A question for mathematicians...

 We're just having a bit of fun buddy.

It's you who's missing the point, ErrantDeeds.

The hypotenuse only looks equal to the catheti on the chessboard just because the diagonals it's made of are longer than the sides of the squares that the catheti are made of.

In other words, your "unit" of one square is diagonally longer than it is horizontally or vertically. It's not constant.

Try and draw a right rectangle with equal catheti on a beach. Now walk along a cathetus keeping constant pace length. Note the number of steps. Now walk along the hypotenuse.

Not surprisingly, you too - like the king on a chessboard - can walk along the hypotenuse in the same number of steps as you did along the cathetus, as long as you take longer steps (i.e.: you change your unit's length when walking diagonally).

 But that's the whole point! The King DOES NOT take longer steps! He may do physically on the board, but not in terms of number of squares! I feel I have been repeatedly clear on this point!

I guess if you wanted to you could pick the king up off the board and stand up and fly him around the room with sound effects and then set him down on an adjacent square on the board and that would be perfectly legal.  But maybe it shouldn't be.

It could be argued that the sound effects may distract the opponent. If you remove those, it might be a legal move.

It's the same number of squares because the hypotenuse is made of diagonals-of-a-square, while the catheti are made of sides-of-a-square.

It doesn't mean that the chessboard has got some "weird dimensions". It's a simple consequence of the fact that a square's diagonal is longer than its sides.

This appears to need clarification.

Distance on a chessboard is measured in SQUARES. Example:

If you are playing as white, if you were to ask "How far is the pawn from Queening", the answer is FOUR SQUARES! The answer is NOT "Well this chess board happens to be made of squares that are of length of side 4cm, so from the centre of the pawn to the 8th rank must be 16cm"! That's meaningless for the context of the game!

Similarly:

The distance from the King on a1 to h1 is 7 squares. The distance from the King on a8 to h1 is ALSO 7 squares! DIAGONALS = ROWS = COLUMNS! I struggle to think how I could make this point more clear.

But the question may ultimately rival Xeno's motion paradoxes and preoccupy philosophers and mathematicians for the next 2000 years.

LOL! Did U never studied about Equilateral Triangle... If no. of squares are equal it becomes equilateral triangle.

 It's not a traingle. It's a square in 1.5 dimensional space. I am utterly convinced of this

Are you kidding or are you really having difficulty understanding that your "measuring unit" is LONGER diagonally, thus explaining why it makes diagonal distances APPEAR the same as horizontal or vertical ones?

 My god man, I don't know how much more clear I can make this. I am aware that the diagonal is physically longer! But since distance on a board, during a game, is solely measured in squares, this length difference is irrelavant! If a piece travels 5 squares, it travels 5 sqaures! Are you aware that I'm talking about an abstraction? And that this abstraction must be physically represented in the real world, in this case in two dimensions, neccesitating a distortion, i.e. the longer diagonal? In chess, the square is the smallest descreet unit of length, and it's physical dimensions are meaningless!

I would say at a fundamental level, movement would have to involve the annihilation of an object at one point and it re-creation at another point some distance from it.  So if an object moves from point A to point B on a line,  it would mean that the object was annihilated and recreated a finite number of times in its "movement" between those points, i.e. it  occupied a finite number of intermediary points in that journey and no others.To move between any two fundamental points in the journey from A to B would involve as stated the object being annihilated completely at the first point and then appearing completely in tact at the second point.  But if you imagine a 3-d grid imposed on the physical universe, with each cube in the grid being of this incredibly minute fundamental distance,  would it take longer to travel between two adjacent cubes on a diagonal than it does to travel between adjacent cubes sharing an edge.  I would say no.  At a macro level, (traveling through multiple cubes)   the diagonal would take longer, but only because of an additional delay imposed at each fundamental point. But two adjacent fundamental points it wouldn't make any difference if it was a diagonal.  So maybe imagine the squares on a chess board being of 0 length and width and think in terms of calculus or something.  Don't ask me to elaborate on this as it may be B.S.

Interesting post. And the above caption is preceisely what I've been trying to say, elegantly put.

Errant deeds wrote:

My god man, I don't know how much more clear I can make this. I am aware that the diagonal is physically longer! But since distance on a board, during a game, is solely measured in squares, this length difference is irrelavant! If a piece travels 5 squares, it travels 5 sqaures!

And your point is...?

Are you aware that I'm talking about an abstraction?

No, you aren't. You are talking about basic euclidean geometry and marvelling at the simple properties of a square.

 ndrw.... How do I find the words.... The fact that it is a square is IRRELAVANT. For the purposes of this argument, which IS an abstraction, they could be circles, triangles, whatever. The point is they are, I say again, the smallest descreet units of length on a chessboard. By now, you either see what I am trying to say or you do not; I can find no more ways to express it.

Look, I'll go back to your first post.

You asked: "How can a triangle have sides of the same legth in two dimensions?".

Answer: it doesn't. You're just measuring different sides with different measuring units.

I don't care if you take the square and call it your "discreet measuring unit".

That's WRONG, because you (and the chessboard!) are not really using the square as the unit: you're using its diagonal for the hypotenuse and its side for the catheti.

Your fallacy lies in considering "the square" as a measuring unit.

"The square" is a 2-dimensional object. The distances you're trying to measure are lines: they are 1-dimensional objects.

So what do you (and the chessboard!) do?

You take the square apart into its 1-dimensional components (the sides and the diagonal) and use those to measure your distances.

Problem is you're using different components for different sides of the triangle! I don't care if we all simplify our language by saying "move by 2 squares" when we should really say "move by 2 square-sides" or "move by 2 square-diagonals". The latter two phrases are what is really taking place when we move a piece on the board.

They could be equal if it was and equilateral triangle. But as someone has stated before, traveling by the diagonals is faster because there is a greater distance from corner to corner on a square then there is from one side to another

The diagram U given represents a triangle not a square. I think ur question is irrelevant. Sorry but dats da fact

No no no! I am not using ANY real length of side of the square as a measuring unit! It is not a fallacy using a square as a measuring unit, since that is the MINIMUM distance any piece (Knight excepted) can move! As far as, say, a King in concerned, the distance from a1-h8 is the SAME as the distance from a1-h1, since it takes the same number of moves! Please try to see it, I can't think of a better way of putting it! Look:

In the movements of chess, the time is the 'tempi'. One piece moves per tempi. Therefore if the King moves, he has moved as a speed of "1 square per tempi".

So,

Speed = 1 square per tempi.

Distance from a1 to h1 = 7 sqaures (not counting the square you are on).

Distance from a1 to h8 = 7 sqaures

Time for both: 7 tempi!

Since, speed = distance/time, the lengths are the same. By length, I mean THE NUMBER OF SQUARES.

Think about this carefully - it does not matter that the square is a two dimensional object. In terms of the moves that a chess piece can make, it is of zero length and zero height, and occupies a point. That's what I mean by 'smallest descreet distance'. The physical 2D distances that you describe are not relevant to this question, which, despite your denial is in the abstract. Of course the physical piece moves further along the 2D diagonal! I know this! But that's the whole point of the thread, that chess, in the abstract, is not 2D, and that the diagonal lengths appear to be longer because they are distorted by the 2D representation.

Think of the Tesseract (look it up on google). It is a "3d" shadow of a "4d" cube. We cannot visualise the real 4d cube (if it even exits), but we can represent it's shadow in three dimensions.

Since I have shown that diagonals in chess ARE the same length as rows and columns in chess, but of DIFFERENT length when measured in 2d, then chess in the abstract is taking place in a different dimensional way than the obvious 2d representation. You seem to be accusing me of saying the different 2d lengths are the same, when I am not.

What more can I say?