But what if we define the default length as the diagonal move, then orthogonal moves are actually smaller than one, so it's actually played in less than one dimensuon.
A question for mathematicians...
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Eberulf wrote:
So for all the squares on the board, the average dimension will be less than 1.5.
Should read:
So for all the squares on the board, the average dimension will be greater than 1.5.
rooperi wrote:
But what if we define the default length as the diagonal move, then orthogonal moves are actually smaller than one, so it's actually played in less than one dimensuon.
No, it would be the same situation. If you move to an orthogonal square but don't perceive any distance moved on one of the two diagonals when doing so, the exact same situation is present.
RainbowRising wrote:
You guys seem to ignore my notion of the definition of a dimention ;)
No, I think, all things considered, you are right. Another problem has been highlighted above, Knight moves/edge of board moves seem to at least throw inconsistancies at the hypothesis, if not downright disprove it.
Bloody good fun, though 
RainbowRising wrote:
Why do knight moves ruin anything? You dont even need 3-d for knight moves (I presume people have said that because of the jumping ability right?). Think of knights as gamma rays :)
What else has been thrown at the theory?
To be perfectly honest with you mate, I have no idea! We've had general relativity, nth dimensional fractals, orbital mechanics... I've completely lost track of it all!
Nov 2, 2009
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#127
I have a math degree from a fine institution but I do not wish to spend time analyzing concentric circles.
RainbowRising wrote:
General Relativity? That's a theory of gravity - how has that got involved here lol!
Ha! Who knows! I think someone used the speed of light to define time, and one thing leads to another...
ozzie_c_cobblepot wrote:
I have a math degree from a fine institution but I do not wish to spend time analyzing concentric circles.
On a scale of 1-10 how would you gauge the difficulty of completing your degree. And what specifically was your degree?
Right... I'm thinking of going to school sometime soon and the only degree I'm even considering taking is some sort of mathematics.
I'm not sure if I'm up to it though.
How about the difficulty in becoming an NM? 1-10
Nov 2, 2009
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#136
paul211 wrote:
rooperi wrote:
paul211 wrote:
a²= b²+c², solving we have: a²= 3²+2² and
a=√9+4 or √ 13, which is 3.606 units of distance.
Do review and if in disagrement I will expand my explanation.
Actually" a²=b²+c², correct, but you use the incorrect values for b and c
a²= 2²+1² = 5.
You are close but not right.
The knight moves from square 1, 2 more square the total distance covered from the initial position is 3 squares and then moves at right angle another square to position himself on the final square.
So the knight moves across a rectangle of a 3 by 2 square, when you join the opposite ends, I think that I am right, a²=3²+2²
so I assume if you start on a number line, and move from 1 to 3, you have traveled a distance of 3 numbers by your logic. Therefore 3-1 = 3.
Nov 2, 2009
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#137
"I actually do both math and physics and after university, I still consider that a math person is the person that enlightens us by giving the solution or explanation of the equation posed by a physicist."
Really. Um, mathematicians, at least earlier in history, never really cared about their math being useful at all. I mean just look at the massive amount of number theory that is cool adn beautiful but not "applicable" to everyday problems.
Nov 2, 2009
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#138
Scarblac wrote:
mottsauce wrote:
RainbowRising wrote:
mottsauce wrote:
MugglesMan wrote:
The distance from a light-squred bishop to a dark square is infinte.
precisely.
Not exactly. It can be described without the needless use of infinities.
I disagree. Opposite colored bishops can never touch. Think of it as a limit.
It's like dividing by zero. 1 / 0 isn't infinity, because 0 * infinity still isn't 1 (if you have 1 apple to divide over 0 people, even if you give all of them infinity apples, you'll still have the 1 apple left over).
In this case, it's not infinity, because even in infinite moves, the bishops won't reach each other.
Taking the limit would only make sense if the bishops would get closer to each other as you get closer to infinity. But they don't.
It's just undefined.
okay, i agree undefined is more correct. BUT the mistake you're making here is a fundamental one: infinity isn't a number, it's an idea.
Nov 2, 2009
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#139
If you start at one, then count to 3. There are 3 numbers. The difference between them is 2. This is really elementary.
Paul, it appears that you are implicitly assuming that the knight travels from the upper left corner of your diagram to the bottom right corner. I think the rest of us are saying the location of the knight is best described as being in the center of the square. If we agree to use euclidean geometry to measure distances, then the knight moves a distance of sqrt(5).
*Mutters under breath - "chess is played in 1.5 dimensions"...*
I do see your point now.
When you move diagonally, you move both horizontally and vertically. Say you could only percieve one dimension. Whether it was the vertical or horizontal dimension you could perceive, if one square diagonally was moved, you would percieve a distance of 1 being moved (i.e. not 2^-.5). So I now fully see your point and think is is uncontroversial as far as it goes. The rules of chess correctly perceive distance movement if that movement is directly horizontal or vertical. If the movement is diagonal however, the rules of chess is only seeing one dimension (whether that dimension is the horizontal or vertical dimension is unknown and irrelevant.) So from a given square, 4 of the eight squares to which you can move chess only perceives 1 dimension. Thus chess perceives 1.5 dimensions.
However you did not consider the case of squares along the perimeter of the the chess board and at the corners. If a king is at A4 for example, only 40% of the sqaures it can move to are on a diagonal. At A1 only 33.33% of the squares it can move to are on a diagonal. So far all the squares on the board the average dimension will be less than 1.5. You or someone needs to come up with the exact figure.
My God... Are you saying that chess is played in VARIABLE dimensions?! That's put a cat among the pidgeons! Still, I'm glad the essence of my point makes some kind of sense