Except that the game doesn't use three spatial dimensions.
To give example, the board could just as well be arranged in a single line with more complex, but equivalent rules. Is it really 2D if it can easily be brought to 1D?
https://en.wikipedia.org/wiki/Dimension
"In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it."
Since we can define an order and enumerate all possible gameplays (not just states -- all possible gameplays) from 1 to however many possible games there are, the game is only 1D.
Which means you can describe any game from start to end with a single number.
Your assumptions about dimensionality are completely incorrect. Just because something can move in different directions doesn't mean these are different dimensions.
Think about idealized ant moving on idealized string. Well, you could say it is only 1D because you can describe its position by giving the distance from selected origin. Now, even if you crumple the string into a ball, the ant might be able to travel in different directions in 3D space, but the movement is still 1D movement on the string that can be described with a single number -- distance from origin.
With regards to chess, you can reduce the game the same way. Let's start from easy to more complicated:
1. You can reduce the board to 1D (line all fields one after another, preserving their names, but giving them order that will let you describe position of a piece with a single number).
2. You can reduce entire game state to 1D. Since there is finite possible game states, you just need a function that will map the board to an integer, so that you can refer to all possible board states with an integer that uniquely identifies it.
3. Just as with #2, you can do the same with an entire gameplay. In this case a single integer can describe not just board state but entire course of the game from start to finish. The game is now 1D.
In general, if the state space of the game is finite or countably infinite (https://en.wikipedia.org/wiki/Countable_set) the game IS 1D.
You are disagreeing with the definition made by somebody else and yet you are defining dimensionality as you please.
I don't find it useful in the slightest.
When I want to describe your movement from point A to B I need to use coordinates that have four dimensions, 3 of space and 1 of time. From our point of view it is not possible to reduce those dimensions (ie. have less numbers that can unambiguously describe your position).
That is not the case of a chess game. In case of chess game you can not only have just one number to describe your position on board, but you can also have one number to describe entire state of the board or just one number to describe entire game, completely.
Number of dimensions is about irreducibility. Of course, you can describe your position on piece of paper with three dimensions. But does it mean that piece of paper is 3D? No, not really. It is embedded in 3D space but the position on the sheet of paper is only 2D. A sheet of paper is a surface where each point might have different 3D coordinates but you might reduce it to 2D when you notice that the coordinates are not really independent.
Chess pieces are 3D objects in real world but as somebody already mentioned, that does not make the game 3D. We say that because when we are talking about the game, we really only care about game states that are independent from physical representation (ie. physical board and pieces). Game played with people as pieces standing on painted tarmac is the same as game played on rendered pieces on computer.
When people refer to 3D Chess they might refer to the fact that this is rendered in 3D on a computer. When the author named his game 5D he might want to refer to two additional dimensions of time.
But when you really want to strictly refer to game state space it is really just 1D and that's what I surmised after saying "strictly speaking". It is not useful to talk to people who may lack math skills that the game is 1D, but this is "strictly speaking" correct description of dimensionality.
If you go into argument about dimensionality you must understand what kind of definition the other person has. If you want to disagree it does not make sense to do it by inventing your own definition.
I personally agree with the description 5D because it points out to the feature of the game (ie. concurrent timelines) which in normal world would be described as fifth dimension.
My question about dimensionality would be about the decomposition of the 2D coordinate of the chessboard into a 1D state space. First of all, all games with discrete positions would be 1D games, correct? Even the 5D chess game, since time and timelines can also be encoded into state space. This would also seem to apply to every computer game as well, since the state space can be taken to be the snapshot of any data at a certain time step in the computer. And finally, if we take that to the limit, as we approach infinitely small time steps, the state space would get infinitely less granular, and we could map any N-dimensional space into 1D. That doesn't seem right, where is the fallacy in this line of reasoning?
> Since we can define an order and enumerate all possible gameplays (not just states -- all possible gameplays) from 1 to however many possible games there are, the game is only 1D.
This argument only makes sense if you stop reading the wikipedia page after the first sentence. The rest of the page talks about coordinates being properties that can change independently of other properties of the system, and other things that contradict what you're arguing.
E.g.:
> 1. You can reduce the board to 1D (line all fields one after another, preserving their names, but giving them order that will let you describe position of a piece with a single number).
A mapping like that would be a bijection of a chessboard, but would not be homeomorphic to one. That is, points that are adjacent in one mapping can be distant in the other, so there's no reason to assume they have the same dimensionality. (To the extent that a set of discrete points has dimensionality, anyway.)
Game rules do not specify how high the knight can jump and whether two pieces standing close to each other can block the knight from jumping.