Let's talk about Pieces

In a play like Chess a Piece if defined by how it moves; i.e. what part of the board it can reach, which squares and how many of them. Ultimately, what defines a piece is its mobility

Just to be precise these are the characteristics distinguishing the figures, or types of pieces. Then we may have a number of copies of any figure and those are the Pieces.

Greater mobility gives the pieces more opportunities; therefore it makes them stronger for the purpose of the game. We all know that the Queen is the strongest figure in chess because it's the one with greatest mobility; while the Pawn is the least strong because its mobility is the lowest one.

Adding one more dimension, we multiply the spaces available for pieces and we also open up new lines and directions that they can use for their mobility. Because of the increased mobility they are hence defined differently: they are in fact Substantially Different Pieces, significantly More Powerful than those we know in chess.

In addition to knowing how to orient ourselves and operate in the three dimensions, not at all an easy task for those used to play chess, we must also learn to handle different and much more powerful types of pieces.

This will require specific training for the capabilities of a human mind willing to engage in these types of games. But this is the same kind of capabilities needed to govern the complexity of the world, today and in the near future.

Here is a method I've just invented for describing piece movement within the 3D matrix (board) that you may utalise or find convenient for describing piece movement. Feel free to improve upon this method.

Vector

Using the notation: (x, y, z), where x describes the horizontal axis (equivalent to the alphabetic "files" in standard chess), y describes the depth axis (equivalent to the numeric "ranks" in standard chess), and z describes the vertical axis (described using greek letters in conventional 3D notation). We can describe a piece's movement using classical vector notation (albeit the movement is described within a 3-dimensional matrix).

Many pieces can move in different ways, so to keep things as thoroughly identifiable as possible, the variables that a piece can move are described explicitly in the direction in which they can move. For example, if a piece's movement were described with the following:

(1, 0, 0)

This would mean that the piece would be able to move 1 space horizontally (left or right) upon movement.

To describe a piece with move versatility and to include other variables of movement direction we would use the & symbol to string movement capabilities together.

Example:

(1, 0, 0) & (0, 1, 0) & (0, 0, 1)

This describes a piece that can move one space horizontally (left or right), one space depthwardly (forward or backward) and one space vertically (up or down).

We use the r symbol to indicate a rider, this may appear within brackets or outside a bracket depending on necessity.

(r, 0, 0) & (0, r, 0) & (0, 0, r),

r(1, 0, 0) & r(0, 1, 0) & r(0, 0, 1),

Both of the descriptions above describe the movement of a Rook in 3D-Grandchess.

Capture capabilities are generally described as the same for movement. However, if the piece can capture in a way that it cannot usually move, we describe this "capture bracket" with the symbol c. If the piece can move in a particular but cannot capture as such, then we describe this movement with the symbol m. Here is a description of how a pawn in 3D-Grandchess operates (excluding en passent, promotion and 2 square first move as I haven't thought that far ahead yet):

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

In the above example we see use of the symbol F. This refers to the fact that the movement/capture can only be completed "forward" in the "y" direction.

To describe movement that is only specific to a particular direction we can use the following:

F = forward
B = backward
L = left
R = right
U = up
D = down

FL = forward-left
FR = forward-right
FU = forward-up
FD = forward-down
BL = back-left
BR = back-right
BU = back-up
BD = back-down
LU = left-up
LD = left-down
RU = right-up
RD = right-down

FLU = forward-left-up
FLD = forward-left-down
FRU = forward-right-up
FRD = forward-right-down
BLU = back-left-up
BLD = back-left-down
BRU = back-right-up
BRD = back-right-down

Using this system we can describe the movement of many pieces in 3D chess boards with clarity. The pieces in 3D grandchess can thus be described as follows (except the pawn in detail and castling stuff because: hard):

King (without castling)
(1, 0, 0) & (0, 1, 0) & (0, 0, 1) & (1, 1, 0) & (1, 0, 1) & (0, 1, 1)

Queen
r(1, 0, 0) & r(0, 1, 0) & r(0, 0, 1) & r(1, 1, 0) & r(1, 0, 1) & r(0, 1, 1)

Rook
r(1, 0, 0) & r(0, 1, 0) & r(0, 0, 1)

Bishop
r(1, 1, 0) & r(1, 0, 1) & r(0, 1, 1)

Knight
(1, 2, 0) & (2, 1, 0) & (1, 0, 2) & (2, 0, 1) & (0, 1, 2) & (0, 2, 1)

Pawn (without en passent and first move allowing for 2 square movement)
mF(0, 1, 0) & cF(1, 1, 0) & cF(0, 1, 1)

Here's a piece suggestion:

Interdimensional-Bishop

r(1, 1, 1)

Here's another:

Super Queen
r(1, 0, 0) & r(0, 1, 0) & r(0, 0, 1) & r(1, 1, 0) & r(1, 0, 1) & r(0, 1, 1) & r(1, 1, 1)

yes, & is used for the boolean AND, OR would be more accurate. I was just being lazy and wanting something shorthand, the letter R I have used to indicate both rider (lowercase) and directional Right (uppercase) so was trying to think of something that wasn't a letter so as to differentiate it in the string description. Perhaps the symbol + might be more appropriate for its boolean associations, but perhaps many might be confused to think that this implied the move could be made "this as well as that" "in the same move". I quite like + actually lol

"Pawn (without en passent and first move allowing for 2 square movement)
mF(0, 1, 0) & cF(1, 1, 0) & cF(1, 0, 1) & cF(0, 1, 1)"

For the GrandPawn I think the following would be more accurate.    ?

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

You can use n instead of r, where n is an integer with  1 ≤ n ≤ 8

I would point out a possible source of confusion.

in geometry the vertical axis is traditionally indicated as the third, as you correctly do in your analysis. However this could create some confusion with move notation, where the vertical axis, the one marked with Greek letters, is the first.

Maybe later we should think how to fix this.

Ah! You are certainly right about the pawn movement, this is my mistake.

As for notation and axis, this makes sense, and I have also come across this issue before in a game called minecraft.

In minecraft the axis are indicated thusly:

But of course my bias as a student of mathematics naturally prefers:

Rotated of course so that the x axis represents my West-East.

In terms of the vectors I have of course written it as (x, y, z). My opinion would be that the 3D chess notation is better to conform to conventions of mathematics than for maths to conform to chess notation. a1α

Chess notation and conventions as a whole are rather messy and need a lot of tidying up in my opinion (but that's not something a lot of people will support I don't believe).

Can I build some 3D Images so we can understant better the pieces movements?

@Hitsar_Pride Don't need to ask bud

Ok, thanks

Correct me if I am wrong

I think the knight moves like this:

Brown: squares where the knight moves or captures

Gray: square where the knight is, because I takes too long to make a knight in 3D

4 different colours for the squares is a little confusing to me; but yeah, okay; that's it

Bishop inside 5x5x5 board

Brown: squares where the bishop moves or captures

Gray: square where the bishop is

For some reason I cant send the Image of the rook movement.

Why? The button "add a picture" doesn't work?

If it's a png maybe try transforming it into a jpg. That could work, I suppose.

Thanks for the Tip

Rook: gray square

Hitsar, your images are really great. You can build pawn, queen and king's images, to complete the set.  ??