A question for mathematicians...

Now, this may sound like an obvious question, but I'm not sure it is:

Q: In how many dimensions is the game of chess played?

A: 2?

Let's see...

Consider a two dimensional triangle abc, thus:

It is obvious that the hypotenuse b is of different length to sides a and c. This is a natural consequence of the geometry of a triangle in two dimensional space.

However, consider the chessboard:

For the white King to move to e3, along the base of the triangle, takes the same amount of time as to move to e6, along the hypotenuse. As the King would be travelling at the same speed in each direction, i.e. one square per tempi, 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.

That made sense to me. Should I worry?

 Nope, it shows what a bright chap you are.

"As the King would be travelling at the same speed in each direction, i.e. one square per tempi, it follows that the distances must be equal."

Here is your mistake.  The king travels "faster" along the diagonals.  It still moves one square per move, but it is moving faster because the distance is greater. 

[edit - just in case you were serious :) ]

The king is still moving further in measurable distance when moving diagonally, as the diagonal of each square is greater than the side. The problem is trying to define speed as movement in terms of squares, which doesn't really make sense. It would be something like describing the speed of a moving car in terms of states crossed per day.

If you want to assign 'speed' to the pieces, you would simply say that the king moves faster in a diagonal direction than in an orthoganal direction. I.e., when moving orgthonally, the king moves at x/tempo, while when moving diagonally, he moves at x*sqrt(2)/tempo.

Sorry if your post was meant as a joke. I didn't read it as one.

The diagonal of a square is not eqaual in length to the vertical or horizontal sides, it is longer. 

This actually does have practical implication in "fairy" problems, where the length of moves are sometimes relevant.

Eg, in a MAXIMUMMER, Black must make the geometrically longest move. The Unit of Measure is the distance between the centres of two orthogonally adjoining squares. For example, a move from a1 to a8 measures 7 (sqrt of 49) and is shorte tham a move from a1 to f6 (sqrt of 50).

Check ths out, great fun...

http://www.chess.com/forum/view/more-puzzles/maximummer

And chess is more than moving kings around. A bishop would move to any available square in one leap; so, to continue in your abstract sort of way, they are all the same distance away?

My conclusion is that chess is not played in 1.5 dimensions and we're kind of whipping a squashed elephant or whatever the phrase is.

very interesting though, keep up the pondering...

 It is. And which postulate?

I think I may have mis stated what I meant. The fact that a square is physically longer accross it's diagonal than its sides is irrelavent. If you take my postulate as true, then the board that you play on is a two dimensional representation of the actual game, in the same way that a cube drawn on a piece of paper is a two dimensional representation of a 3d object. We know that the angles in a cube are all ninety degrees, but drawn on a 2-d surface, many of these angles are distorted to, say, 45 degrees. Is it not conceivable that in 2d space, the fact that the diagonal of a square is longer than the sides is a similar 2d distortion of a 1.5d object? In 1.5 dimensions, are the diagonal and side the same length?

Many say that a King moves faster along the diagonal, which is of course true. In 2 dimensions! In 1.5, he moves the same speed! The only measure of time that means anything at this level of abstraction is the tempi, and a king moves at a rate of '1-square-per-tempi'. A bishop has a range of speed of between 1 and 7 squares per tempi, i.e. the maximum range of movement in one go. The physical distance and speed in a 3d world are irrelavant! I say 1.5 dimensions, because it would be necessary for the angle between a row and a diagonal to be 90 degrees, and the angle between a row and a column to be 180 degrees. Thus one of the dimensions is halved.

Warp those minds people.

ED.

 Wrong sir. A traveller in a state can remain there with other people and move around within it. Furthermore, all states are of different size and take different amounts of time to traverse. A square on the chessboard can only be occupied once. The square is the smallest discreet unit of space and time. The next time that you are faced with a pawn race to win a game, tell me then that speed of movement in terms of squares doesn't make sense!

 Why? Says who?

You can move up-down, and left-right movement = 2 dimensions.

The speed or amount of time it takes to complete a tempi has no correlation on the fact that the pieces are moving on a 2 dimensional plane. 

Hausdorff would argue otherwise

Bondiggity - What I am saying is that the pieces may be physically moving in two dimensions, but that is merely a convenient representation.

I'm quite oddly passionate about this. I'm possibly a bit wierd. Can someone tell me I'm not mad please. Surely someone can see it as well!

No, its completely irrelevant. You are making the assumption that a piece moves x/8  per tempi, where x is the length of the board. This is an unreasonable assumption, as the rules state the the piece can move farther if moving diagonally. 

Dimensions are merely a way to describe an object in space. In none of the definitions I have looked for does it say it need be an integer.

 What's completely irrelevant?

 And, mathematically, can two linearly independant vectors exist in a half dimension? Your definition appears not to preclude the possibility.

I mean I understand what you're saying. But it seems that it requires some assumptions that just aren't true. You have to really assume that each piece is only allowed to move a finite distance in a tempi. But this is obviously not true under the rules of chess. If this was the case, then the piece obviously wouldn't be able to move diagonally within the rules of 2-D movement, but all this seems irrelevant since the underlying assumption of fixed distance movement isn't the case.

Am I making sense or just rambling?

 The vector is not halved. The dimension it exits in is!