A question for mathematicians...

 Perfect sense, and i see what your getting at. Each piece is allowed only finite movement per tempi. As I said, take the Bishop. It has a range of 'speed' of between 1 and 7 'squares per tempi'. In any given instance, the maximum number of squares it could move is seven, assuming it were at the corner of the board and had a clear diagonal ahead of it. If a chess board were 20x20 squares, a bishop would have a maximum speed of 20 squares per tempi, depending on its starting location. The finite distance, and therefore speed, is a consequence of the limited space on a board. I stress though, that this is not actual speed, e.g. the speed that you could physically move a piece with your hand, but tempi-based speed, which exists only in the abstract.

Your use of 'dimensions' puzzles me.  Do you live in a world of 3 dimensions + time for a 4th?  Your logic about relating dimensions to speed (# squares moved per tempi) is confusing you.  Does an airplane that moves faster than you operate in a different number of directions?  Chess has two dimensions, + time; then you have as stated before, the up-down dimension, and left-right dimension.  Diagonal is not a new dimension.  A piece can stay on the same square too, if a different piece is moved, so that's the time dimension. 

Good luck confusing yourself more. You're quite caught up in maintaining that your 1.5 d idea is correct.  I can see your argument, sort of, but I see it in the light of you conflating the usage of the concept of dimensions.  :)

 Ha! No doubt. Well, it was fun while it lasted...

 Hmm.. what about fractal dimensions for instance!?? .. They are not integers..

http://en.wikipedia.org/wiki/Manhattan_distance#Measures_of_distances_in_chess

I'm not a mathematician, but this seems to be about different definition of distance, not dimension.

in no way does it make it 1.5d

Chess does not follow the axioms of Euclidean geometry or algebraic geometry.

You can create a set of axioms which follows chess rules and call it "Chess Geometry", but I don't know how you will define time, speed or dimensions in the new set of axioms. I also don't see any need to define these things as they are quite useless, both to the progress of chess and to the progress of mathematics.

This doesn't have to do with the chess side of this discussion, but some spaces (fractal sets, for instance) have non-integer dimension.  http://en.wikipedia.org/wiki/Hausdorff_dimension

As far as a chess board goes, rather than considering a number of dimensions other than 2, it makes more sense to think of it as two dimensional with a different metric.  Metrics are functions that measure (define) distance.  The distance between two squares (as far as the king's movement is concerned) is given by the maximum of the horizontal and vertical distances between those squares.

:) A lively discussion indeed! I like the idea of metrics. Perhaps you can't have a non-integer dimension, at least in this regard.

Good discussion though. Beats the pants off of the continual rants about cheating/resignation etiquette etc.

There is a finite number of squares. You only need a single number to refer to a square (you can simply number them 1-64). Therefore, if you want to talk about dimensionality at all, chess must be 1-dimensional.

That's in a vector space. You can't seriously claim that a chess board constitutes a vector space.

 Maybe so... my brain is starting to hurt. If chess were one dimensional, that would satisfy the diagonal/row similarity paradox, but raises a further problem. One dimension as described could only be traversed in sequence, i.e. 1,2,3,4...63,64. However, pieces are not limited to this range of movement. If it were 1d, pieces would jump through wormholes!

  I'll keep you posted!

 Thank you ScarBlac!!

Hey, that's exactly what a knight already does in the 2D representation, so it's not that strange.

You can view chess this way -- I once wrote a chess program that just used squares 0 to 63, and a set of tables that said a king on square 1 could move to squares 0, 2, 7, 8 and 9, et cetera. Why bother with a 2D array? It's not actually relevant to the game, and it makes the programming more complicated.

But I don't think you can really talk about dimensionality of a finite set. At least it's just wrong to state that the set can only be traversed in one direction. The set doesn't even HAVE a defined direction! Instead of naming the squares "0" to "63", I could just as well have named each square after a different Muppet.

Scarblac, your post about only needing the numbers 1-64 to label the chessboard reminded me of the space-filling curve, a 1-dimensional curve that entirely fills a 2-dimensional space.  It's not exactly relevant, but interesting nonetheless.

http://en.wikipedia.org/wiki/Space_filling_curve

Ok.

For a set Vto be called a vector space,

1) You need to have an addition operator +, defined so that a + bin V for every a and b in V, and a + b = b + a for all a and b; and a + b = a + c if and only if b = c (I might have forgotten something, doesn't matter here)

2) There must be a unique zero vector (named 0) in V, so that a + 0 = 0 + a = a for all a in V;

3) There must be a multiplication defined so that vectors can be multiplied by scalars from some field (usually the real numbers), so that 0 * a = 0, f * a in V for all f in the field and a in V -- et cetera.

So you're claiming the chess board is a vector space, and you're asking mathematicians. Then you need to show what the zero square is, and how the multiplication and addition work. Otherwise, talk of vectors is just not applicable.

Well for the chess board to be a vector field, you would have to be able to assign a vector to every point. I guess that is possible if you simply view the board as a standard coordinate system. But doesn't it loose its relevance for the chess game then? maybe we should view the chess board as a 'puntured' plane then.. sort of like R^2_+/N^2 where the _+ means that we are looking at the part of the plane where both coordinates have positive values.. But I guess this is topologically equivalent to a torus(!?) .. hehe okay, now I am getting a head-ache...

And the reason you need to is of course that without addition and multiplication, the "linearly independent" of your quote makes no sense.

No, because then there would have to be a square (pi, e), and there isn't.