3 ms·

I think the difference is that, in the ant example, it would appear that the ant can teleport if you are looking at it from a 1D perspective, while in the 3D ba

by shard 6y ago

I think the difference is that, in the ant example, it would appear that the ant can teleport if you are looking at it from a 1D perspective, while in the 3D ball, the ant would be moving through space in a continuous fashion. Now since we know that ants cannot teleport, the 1D perspective does not align with reality. I think that's the distinction between being able to unroll certain movements from higher dimensions into lower dimensions. Now the question is, do we want to define chess movements as continuous or teleportation? If it is defined as teleportation from the beginning, then unrolling it into 1D would not change aspect of movement. But if we define it as continuous movements, then unrolling it into 1D would break that rule.

lmilcin 6y ago
When we use dimensions we are not talking about actions but about state space. Action implies time and time is just one of the dimensions. 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.
- shard 6y ago
  Ah, then when you were referring to the ant example, you are assuming the constraint that the ant is moving only along the string. You did not specify that constraint in your original phrasing, and I imagined it as -- if the string was rolled into a ball, the ant can travel in any direction on the ball. This would cause it to appear to teleport if observed from a 1D point of view. 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?

by shard 6y ago

lmilcin 6y ago

When we use dimensions we are not talking about actions but about state space. Action implies time and time is just one of the dimensions. 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.

shard 6y ago
Ah, then when you were referring to the ant example, you are assuming the constraint that the ant is moving only along the string. You did not specify that constraint in your original phrasing, and I imagined it as -- if the string was rolled into a ball, the ant can travel in any direction on the ball. This would cause it to appear to teleport if observed from a 1D point of view. 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?