A question for mathematicians...

 Why, what a useful addition to the discussion.

Then you have a rather fundamental problem, since a triangle (i.e.: the point of your question) only exists on a 2-D plane.

ErrantDeeds: I like this topic, by the way.  Apparently so do a lot of other people.  I haven't read every post on this thread, but I'll chip in anyway.

I'm not sure you can talk about games being played in a certain number of dimensions.  (I also think that if you can, the number of dimensions must necessarily be a whole number.)

At each point in chess a player has x choices available.  Whether he moves his bishop seven squares or one might be two of those choices, but the distance a piece moves is not germane to the outcome of the game.  Time, as you've said, can be measured in tempi, but I don't think there is any reason for measuring distance as literally as you want to measure it. 

Imagine a chessboard where some files are 'narrower' than others, some ranks are 'thicker' than others.  This does not change the nature of the game at all, even though it changes literal distances.  If geometrical distance were relevant to the nature of the game, then the game would be different if we monkeyed with such things.

I think that if you want to measure distance in chess, it ultimately has to come back to the time it takes for each piece to get to a square.  So, the distance between one square and another is a different distance for each piece.  a1 to f3, on an open board, for instance, is two moves travel away for a rook or queen, five for a king, three for a knight, and an infinite distance apart for a bishop, who can never get from a light square to a dark square.  This all depends on the position as well, of course.

Despite all this talk of distance, in the end I think chess is two-dimensional in physical space, and one dimensional in the abstract.  There are only three possible outcomes of a game, and they are lined up in a row neatly on one dimension: 0, .5, and 1.  Chess is just the maximization of this one-dimensional outcome for either side.

Sorry if a lot of this has been said before.

Minimal changes necessary to turn it into a bonafide 2-d game I guess:

1) Push each successive row  to the right by  half the width of a square.  This will make the distance between the midpoints of any two adjacent squares the same. 

2) A bishop starting on a given color has to stop on that color as well.  This new rule is necessary as diagonals in one direction can be multicolored now.

3)Really the movement of all the other pieces will be the same but translated throught the new board layout. So for example, now there are not columns as such, so a rook has to zigzag from white to black  when not moving on a row  (and thus a pawn zigzags as well).  Queens can move like the new bishop as well, and a king as always can move to any adjacent square.

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edit:

3) is incorrect;  rooks, etc would not zigzag and the new board layout would make queens and rooks indistinguishable

why are you comparing it with MATHEMATICS?

ErrantDeeds: First of all, nice thread. I love it when something like this comes up. Secondly, well done for arguing your case. I'm pretty sure you know you're wrong (you pretty much admitted it back on page 2!) but it's always fun to see people scramble around, trying to disprove you in 101 different ways.

Anyway, at this moment in time, I have just one question for you:

How many 'squares' does a knight move, and how would this be represented on the 1.5 dimensional plane? (Which, as we all know, is impossible, but let's go along with it anyway! )

The Knight is a leaper with a distance SQRT 5. (sorry, don't know how to type a square root symbol)

That's the real answer, now I want ErrantDeed's made-up and hopefully incredibly entertaining explanation. 

Sorry, I cant compete....

Well obviously that is just blatantly wrong. I mean ONE SQUARE IS THE SMALLEST UNIT OF MEASURE ON A CHESS BOARD. So it is impossible to travel non-integer amounts. Therefore the knight travels through the 5th dimension, gets some Dairy Queen, and lands on his destination. 

 Ha! The KNight may well be the achilles heel of the whole premise. Great posts people. Unfortunately, I've ran out of time tonight, But I shall return tomorrow with my thinking cap

Regardless of what you say the chess board is only measured in length and width (excluding depth). The vertical thing isn't a dimension; you can only have a dimension of length, width, and depth (excluding the string theory but we won't get into that). You can't have a .5 dimension - you can either measure it or you can't - there is no in between idea.

"How can a triangle have sides of the same legth in two dimensions?"

why would this be impossible? If a triangle is equallateral it is still 2-d not 1.5-d. You have to remember dimensions are just ways of being able to measure something which means the chess board is 2 dimensioned.

I think that errantdeeds would argue that a knight moves two squares: once orthogonally, once diagonally. 

ED, do the black pieces make a circle of radius three around the white king?

yes. for those geometrically challenged, a circle is defined as "all point on the perimeter are equidistant from the center".

though there will probably be fights over the concept of distance...again.

you COULD suggest that polar coordinates are used in chess (angle and radius), rather than cartesian plane coordinantes (i.e., x and y coords)

by the way, i REALLY like this thread. people who we expect to get mad when we aren't convinced they MUST be right (read: rainbowrising, who i DO support in his battle against Demetrios the Inept) are here, thoughtful people are here...wonderful!

My god, that is indeed a circle!

Bloody fantastic posts people. General relativity, Einstein, twisted dimesions... Great stuff.

Reading back through the posts, the general consensus appears to be that, even in the abstract, a non-integer dimension is not possible, or at least not appropriate in these circumstances.

Still, I shall cling to my hypothesis like it were my dying dog... 1 man, against the tide... some called him a maverick... some called him insane...

surely chess is played in three dimensions, and not on planes, so triangle properties do not aplly.

you mean 186,200 miles per second, which is indeed equal to steven^2's 300,000,000 meters per second

http://www.thechessworld.com/chess_wallpaper/set4/images/chess-globe-v3_jpg.jpg

interesting

In mathematics, metrics other than the Euclidean metric are often used. One metric on the 2-dimensional plane (think chessboard) is given by the maximum of the vertical and horizontal distances. In this metric a "circle" (all points at a given distance from a given point) is square, just like on the chessboard in fiver's post #124.