4 ms·

Well, strictly speaking, it has no dimensions as long as all possible states can be enumerated. To give example, the board could just as well be arranged in a

Well, strictly speaking, it has no dimensions as long as all possible states can be enumerated. 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?

Arnavion 6y ago
There's no "strictly speaking"s about it. https://news.ycombinator.com/item?id=24066521 https://news.ycombinator.com/item?id=24066521
- lmilcin 6y ago
  If you really insist, I can correct myself and you, too. https://en.wikipedia.org/wiki/Dimension 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 https://en.wikipedia.org/wiki/Countable_set) the game IS 1D.
  - Arnavion 6y ago
    Congratulations on having invented a definition that nobody finds useful. If you don't mind, the rest of us will continue treating the game as having four dimensions.
    - lmilcin 6y ago
      Many people find it useful. They are called mathematicians. 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.
  - 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.

Arnavion 6y ago

There's no "strictly speaking"s about it. https://news.ycombinator.com/item?id=24066521 https://news.ycombinator.com/item?id=24066521

lmilcin 6y ago
If you really insist, I can correct myself and you, too. https://en.wikipedia.org/wiki/Dimension 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 https://en.wikipedia.org/wiki/Countable_set) the game IS 1D.
- Arnavion 6y ago
  Congratulations on having invented a definition that nobody finds useful. If you don't mind, the rest of us will continue treating the game as having four dimensions.
  - lmilcin 6y ago
    Many people find it useful. They are called mathematicians. 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.
- 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.