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.
I does appear to refute my initial argument, and now my mind is blown...
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?
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.
I'm gong to move my bishop at the same speed if I move him one square as if I move him 3. I stress to you that constant speed with an undefined amount of time to complete a tempi is a more reasonable (2-D) way to view this game of ourss.
Ha! Magic. This is the kind of discussion I usually have at 4am after a big party, closely followed by an appraisal of the relative morality of the Empire in Star Wars.
ScarBlac, you're a genius. Am I correct in saying that a chessboard is does not contain vectors as there is no point that could reasonably be called the zero square? Or have I misunderstood?
Well, yes. That alone is sufficient to prove that it can't be a vector space.
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?
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.
I'm gong to move my bishop at the same speed if I move him one square as if I move him 3. I stress to you that constant speed with an undefined amount of time to complete a tempi is a more reasonable (2-D) way to view this game of ourss.
Surely, if, in one move, you move a bishop 3 squares, it has moved faster than if you move it one?
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?
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.
I'm gong to move my bishop at the same speed if I move him one square as if I move him 3. I stress to you that constant speed with an undefined amount of time to complete a tempi is a more reasonable (2-D) way to view this game of ourss.
Surely, if, in one move, you move a bishop 3 squares, it has moved faster than if you move it one?
Distance = (rate)(time)
You could have just taken 3 times as long to move it. Or 6 times as long and moved it half as fast. The point is that although tempi isn't a measurement of absolute time. You could take 3 seconds to move a bishop, or 1 second to move a bishop, in either case it is still one tempi.
This thread is absolutely ridiculous. To answer the OP's question, if the length of each square is 1 inch, the length of the diagonal is sqrt(2) inches. So the king travels farther along the diagonal than it does horizantally or vertical. And to answer the last question posed about moving a bishop three is much faster than one. It depends how long it takes to move the bishop three squares. If the velocity is constant, it is the same. However, if there is acceleration when moved, the bishop moves faster.
This thread is absolutely ridiculous. To answer the OP's question, if the length of each square is 1 inch, the length of the diagonal is sqrt(2) inches.
You're missing the point. It doesn't matter if a sqaure is a square inch or a square mile, A King moving can only move 1 square! The smallest unit of length on a chessboard is one square, regardless of what it actually measures!
Let's try it this way.
The definitions of distance within a curtain dimension are always determent by the point of view.
The King is allowed to move 1 square in any chosen direction, meaning the definition of distance is in squares (not yards, meters or miles) so the King moves always 1
If you walk a circle and arrives back to the starting point the distance is 1 lap, on the point of view the distance is 0 because it’s the same place.
This thread is absolutely ridiculous. To answer the OP's question, if the length of each square is 1 inch, the length of the diagonal is sqrt(2) inches.
You're missing the point. It doesn't matter if a sqaure is a square inch or a square mile, A King moving can only move 1 square! The smallest unit of length on a chessboard is one square, regardless of what it actually measures!
You two are just talking about different things. Killabeez is talking about the distances on the chessboard in terms of the physical world, and Errantdeeds is talking about abstract distances in the chess 'world'.
This thread is absolutely ridiculous. To answer the OP's question, if the length of each square is 1 inch, the length of the diagonal is sqrt(2) inches.
You're missing the point. It doesn't matter if a sqaure is a square inch or a square mile, A King moving can only move 1 square! The smallest unit of length on a chessboard is one square, regardless of what it actually measures!
You two are just talking about different things. Killabeez is talking about the distances on the chessboard in terms of the physical world, and Errantdeeds is talking about abstract distances in the chess 'world'.
Indeed i am sir, which is the foundation of the whole argument!
Hehe, except I already told you why it can't be so Errant!
Indeed! We'll have to agree to disagree.
This is actually a clever question. The key here that nobody has mentioned is that when the king travels on the diagonal only half of those squares (the portion
to the right and below the red line) is really part of the "fake" triangle.
In other words, it is not really a triangle but a series of stairs.
We all saw 3-D chess in the original Star Trek.
When you play over the board, you play the chess game in 4 dimensions, it is that simple.
You have a 3D board in front of you, width, height and length and you play the game over a time period, the 4th dimension, as in Einstein general equation.
Does time and dimensions contraction occur, well no naturally as we are not playing the game at the speed of light , which is 186,000 miles an hour or 7 times around the earth in 1 second.
I will here avoid to discuss this equation, not that I am not knowledgeable enough, but for the simplicity of my intervention and explanation.
There could be a 5th dimension or an n ieth dimension if parallel universes exists, but this is not the subject of this forum.
Interestingly my clock always runs faster than my opponents just because I'm moving more slowly. 
also the speed of light is a fair bit faster than you said, at 3x10^8m/s, and extra dimensions are not really anything to do with parallel universes.
^ ...and Norwegians claim that Finns are drunks 
Square is the basic measure of length.
a squared + b squared = c squared where "a" is asinine "b" is boring and "c" is crap :) what in the world are you all arguing about. the original question is a brain-fart of a postulate thats gone way to far! we dont need to bring in calculus, analytical geometry, theoretical physics, or any other mathematics into this discussion. the simplest solution is the best solution, pieces move square-by-square on the board. the king is moving the same amount of squares no matter what direction its moving on. dosent matter if its in a diagonal, vertical or horizontal manner.
ScarBlac, you're a genius. Am I correct in saying that a chessboard is does not contain vectors as there is no point that could reasonably be called the zero square? Or have I misunderstood?