Chess is played in two dimensions. This is more obvious in Chinese chess, which is played with discs rather than upright pieces. It is also the reason we can play chess on a flat screen. It would be nice to think that chess is three-dimensional because the knights can hop over things, but it is just as easy to define the knight's move as passing through things.
A question for mathematicians...
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Ricardo_Morro wrote:
Chess is played in two dimensions. This is more obvious in Chinese chess, which is played with discs rather than upright pieces. It is also the reason we can play chess on a flat screen. It would be nice to think that chess is three-dimensional because the knights can hop over things, but it is just as easy to define the knight's move as passing through things.
I was about to say chinese chess has a knight too, but i forgot it can't pass through things; rather, it gets blocked if a piece is in front of it =)
That's true, but the cannons in Xiangqi (Chinese chess) do jump over pieces (in fact, they are required to.)
For the white King to move ... takes the same amount of time as to move ... along the hypotenuse ... it follows that the distances must be equal. Diagonals = Horizontals = Verticals!How can a triangle have sides of the same legth in two dimensions?
My conclusion is that a game of chess on a chess board, in an abstract sort of way, is played in 1.5 dimensions. However, I am not a mathematician, and I am eager to be proved wrong.
ED.
In a way, you are wrong. You are travelling dimensionally faster while going on a diagonal (the squareroot of 2, for squares). In another way, you are right: You ARE moving only 1 square per turn. So now the question is, would chess be eaier played on a natural circle, or is it already (remember what a circle is)?
I'm not really sure if I should be responding to Paul or not, since he doesn't actually have any points. However, I suppose I will try to clarify.
1. Math combined with physics is a simplified model of reality, as Richard Feynman himself said. For example, when you toss a ball in the air, do you think Mother Nature whips out her calculator and says to herself, "Let me think, the acceleration of gravity is 9.8 m/s squared, therefore..." No, of course not. That's how humans do it, because it's easy and we get the same answer as real life to a sufficient precision. Equations don't explain reality, they simply model reality. The equations exist because that's what we observe reality to be. As Feynman famously said, asking "why" is a futile question, "I don't tell you why it is. All I know is that if you use these equations you will be able to predict what will happen in the future." Gravity, for example, is inversely proportional to the square of the distance between two objects. Why? Who knows. But that's what it's been measured to be.
Of course, that's really physics, not math. Math by itself is nothing more than a series of axioms and logical deductions following from those axioms. In general, we choose which axioms to use based on what we observe in the universe around us. Again, no explanation, simply a nice model.
2. Of course there is a historical reason why the pieces are the height that they are, but if you've ever played chess on a computer, where each piece is exactly the same size, you know that the height of the pieces has no bearing on the outcome of the game.
3. Of course chess is played over a certain period of time. What's your point? A mate in five moves will always be a mate in five moves regardless of what time it is.
4. What "judgment" have I rendered?
5. I guess you don't know how to count, because you skipped directly to 6.
6. You don't think describing positions on a chessboard in terms of radians would be weird? Huh. To each his own.
To think that you have "disproved" anything really shows a lack of understanding of what a proof is. I'm sure you could teach math for free, but I pity the fool who would have to sit there and listen.
The only point I was trying to make is that chess is based on rules. Rules which have nothing to do with dimensions. That's why the question the original poster asked is misleading. We like to represent the rules of chess using two dimensions, but that's not the same thing. As the original poster pointed out, one consequence of using two dimensions to represent the rules of chess is that you have strange things such as the king moving at different speeds depending on which direction he's going, diagonally or horizontally or vertically.
Nov 3, 2009
0
#167
pbrocoum wrote:
I'm not really sure if I should be responding to Paul or not, since he doesn't actually have any points. However, I suppose I will try to clarify.
1. Math combined with physics is a simplified model of reality, as Richard Feynman himself said. For example, when you toss a ball in the air, do you think Mother Nature whips out her calculator and says to herself, "Let me think, the acceleration of gravity is 9.8 m/s squared, therefore..." No, of course not. That's how humans do it, because it's easy and we get the same answer as real life to a sufficient precision. Equations don't explain reality, they simply model reality. The equations exist because that's what we observe reality to be. As Feynman famously said, asking "why" is a futile question, "I don't tell you why it is. All I know is that if you use these equations you will be able to predict what will happen in the future." Gravity, for example, is inversely proportional to the square of the distance between two objects. Why? Who knows. But that's what it's been measured to be.
Of course, that's really physics, not math. Math by itself is nothing more than a series of axioms and logical deductions following from those axioms. In general, we choose which axioms to use based on what we observe in the universe around us. Again, no explanation, simply a nice model.
2. Of course there is a historical reason why the pieces are the height that they are, but if you've ever played chess on a computer, where each piece is exactly the same size, you know that the height of the pieces has no bearing on the outcome of the game.
3. Of course chess is played over a certain period of time. What's your point? A mate in five moves will always be a mate in five moves regardless of what time it is.
4. What "judgment" have I rendered?
5. I guess you don't know how to count, because you skipped directly to 6.
6. You don't think describing positions on a chessboard in terms of radians would be weird? Huh. To each his own.
To think that you have "disproved" anything really shows a lack of understanding of what a proof is. I'm sure you could teach math for free, but I pity the fool who would have to sit there and listen.
The only point I was trying to make is that chess is based on rules. Rules which have nothing to do with dimensions. That's why the question the original poster asked is misleading. We like to represent the rules of chess using two dimensions, but that's not the same thing. As the original poster pointed out, one consequence of using two dimensions to represent the rules of chess is that you have strange things such as the king moving at different speeds depending on which direction he's going, diagonally or horizontally or vertically.
I'd like to say I liked the way you presented your information. Very smart, yet easy to follow. I myself previously was stating that nowhere in the rules of chess does it say a piece has to move at constant speed, but in no way could do it as eloquently as you.
As far as Paul goes, I wouldn't take anything he says to heart. He is hypercritical of perfectly good ideas and thoughts, and usually his criticism is just flawed in logic.
Also I thoroughly enjoyed your video on conditional probability. Im studying probability this year and it was a refreshing way to present it (even if I already knew all the concepts).
LOL This whole discussion is SOOOOO far over my head!
I have enjoyed to follow you guys exposing the ultimate truth of chess and would like to add my view while this thread is still alive.
I agree with pbrocoum that 3d boards and 2d diagrams are just abstract ways to interpret set of rules in dimensions we live in. Chess could be played as well on a surface of soccer ball or perhaps with some unseen insect that behaves according to adapted set of rules when stimulated.
Just like our 4d space consists of 1 dimension of time and 3 dimensions of distance I would like to think that chess space has 2 dimensions of time. Tempo being one dimension and another being the time we live in. I’m saying latter because time on a chess clock has an effect at least on a quality of game, which means that our time dimension interacts with chess space and therefore can not be departed from it.
I’m not able to tell how many other dimensions are in chess. One could say that you can reduce 2 dimensions of board to 1 by numbering squares from 1 to 64. If that holds, doesn’t it mean that you can take any 2d plane, number every point of it and claim that it is actually one dimensional line? Perhaps it works with chess board but I wouldn’t try it with Mona Lisa, what ever that means.
If there are other than two dimensions of time they would be dimensions where 64 positions (“squares”) besides. Number of these dimensions would be defined by how 64 positions are arranged in relation to each other. It seems that we can’t get any information of their arrangement and won’t be able to know exact number of dimensions because we don’t live in chess space (at least yet, I’m actually starting to feel like Fischerized while reading this post).
Probably there are no other dimensions because after all chess space is in our minds …just 32(?) imaginary chess particles with 64(?) possible quantum states changing in time.
I would say that there are at least 2 dimensions in chess. I’m sorry ErrantDeeds.
Nov 4, 2009
0
#170
Definitely an interesting topic. I think that there's one concrete way to settle the minimum dimension of chess though.
Using the idea of the square as a unit of distance, if you take any 2x2 group of squares from a normal chessboard, they are all equidistant under that measurement.
There's a common proof involving Lebesgue Measure which shows that the maximum number of elements which can be equidistant is 2^n, where n is the dimension.
Since we have a set of 4 equidistant points, the dimension must be at least 2.
Granted I haven't looked at the proof enough to be sure it accounts for non-integer dimensional spaces, but since we have a finite (64) number of spaces involved, there's no way to arrive at a non-integer dimension.
Eiwob wrote:
ClintBeastfood wrote:
I’m not able to tell how many other dimensions are in chess. One could say that you can reduce 2 dimensions of board to 1 by numbering squares from 1 to 64. If that holds, doesn’t it mean that you can take any 2d plane, number every point of it and claim that it is actually one dimensional line? Perhaps it works with chess board but I wouldn’t try it with Mona Lisa, what ever that means.
I think there's a difference between a chessboard and a "normal" 2d plane. In a normal 2d-plane (and in a 1d-line) there's an infinite number of points, while on a chessboard, you can argue that there really are only 64 points. At least that's what I think pbrocoum means when he says that you could simplify the board to 1d.
We could get rid of infinite number of points by filling 2d-plane with numbered squares. Anyway, pbrocoum's point is clear and my comparision between chess board and any 2d plane was more or less provocative.
Yeh Eiwob, infinity numbers beginning from 1 - infinity. They'd be numbered, but you can't divide them really into anything but infinity.
Its wierd (my opinion), infinity divided by 5 = infinity, but its true! Think about it.
When i say divide in the 2nd line, i don't mean that. I mean that you know, "get rid of" infinity. To get rid of infinity, you attempt to subtract or divide. You can't make inifinity go into not infinity by doing that.
Nov 7, 2009
0
#175
Cosine wrote:
rubixcuber wrote:
Definitely an interesting topic. I think that there's one concrete way to settle the minimum dimension of chess though.
Using the idea of the square as a unit of distance, if you take any 2x2 group of squares from a normal chessboard, they are all equidistant under that measurement.
There's a common proof involving Lebesgue Measure which shows that the maximum number of elements which can be equidistant is 2^n, where n is the dimension.
Since we have a set of 4 equidistant points, the dimension must be at least 2.
Granted I haven't looked at the proof enough to be sure it accounts for non-integer dimensional spaces, but since we have a finite (64) number of spaces involved, there's no way to arrive at a non-integer dimension.
That doesn't seem to be right. If the dimension is 2 then that says that there can be 2^2 = 4 equidistant points but I don't see how 4 equidistant points can exist in a plane. I thought the maximum number of equidistant points was n + 1 where n is the dimension.
Isn't a circle an infinite number of equidistant points? And the same with a sphere?
bondiggity wrote:
Isn't a circle an infinite number of equidistant points? And the same with a sphere?
No, equidistant means all points are the same distance away from every other point.
Instead of seeking the dimensions of your chess plane, you should be looking for the manifold hat have these properties. You must still take into account that the chessboard isn't really an analytic object but a finite algebraic object.
Nov 7, 2009
0
#178
rooperi wrote:
bondiggity wrote:
Isn't a circle an infinite number of equidistant points? And the same with a sphere?
No, equidistant means all points are the same distance away from every other point.
Hmm, I just don't see how that works. Is this only applicable to subsets? I don't see how it works using the entire dimension.
See, I knew it was an interesting topic! This really got out of hand...
Nov 11, 2009
0
#180
Cosine wrote:
That doesn't seem to be right. If the dimension is 2 then that says that there can be 2^2 = 4 equidistant points but I don't see how 4 equidistant points can exist in a plane. I thought the maximum number of equidistant points was n + 1 where n is the dimension.
That number is an upper bound. There can be at most 2^n equidistant points in a space with dimension n, but there won't necessarily be that many. n+1 is generally considered to be the lower bound, but it isn't proven.
Here's a reference if you're interested:
http://konradswanepoel.wordpress.com/2008/04/17/equilateral-sets-in-normed-spaces-upper-bounds/
And on the lower bound:
http://konradswanepoel.wordpress.com/2008/05/23/equilateral-sets-in-normed-spaces-ii-lower-bounds/
varal wrote:
Instead of seeking the dimensions of your chess plane, you should be looking for the manifold hat have these properties. You must still take into account that the chessboard isn't really an analytic object but a finite algebraic object.
Did you see my post? You can create a normed vector space on the structure of a chessboard and then because of the equidistant squares, show that the dimension must be at least 2.
A given chess piece has properties that are completely independent of where it is on the board or at what time it is at that location. So the dimension of "piece" has to be considered as well. So let's say the relevant dimensions that must be considered are square, time and piece.