Chess and the Cartesian plane

It has recently occurred to me that, given the fact that we currently represent moves in chess using algebraic notation, which is essentially (in part) a system of coordinates, it is possible to represent the chess board using a Cartesian coordinate system. Specifically, files a-h correspond to 1-8 on the x-axis, and ranks 1-8 the same numbers on the y-axis. Under such a system, we can observe that the movements of the chess pieces model various mathematical functions:

1. Rook moves model the function x=n or y=n, where n is a natural number between 1 and 8, inclusive.

2. Bishop moves model y=x+n or y=-x+n, where n can be negative, 0 or positive natural numbers (there are additional restrictions, but I have not as yet given enough thought to the issue to describe them with complete accuracy).

3. When a knight is situated on the square represented by the coordinates (p,q), all of the squares that it can reach on the next move (on an empty board) lie on a circle with (p,q) as its origin and a radius of √5. In other words, they are all points on the function (x-p)^2 + (y-q)^2 = 5.

Given these observations, it may be helpful to think about what representing chess on a Cartesian plane can contribute to the study of chess. As I am not an expert in chess theory, this is not a question on which I have much to say as of now. Nevertheless, I can think of a situation where the Cartesian plane can have practical applications, namely in blindfold chess:

Suppose that you have a bishop on b1, and you intend to move it in the direction of the long diagonal (in other words not to a2). However, on his last move your opponent moved his knight to d6, and you do not want to put the bishop on a square where it can be taken by the knight. What are the squares that you must avoid? Using Cartesian coordinates, it is not difficult to find the answer. All you need to do is solve the system of equations

     y=x-1

     (x-4)^2 + (y-6)^2 = 5

and you will find the two results (x=5, y=4 and x=6, y=5) that correspond to the coordinates of the squares to avoid.

Now, I know that there are many people here who are well-versed in chess theory. And to those people (but not only to those people - beginners like myself are also welcome to respond) I ask: what are some of the ways in which you think the use of Cartesian coordinates can benefit the study of chess? Tell us your thoughts in the comments below!

 So would White start on the negative Y-axis, and Black on the positive, with 0 being the middle of the board? 

I don't believe it's worth the trouble to try to make a go of this idea. About a year ago somebody posted a similar idea, where they rotated the board and tried to find some use in doing so, I rebutted the idea, but was told to shut up by maybe a friend of the poster. Very foolish thing to do. I had some ideas that probably *would* have worked if they had been civil and wise, whereupon they could have won a science fair or come up with a publishable project, but it was much easier to insult and run. If anyone out there is still trying to make such ideas work, let me give you a hint of how to proceed: create a new type of geometry that is applicable to chess.

Well, my conception, as I described in the first paragraph of the original post, is: "Specifically, files a-h correspond to 1-8 on the x-axis, and ranks 1-8 the same numbers on the y-axis." But I suppose your method would also work.

Was it something along these lines?

https://www.chess.com/forum/view/general/why-are-2d-boards-arranged-the-way-they-are

That wasn't it. I responded to the thread I mean on 6-27-2016.

It sounds more canonical indeed, and we should give the position of the piece as the center of the square, id est, a1 is (-3.5 , -3.5), h1 is (3.5 , -3.5), h8 is (3.5, 3.5), f3 is (1.5,-1.5) etc.

Maybe some sort of continuous chess (http://www.chessvariants.com/other.dir/continuouschess.html) is a relaxation of chess and is much easier to solve, and maybe we can, using some sort of randomized algorithm, transform a solution of continuous chess into one of chess.

I am intereted

But what if people are sloppy, and don't place their pieces at the center of the square?

SHAMEFUL! And you're trying to teach us math applied to chess? That's not a function!!

Comment delivered by Math Nazi Inc.

Did you really look it up just to post the date here?

Oh, right... a function can give at most 1 output for any given input. I forgot about that. Thanks for reminding me!

You are correct, it is not a function, it is the equation of a circle. So it does still work, technically.

Several good points in your idea macer.  

First, the idea of using the x-y axis is brilliant if not for two problems...1) It would take to much paper. and 2) Chess players have enough on their mind trying to plan (and win) the game, they don't need the extra stress of putting the correct formula down.

Secondly, for every move the position would either increase or decrease in your favor.  So black would start at the top of the y axis and white would start at the bottom (naturally).  But then comes the problem of not using a engine...someone in the top 10 would have to be at every game to place the score correctly until one side gets above the x axis and the game ends. 

I think you missed the point entirely....1. He isnt suggesting changing of how we notate, hes just proposing that cartesian planes could be used in how we think about chess. 2. Even if the notation were changed, it would not mean we are writing "formulas". All it does is change the column letter to a number. For example, if as white the origin is the bottom left corner, then Nf3 would instead be N6,3...or something similar.

As for your second point...i have no idea what you are talking about...

This is a very interesting application. I actually had always been wondering if there was an immediate and direct way to calculate if tactics exist within a position, rather than relying on algorithmic if/then logic enhanced by pattern recognition. 

1. Rooks move along slopes that are either 0 or undefined. 

2. Knight movement can be defined as the intersection of a circle whose center is the current knight position and radius sqrt(5), with integral coordinates. 

3. Bishops move along +/- 1 slopes. 

4. Queens take on both Rook and Bishop movement. 

5. Kings would take on some kind of unit function. 

The immediate applications for defining explicit functions that govern piece movement can adequately explain certain endgame principles such as Philidor and Lucena (Placing your rook to block the enemy rook's line of influence). You should then be able to generalize these applications to appropriate middlegame strategy, possibly even early game. 

I'm not sure if any of the current engines employ any of these lattice interpretations of the board and pieces. It seems intuitive enough, but I was always under the impression that they just have a top move for pretty much every position you can encounter based on comprehensive databases such as chessbase. 

It makes it a lot easier to use all positive -or sometimes starting with 0 instead of 1, especially since you are doing things like square roots. I wrote the first-ever animated computer adventure game, Castles of Darkness (you can find me on the Giant List of Classic Game Programmers) in 1981 for the Apple II computer, using Steve Wozniak's personal machine language assembler -he kindly provided me with the code for it. For the graphics, written on graph paper in binary then converted to hexadecimal, it would have been a tremendous headache if the x and y coordinates did not start with 0 and then be all positive.