Is Chess 3 dimensional?

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Date: 07 Nov 2005 11:02:15
From: Fred
Subject: Is Chess 3 dimensional?

We can describe the position of piece on the board as the function of
the x axis and a function of the y axis.

So so far we have a two dimensional game. But enter the bishops which
are restricted to either dark squares or light squares. It is necessary
to add a function which to describe the color of the square.

I understood that 3 functions = 3 dimensions.

Fred

Date: 18 Nov 2005 23:35:10
From: Johnny T
Subject: Re: Is Chess 3 dimensional?

Fred wrote:
> We can describe the position of piece on the board as the function of
> the x axis and a function of the y axis.
>
> So so far we have a two dimensional game. But enter the bishops which
> are restricted to either dark squares or light squares. It is necessary
> to add a function which to describe the color of the square.
>

Chess requires 3 dimensions to describe the state of the game.

This assumes that rules (like how pieces move), are separate from the
understanding or state of the game.

The first dimension are the positions of the pieces. It turns out that
the whole two dimensional thing, is not necessarily valid, and is only
truly useful for human understanding. 1 dimension works well enough
for computers. For moves it is only a matter of making the math one
dimensional.

The second dimension is the piece itself. Whether it is a pawn or a
king. The description of the piece is a vital part of describing the
state of the game.

The third dimension is time. In order to understand the state of the
game you need to know if it is white or blacks turn, you have to know
whether or not castling has happened or is possible, you have to know if
en-passent is possible, and you have to have a 50 move timer for draws.
All of these are time based pieces of knowledge, and are required for
understanding the state of the game.

And you know what this is exactly what you find is stored in computer
representations of the game. The positions of the pieces, and all of
the time elements.

Date: 21 Nov 2005 12:50:04
From: Alexander Belov
Subject: Re: Is Chess 3 dimensional?

Position in chess can be encoded into 173 bit or less, so it is just a value
of {1, ... ,2^173} - one point. The whole game is determined by a history of
all the moves in the game. So the game is a path in the game tree from the
root (starting position) to the considered position.

"Johnny T" <not@home.com > wrote in message
news:437ed5b4$0$1763$8b463f8a@news.nationwide.net...
> Chess requires 3 dimensions to describe the state of the game.
>
> This assumes that rules (like how pieces move), are separate from the
> understanding or state of the game.

Date: 21 Nov 2005 17:21:59
From: Johnny T
Subject: Re: Is Chess 3 dimensional?

Alexander Belov wrote:
> Position in chess can be encoded into 173 bit or less, so it is just a value
> of {1, ... ,2^173} - one point. The whole game is determined by a history of
> all the moves in the game. So the game is a path in the game tree from the
> root (starting position) to the considered position.

So?

And in many <most > positions the variety of possible trees to get there,
is largely meaningless in understanding a given position.

And if you have a game tree, you can gain temporal information implicitly.

And the whole 173 bits, thing, again so? Whereas the 1 dimensional
chess representation is actually a bit more interesting, as the chess
moves and rules are also 1 dimensional inside most modern chess engines.
Though, there are some that operate in two dimensional arrays.

But maybe I am missing the whole, point. Did you have a point?

Date: 19 Nov 2005 03:58:24
From: Major Cat
Subject: Re: Is Chess 3 dimensional?

Johnny T wrote:
>
> Fred wrote:
> > We can describe the position of piece on the board as the function of
> > the x axis and a function of the y axis.
> >
> > So so far we have a two dimensional game. But enter the bishops which
> > are restricted to either dark squares or light squares. It is necessary
> > to add a function which to describe the color of the square.
> >
>
> Chess requires 3 dimensions to describe the state of the game.
>
> This assumes that rules (like how pieces move), are separate from the
> understanding or state of the game.
>
> The first dimension are the positions of the pieces. It turns out that
> the whole two dimensional thing, is not necessarily valid, and is only
> truly useful for human understanding. 1 dimension works well enough
> for computers. For moves it is only a matter of making the math one
> dimensional.
>
> The second dimension is the piece itself. Whether it is a pawn or a
> king. The description of the piece is a vital part of describing the
> state of the game.
>
> The third dimension is time. In order to understand the state of the
> game you need to know if it is white or blacks turn, you have to know
> whether or not castling has happened or is possible, you have to know if
> en-passent is possible, and you have to have a 50 move timer for draws.
> All of these are time based pieces of knowledge, and are required for
> understanding the state of the game.
>
> And you know what this is exactly what you find is stored in computer
> representations of the game. The positions of the pieces, and all of
> the time elements.

The FEN conventions aim to address the problem
of unambiguously specifying the state of a game
of chess at any point in time. However, these
conventions appear to be silent on the topic of
elapsed time in the context of time controls...
How many "dimensions" do the FEN conventions
imply or require? 8 >)

Major Cat

Date: 21 Nov 2005 09:35:49
From: David Richerby
Subject: Re: Is Chess 3 dimensional?

Major Cat <epikuros@istar.ca > wrote:
> The FEN conventions aim to address the problem of unambiguously
> specifying the state of a game of chess at any point in time. However,
> these conventions appear to be silent on the topic of elapsed time in
> the context of time controls...

FEN is concerned only with presenting the position and defining what moves
(castling / en passant) are legal. After all, you never see chess
puzzles, ``White to play and win, bearing in mind that he has 35 minutes
to reach the time control at move 40 and Black has 21 minutes.'' :-)

Dave.

--
David Richerby Strange Game (TM): it's like a family
www.chiark.greenend.org.uk/~davidr/ board game but it's totally weird!

Date: 21 Nov 2005 08:54:26
From: Major Cat
Subject: Re: Is Chess 3 dimensional?

David Richerby wrote:
>
> Major Cat <epikuros@istar.ca> wrote:
> > The FEN conventions aim to address the problem of unambiguously
> > specifying the state of a game of chess at any point in time. However,
> > these conventions appear to be silent on the topic of elapsed time in
> > the context of time controls...
>
> FEN is concerned only with presenting the position and defining what moves
> (castling / en passant) are legal. After all, you never see chess
> puzzles, ``White to play and win, bearing in mind that he has 35 minutes
> to reach the time control at move 40 and Black has 21 minutes.'' :-)
>

Very, very true. FEN is for old-fashioned
analysts (the F part, for sure)... However,
its modern(ist) extension, EPD, allows for
the accommodation of a truly diverse set of
"chessic" descriptors. The same can be said
about the _full_ PGN specification. I guess
that, among other things, "gladiatorial"
interests have been deemed important enough
for these relatively recent sets of convent-
ions to cover! 8 >)

> Dave.
>
> --
> David Richerby Strange Game (TM): it's like a family
> www.chiark.greenend.org.uk/~davidr/ board game but it's totally weird!

Major Cat

Date: 21 Nov 2005 15:56:25
From: David Richerby
Subject: Re: Is Chess 3 dimensional?

Major Cat <epikuros@istar.ca > wrote:
> However, [FEN's] modern(ist) extension, EPD, allows for the accom-
> modation of a truly diverse set of "chessic" descriptors. The same can
> be said about the _full_ PGN specification.

The full PGN standard doesn't actually have any way of representing the
time taken for each move, only for quoting the time control of the game.

Dave.

--
David Richerby Impossible Tree (TM): it's like a tree
www.chiark.greenend.org.uk/~davidr/ but it can't exist!

Date: 21 Nov 2005 12:49:55
From: Major Cat
Subject: Re: Is Chess 3 dimensional?

David Richerby wrote:
>
> Major Cat <epikuros@istar.ca> wrote:
> > However, [FEN's] modern(ist) extension, EPD, allows for the accom-
> > modation of a truly diverse set of "chessic" descriptors. The same can
> > be said about the _full_ PGN specification.
>
> The full PGN standard doesn't actually have any way of representing the
> time taken for each move, only for quoting the time control of the game.

Agreed. Going back to the issue of dimensions,
I cannot see why one may want to introduce this
concept in chess. Any "adequate" descriptor stand-
ard should be able to stand on its own structure
and merit.

>
> Dave.
>
> --
> David Richerby Impossible Tree (TM): it's like a tree
> www.chiark.greenend.org.uk/~davidr/ but it can't exist!

Major Cat

Date: 09 Nov 2005 12:15:53
From: Fred
Subject: Re: Is Chess 3 dimensional?

This site explains it all

http://quantumfuture.net/quantum_future/kaluza.htm

It comes out to 42!

Date: 09 Nov 2005 10:19:19
From: Fred
Subject: Re: Is Chess 3 dimensional?

I admit that the maths are a bit tricky.

I never got beyond orthogonal polynomials, multiple regression analysis
and trying to solve Fermat's last theorem.

Have you looked at the de Broglie relation at all, or are you the Dude?

Date: 09 Nov 2005 10:02:13
From: Fred
Subject: Re: Is Chess 3 dimensional?

"Maxwell showed that if electro-magnetic action were assumed to travel
as a disturbance in ether, as Faraday imagined, then it would be
propagated through ether in the form of transverse waves and its speed
would be equal to the speed of light. Maxwell's equations were taken
for a time to prove that the propagation of light through ether
involved a passage of electric and magnetic forces rather than material
vibrations (Jeans, Growth 287). The theory of ether was eventually
proven to be incorrect, but the attempt to explain it in terms of
electro-magnetic phenomena was a most important development: it showed
the first sign of moving away from the purely mechanical standpoint of
nineteenth- century science (Hull 240)."

But Maxwell required ether!

Date: 08 Nov 2005 12:31:09
From: Fred
Subject: Re: Is Chess 3 dimensional?

Thank you very much for your time spent on my question. very helpful.

But how about this one?

"The light speeds up when it exits the prism because there is
less resistance to its propagation. Just like if you're walking in
water,
you slow down in deeper water because it's harder to walk and speed up
again when it gets shallower."

That's like a sound wave - but that speeds up in water!

If light if propagated as you say, and it recovers it's velocity, there
must be a medium for it to be propagated through.

I prefer string theory.

Fred

Date: 09 Nov 2005 17:22:17
From: The Man Behind The Curtain
Subject: Re: Is Chess 3 dimensional?

Fred wrote:
> Thank you very much for your time spent on my question. very helpful.
>
> But how about this one?
>
> "The light speeds up when it exits the prism because there is
> less resistance to its propagation. Just like if you're walking in
> water,
> you slow down in deeper water because it's harder to walk and speed up
> again when it gets shallower."
>
> That's like a sound wave - but that speeds up in water!
>
> If light if propagated as you say, and it recovers it's velocity, there
> must be a medium for it to be propagated through.
>
> I prefer string theory.
>
> Fred
>

Dude, I don't think you even *understand* string theory. Stop
flattering yourself.

John

--

Von Herzen, moge es wieder zu Herzen gehen. --Beethoven

Date: 08 Nov 2005 10:36:06
From: Fred
Subject: Re: Is Chess 3 dimensional?

I think we all agree that Chess is a two Dimensional Game played on a
chequered board.

To describe the position of a piece you need the coordinates of the y
axis and the x axis - that is certainly 2 dimensional. But the
coordinates do not tell you if it is a dark square or a white square.

If your opponent has lost his "black squared" bishop you can move
all your pieces to the black squares and be safe from his white squared
bishop.

So all your pieces are now playing on black squares - in the same 2
dimensions as his white squared bishop, which is now completely useless
(apart from stopping advancing pawns). The white squared bishop is
therefore in a different dimension.

Just like the 5th dimension in M-Theory - your pieces are on the
black squares, completely unaware of that bishop on the white squares.
They are only aware of it from the vibrations when it is moved!

Date: 09 Nov 2005 09:20:09
From: David Richerby
Subject: Re: Is Chess 3 dimensional?

Fred <frebak@hotmail.com > wrote:
> To describe the position of a piece you need the coordinates of the y
> axis and the x axis - that is certainly 2 dimensional. But the
> coordinates do not tell you if it is a dark square or a white square.

Yes they do. a1 is always black, b1 is always white, c1 is always black
and so on.

> If your opponent has lost his "black squared" bishop you can move
> all your pieces to the black squares and be safe from his white squared
> bishop.
>
> So all your pieces are now playing on black squares - in the same 2
> dimensions as his white squared bishop, which is now completely useless
> (apart from stopping advancing pawns). The white squared bishop is
> therefore in a different dimension.

No it isn't. Just because things can't reach each other doesn't mean
they're in different dimensions. There are plenty of real-world examples
of this. If you and I are standing on opposite sides of a deep, fast-
flowing river and there's no bridge, we can't get to each other but we're
still in the same dimension. If there's a goat standing 10m away from
you, tethered to the ground by a 5m rope, it can't get to you but you're
still in the same dimension.

Dave.

--
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Date: 08 Nov 2005 20:29:01
From: Richard Delorme
Subject: Re: Is Chess 3 dimensional?

Fred a �crit :
> I think we all agree that Chess is a two Dimensional Game played on a
> chequered board.
>
> To describe the position of a piece you need the coordinates of the y
> axis and the x axis - that is certainly 2 dimensional.

I assume a position is defined by a couple of numbers (x, y), ie A1 - >
(1, 1); A2 - > (1, 2); ... H8 -> (8, 8).
Let z = x * 8 + y
z is a unique number that take into account all possible coordinates,
while reducing the board dimensions to one. Most of the computer
programs work in a similar way with a single dimension to access the
square of the board.

> But the
> coordinates do not tell you if it is a dark square or a white square.

Wrong. Knowing the coordinate of a square is enough to tell its colour.
Just look at the parity of the coordinates (x, y). If x and y have the
same parity the square is black, otherwise it is white.

> If your opponent has lost his "black squared" bishop you can move
> all your pieces to the black squares and be safe from his white squared
> bishop.
>
> So all your pieces are now playing on black squares - in the same 2
> dimensions as his white squared bishop, which is now completely useless
> (apart from stopping advancing pawns). The white squared bishop is
> therefore in a different dimension.
>
> Just like the 5th dimension in M-Theory - your pieces are on the
> black squares, completely unaware of that bishop on the white squares.
> They are only aware of it from the vibrations when it is moved!

Of course you can do that, but computers (and humans too) can
determinate the colour of a square so fast this is unneeded, and
probably less efficient than computing the parity of the coordinates.

--
Richard

Date: 08 Nov 2005 09:45:20
From: Fred
Subject: Re: Is Chess 3 dimensional?

That is the old theory of light - It fails because is does not
explain why light should speed up once it has been slowed down.

Everybody should be into M - theory once called String Theory.

"Theodore Kaluza in the 1920s was to write down a five dimensional
theory of gravity. In five dimensions, the gravitational field has 15
independent numbers, which can be arranged in a five dimensional array
(see fig.4). Kaluza then re-defined the 5th column and row of the
gravitation al field to be the electromagnetic field of Maxwell. The
truly miraculous feature of this construction is that the five
dimensional theory of gravity reduces down precisely to Einstein's
original theory of gravity plus Maxwell's theory of light. In other
words, by adding the fifth dimension, we have trivially unified light
with gravity. In other words, light is now viewed as vibrations in the
fifth dimension. In five dimensions, there is "enough room" to
unify both gravity and light."

Quoted from:-

http://www.mkaku.org/articles/hyper_sci_odyssey.shtml

Date: 09 Nov 2005 09:55:41
From: David Richerby
Subject: Re: Is Chess 3 dimensional?

Fred <frebak@hotmail.com > wrote:
> That is the old theory of light - It fails because is does not
> explain why light should speed up once it has been slowed down.

As far as I'm aware, Maxwell's theory of light has no difficulty
explaining why light speeds up on leaving a prism, though it's been a
few years since I did any physics. Light is a series of oscillating
electrical and magnetic fields. It propagates by an electrical field
inducing a magnetic field, which induces another electrical field and
so on. The propagation speed just depends on how `easy' it is for one
kind of field to induce the other, so just depends on the electromag-
netic properties of the medium.

Dave.

--
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Date: 08 Nov 2005 11:28:06
From: Claudio Grondi
Subject: Re: Is Chess 3 dimensional?

I have tried to understand your concept and came to the conclusion, that
maybe you consider restrictions for movement within a plane as an obstacle
which is in your eyes something 3-dimensional. I suppose, that if you
consider it deeper in the same way you did it for the bishops, you will
detect, that all of the chess pieces are limited in their movement, so not
only the bishops adds further "dimensions" to the game.

By the way: longer time ago I had investigated if there exist 3D games which
can't be reduced to 2D games (i.e. it is not possible to play them when
given a set of rules how to handle a 2D visualisation of the game status to
achieve same effect as playing the 3D game) and found, that
there are NO 3D GAMES
which can't be turned into 2D games (e.g. the Rubics Magic Cube can also be
reduced to a 2D model).

In my eyes chess is definitely a 2D game which can be even reduced to a 1D
game when ordering all of the squares of the board into one line converting
the 2D game rules to 1D rules, so that such "linear board" and 1D rules can
be used to play it.

Maybe there is a smart way of converting the 2D board and its rules to a 3D
version which can be helpful adding some more insight when considering
positions and deciding about next moves, but to be honest I can't imagine
that 3D can be of any value here.

Claudio

"Fred" <frebak@hotmail.com > schrieb im Newsbeitrag
news:1131388199.832220.55870@g43g2000cwa.googlegroups.com...
> We can describe the position of piece on the board as the function of
> the x axis and a function of the y axis.
>
> So so far we have a two dimensional game. But enter the bishops which
> are restricted to either dark squares or light squares. It is necessary
> to add a function which to describe the color of the square.
>
> I understood that 3 functions = 3 dimensions.
>
> Fred
>

Date: 08 Nov 2005 14:53:31
From: David Richerby
Subject: Re: Is Chess 3 dimensional?

Claudio Grondi <claudio.grondi@freenet.de > wrote:
> I have tried to understand your concept

Don't try too hard -- the OP's reasoning is incorrect.

> and came to the conclusion, that maybe you consider restrictions for
> movement within a plane as an obstacle which is in your eyes something
> 3-dimensional. I suppose, that if you consider it deeper in the same way
> you did it for the bishops, you will detect, that all of the chess
> pieces are limited in their movement, so not only the bishops adds
> further "dimensions" to the game.

Restricting movement can only *decrease* the number of dimensions. For
example, the surface of the earth is a two-dimensional surface in 3D space
because, on the surface, you can only move north/south or east/west:
if you move up/down, you'll leave the surface.

> By the way: longer time ago I had investigated if there exist 3D games
> which can't be reduced to 2D games (i.e. it is not possible to play them
> when given a set of rules how to handle a 2D visualisation of the game
> status to achieve same effect as playing the 3D game) and found, that
> there are NO 3D GAMES which can't be turned into 2D games (e.g. the
> Rubics Magic Cube can also be reduced to a 2D model).

Any game with discrete positions can be reduced to a format where
dimensionality doesn't really exist by going to the game graph. That is,
consider positions in the game as being totally abstract so playing 1.e4
doesn't involve moving a pawn `forwards' but is just a change from one
position to another. Alternatively, you can arrange the set of discrete
positions in any number of dimensions you choose.

3D games can trivially be reduced to 2D games: this is exactly what your
eyes do when you look at the game.

> Maybe there is a smart way of converting the 2D board and its rules to a
> 3D version which can be helpful adding some more insight when
> considering positions and deciding about next moves...

It's certainly possible.

> ... but to be honest I can't imagine that 3D can be of any value here.

Me neither.

Dave.

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Date: 08 Nov 2005 14:56:56
From: David Richerby
Subject: Re: Is Chess 3 dimensional?

David Richerby <davidr@chiark.greenend.org.uk > wrote:
> Restricting movement can only *decrease* the number of dimensions. For
> example, the surface of the earth is a two-dimensional surface in 3D
> space because, on the surface, you can only move north/south or
> east/west: if you move up/down, you'll leave the surface.

Or, to give a chess-based example, knights and kings can be considered as
one-dimensional pieces (it would be better to say `considered as having
only one degree of freedom) but bishops, queens and rooks are 2D. When
moving a B/Q/R, you get to choose direction and distance. However, kings
and knights move a fixed distance (one square for kings, sqrt(5) squares
for knights) in each direction so you only get to choose the direction
(e.g., as a compass bearing).

Dave.

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Date: 08 Nov 2005 09:45:03
From: David Richerby
Subject: Re: Is Chess 3 dimensional?

Fred <frebak@hotmail.com > wrote:
> We can describe the position of piece on the board as the function of
> the x axis and a function of the y axis.
>
> So so far we have a two dimensional game. But enter the bishops which
> are restricted to either dark squares or light squares. It is necessary
> to add a function which to describe the color of the square.
>
> I understood that 3 functions = 3 dimensions.

No.

The position of the piece on the board is just its position on the two
axes -- there is no function here. The colour of the square is a function
of its position. That is, the colour is completely determined by the
position: the square a1 is always black, for example.

There are only two dimensions.

Dave.

--
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your lap!

Date: 07 Nov 2005 14:21:23
From: Fred
Subject: Re: Is Chess 3 dimensional?

MorphyFischer wrote:
> The color is a redundant parameter so it is not a dimension at all. No
> matter what square any piece is on, there is a color below it. The
> rooks can be on a light or dark square, so can the queens, so does this
> add more dimensionality by your logic? And what happens when all of the
> bishops are traded, does the game go back to being 2-dimensional by
> your standard? A dimension is defined by being perpendicular to all
> other spatial coordinates, it has nothing to do with piece movement
> restrictions.

Some multidimensional graphs utilise color to illustrate multiple
dimensions

e.g.
http://www.cg.tuwien.ac.at/research/vis/dynsys/ndim/

I stall believe color adds a third dimension to the two dimensional
chess board.

You could use a chess board with square holes for the black squares.
That would be a three dimensional board, with no need for color.

Yes once the bishops are traded the checkered squares are no longer
required and mathematically the game can be described in two
dimensions.

On dimensions:
Light has been described as vibrations of the forth spatial dimension.
Physics teachers are fond of demonstration prisms - Light is slown down
by the glass and is split into the colors that make up white light.
But then they cannot explain why the light speeds up again once it
exits the prism!

Date: 08 Nov 2005 09:49:45
From: David Richerby
Subject: Re: Is Chess 3 dimensional?

Fred <frebak@hotmail.com > wrote:
> I stall believe color adds a third dimension to the two dimensional
> chess board.

In that case, please explain how I can move a piece on a1 from black to
white while keeping it on a1.

> Light has been described as vibrations of the forth spatial dimension.

Er... no. Light is an electromagnetic field in the three spatial
dimensions we know and love.

> Physics teachers are fond of demonstration prisms - Light is slown down
> by the glass and is split into the colors that make up white light.
> But then they cannot explain why the light speeds up again once it
> exits the prism!

Er... no. The light speeds up when it exits the prism because there is
less resistance to its propagation. Just like if you're walking in water,
you slow down in deeper water because it's harder to walk and speed up
again when it gets shallower. There is no magic here.

Dave.

--
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Date: 07 Nov 2005 11:19:04
From: MorphyFischer
Subject: Re: Is Chess 3 dimensional?

The color is a redundant parameter so it is not a dimension at all. No
matter what square any piece is on, there is a color below it. The
rooks can be on a light or dark square, so can the queens, so does this
add more dimensionality by your logic? And what happens when all of the
bishops are traded, does the game go back to being 2-dimensional by
your standard? A dimension is defined by being perpendicular to all
other spatial coordinates, it has nothing to do with piece movement
restrictions.