suggest a way to play that would decrease the ammount of time needed to play?

I consider 8x8x8x8 as the "natural" evolution in 4 dimensions. 4x4x4x4 is surely possible; only consider promotion occurring on the 4th rank and pawns on 2nd rank can reach 4th rank in one move. That could become a continuous popping-up of queens like in a pad of pop corn.

I always try to imagine how a "normal" player can visually deal with the structure, the space, the pieces, etc.

So, in case of a 8x8x8x8 4D-chess game, we have a row of 8 big cubes, each of them like the cube in 3D-GrandChess, with alternating colours of the small cubes inside them. Cube 5 is exactly the same as in 3D-GrandChess. The other large cubes have another queen instead of the king.

Therefore for each player we have: 1 king, 31 queens, 160 rooks, 160 bishops, 160 knights, plus 512 pawns lined up on the whole second rank of all cubes.

The movement of a pawn is quite intuitive.

mF(0, 1, 0, 0) & mF(0, 2, 0, 0) & cF(1, 1, 0, 0) & cF(0, 1, 1, 0) & cF(0, 1, 0, 1)

mF(0, 1, 0, 0) & mF(0, 2, 0, 0) means one or two steps forward for simple advancement;

then cF(1, 1, 0, 0) & cF(0, 1, 1, 0): for capture we have the same quadrifork or quadrident as in 3D-GrandChess

cF(0, 1, 0, 1): the last type of capture however appears less intuitive. A pawn in cube 4, positioned in δf2.4 for instance can capture an opposing piece in δf3.5 or δf3.3 That is: it advances one step while disappearing from cube 4 and appearing in one of the adjacent cubes.

Of course players have to consider the "influence" of the pawn, including the squares where it can capture, so including the ones in the adjacent cubes.

Another "simple" piece is the rook. It moves within a cube the same way as in 3D-GrandChess; but it can also move along the 4th dimension. So a rook in αε1.3 can potentially reach any αε1 square in any cube provided all αε1 squares in between are empty. In this case too the rook disappears from cube 3 and appears at the same position in another cube along the row.

Bishops walk the diagonals. Let's have a bishop in the first corner αa1.1 and take it into the opposite end of a 4D diagonal. I can put it in αh1.8 or αa8.8 or θa1.8. None of these movements conveys the idea of a diagonal at an intuitive level. The bishop can also reach any intermediate square on the three lines in an intermediate cube. Then you have queens and kings. 

When you think your strategy during a game you should always keep in mind the influence of each piece on the squares it can reach, in order to devise your moves. I think for a human player it could be difficult to consider the influence of a pawn on the adjacent cubes. But imagine keeping in mind the influences of 512 of your pawns, plus 512 of your opponent's; then as many big pieces for you and the opponent. All of them have complex influences in the cube where they are and in the other cubes along the big raw.

I think, my friends, that this is a theoretical proposal; but if we hope people will be playing it, we have to wait a bit of time. Let's have thousands of people who want to engage in three-dimensional reasoning; or millions maybe. Afterwards, to the more daring of them we could propose this.

If the proposal applies to the thinking machines of the future, then it could be a very good one.

Yep, definitely something for the super nerds like ourselves out there xd.

actually my best trick for bishop is: just forget it moves diagonally!

it moves the same number of steps in two directions, or it moves like two rooks in different directions with same number of steps.

imagine 9D chess XD its a cube of cuecubes

btw, does someone know of a website with Hebrew keyboard?

In my Apple computer I can open keyboard preferences, then I click + and there are 3 Hebrew keyboards I can choose from

Some secret info for the few super nerds like ourselves. After a good thinking, I've devised a way to perform castling in a 4D-Granchess type space as proposed in this thread.

Obviously, this is only a hypothetical exercise; as we don't expect many people to be interested in playing such a game any time soon.

However my belief is that a complete castling could be made in 3 moves, corresponding to the 3 axes x, z and t. But depending on the order in which the three moves are performed, we can end up with as many as 48 different types of full castling.

Anyone interested in the revelation of this secret? I can explain hyper-castling in full detail.

I didn't expect to get so much enthusiasm about Castling in 4D.

Well, after a bit of daydreaming, I came to the conclusion that the fourth dimension can rightly represent Time. It's obviously only a representation of time, just as the other three dimensions represent space. It too is in fact a spatial representation, but perhaps useful for practising Einstein's space-time reality.

So we have 8 big cubes, each of them like the cubeboard in 3D-GrandChess. They are like 8 positions in time for the same cube. I imagined them in a row; but we can also see them in a file, in a stack or in some other curious and imaginative arrangements: e.g. with their little cubes inserted into each other like matryoshka dolls: a cubic collection of 512 matryoshka sets made of 8 small cubes each.

Ok, cube 1 is the origin of time; cube 8 is the end of time. In addition to the other three dimensions, pieces can threaten each other through this fourth dimension, i.e. through time. A piece can be threatened from two directions in time: from an earlier cube or a later one: i.e. from past or future. Unless it is in cubes 1 or 8, where possible threats only can come from one direction: from future in cube 1 and from past in cube 8. Thus castling in the fourth dimension means transferring the king safely to cube 1 or 8, where his back is defended; so I'm introducing a third type of castling move: the Transfer Castling, where the white king is transferred from εe1-5 to εe1-1 or εe1-8, provided that all boxes εe1 are empty in between. The piece located in εe1-1 or εe1-8 is a queen and in the same move it goes to εe1-2 or εe1-7 respectively. So yes my friends we do castling with a queen instead of a rook.

After that, we have our king at the center of white's side in cube 1 or 8. So we can perform regular pre-castling and castling as in 3D-GrandChess; where castling is also the same as in chess, with the only difference that the king has been moved before for transfer castling and pre-castling. Same reasoning for the black king.

But we can also change the order of the three castling moves at the player's convenience. So we might have a) transfer  - pre-castling - castling as seen above, or b) pre-castling - transfer  - castling, or  c) pre-castling - castling - transfer. 

Both pre-castling and castling can be performed on the x or z axes, depending on which is done first; transfer on the time axis can be performed before, in between or after the other two; Therefore, it may involve a) a queen, b) a rook, c) a knight, as well as the king. 

These pieces must never have been moved before, and all other castling rules must be observed; e.g. the king must not pass over threatened boxes.

if a player succeeds in making all three moves during the game, he gets a complete castling, with the king finally positioned near a corner at the origin or the end of time, in a very well protected and defensible position. 

As we may have 4 possible pre-castlings, 2 castlings and 2 transfers for each of them, and 3 different sequences of the three moves, it follows that a complete castling in 4 dimensions can take place in 48 different ways.

So much reasoning only to explain one move. It'll be useful after we'll have millions of people playing 3D-GrandChess, maybe.

Interesting. Can you explain how pre castling works?

Pre-castling in 3D-GrandChess is the only rule that differs from those in Chess and is explained in video n. 3 (special moves) at this address:

https://www.youtube.com/channel/UCOeIXNTxzW_IC1T1Mlxolew/videos

In practice in 3D the king stands in the middle of its face of the cube: white king in εe1, black king in εe8. they need one pre-castling move to go to an edge of the chessboard as the king stays in chess. So it reaches one of the 4 boxes on the edge, moving three or four steps straight like a rook, thus taking the place of the rook on the edge. These rooks are for White in εa1, εh1, αe1, θe1; for Black in εa8, εh8, αe8, θe8. The chosen rook moves to the next box in the same line.

All boxes in between must be empty, king and rook must never have been moved before and all 4 or 5 boxes visited by the king must be free of threats by the opponent.

After pre-castling, the position of the king is like in chess; so at a later move it can castle like in chess with one of the two rooks in the corners next to him. That could be long or short castling as in chess. Only difference: the king has been previously moved, but only to perform pre-castling.

Having 4 types of different pre-castlings and two possible castlings for each of them, the total number of complete castlings in 3D GrandChess is 8 for each player.

In 4D we can have the same pre-castling, the normal chess castling and its specific transfer (time shift) castling. So if d is the number of dimensions, a complete castling always needs d-1 moves.