A chess math daydream

Yesterday I had a coupon, where I can buy a Culver's Buttery Double Cheeseburger and get another one free, so of course, I had to do it! (I did it for a friend.) I ate one and put the other one in the fridge, and then I was discussing with my friend (the same one as before) that we could share it.

So I imagined different ways to cut it. We could just cut it down the middle vertically like an ordinary sandwich cut in half.  Or she could have one patty and I could have the other patty. Or if we couldn't decide then we could divide it top and bottom, and cut it vertically, and I could have the bottom of one half and the top of the other half.  It would make a checker pattern, like a chess board.

She ended up cutting it into quarters and she took two opposite quarters and gave me the other two. So it looked like a chess board.

I started thinking, what if knights would rotate? Instead of just having a position, what if each chess piece would have more dimensions?  Knights probably start facing forwards, but what if they rotate when they jump to another square? 

My mathematical imagination is not very specific at all about this, yet.  But the chess board likely has some algebraic structures that I have not recognized yet, and may have never read about yet.

I am trained, somewhat, as an applied mathematician, although I am out of practice.  And I was not very careful or studious as a graduate student, and I did not focus on algebra as a discipline (there was scarcely any part of math that I can be said to have really focused on).  But I think it would be great if quaternions or some algebraic structure could be applied over chess to represent the powers of each piece on the board, to enable smoother computations, rather than brute force computations.

I wondered if anyone else would like to discuss it here.  I tend to post out of pride or for entertainment. I've explained to someone in the past week that being entertaining is a matter of making up nonsense.  So you are not obligated, as intellectuals, to participate. There are much more important things, like confronting Burma with the ethnic cleansing of the Rohingyas ...

Cavatine

Love it! Keep us posted

There is more scope for maths in the game of go, where endgames are a nice source of examples. The theory of surreal numbers arose out the of the study of go endgames.

There is a novelette on the subject, of which this is the first entertaining part:

How two ex-students turned on to pure mathematics and found total happiness

and there is a library copy here: https://archive.org/details/SurrealNumbers

There should be infinite ways to split up the burger, I do hope there are veggies. Nice read and thoughts, experimental. 

Speaking of infinity, on the surreal numbers wiki page (thanks @Elroch) the section on infinity is surprisingly small . Seems like it would go on forever. 

"Don't let anyone tell you infinity isn't a number once you're equipped with knowledge about surreal numbers." - Faithful Mathematician

The beauty of chess is in its Inbalances. While the end is to checkmate the opponent yet it is the battle of unequal and incongruence.