Author: Werner Kraft
Date: 02:44:50 05/20/05
Basics of Group Theory for Chess Players ( ca 800 words) I try to give here a basic introduction to Group Theory. I will use plain , jargon free English to the best of my abilities. The whole text is based on an article in the McGraw-Hill Encyclopedia of Science and Technology, page 240ff and hopefully inspired by the spirit of "advanced hippie-ology " A set of chess moves ( the operation of chess figures ) is called a group. Lets call the group of all legal chess moves in the universe G , or , more graphic : GOB ( Gods Opening Book) . So , each chess move ( m(1), m(2) , m(3) ¡K must belong to GOB.... There are four requirements , that have to be fulfilled , if chess moves really should form a group: Closure , Associative Law, Identity element and Inverse. 1. Closure The "product " , or better the sum of two chess moves , for instance m(1) = e2-e3 and m(2) = e3-e4 gives another element m(3) in the group m(1) + m(2) = e2-e3 + e3-e4 = m(3) = e2-e4 Or , more intuitively " if you push your pawn in the opening one field ahead , and then on the next move another one, you have the same result as if you do the whole thing in one go" But you lost a tempo - that is why chess moves only follow the closure law, if we disregard the time component. Question for the chess-intelligentsia : try to work out situations , where chess moves completely follow the law of closure ( that means : include loss/gain of tempo ) . Is there a possibility for ¡§surreal moves ¡§¡K moves within the incredibly large number of ¡§Gods Opening Book ¡§ ( you¡¦d have to discover ¡§ Borges Library ¡§ ( that is the complete number of books , that could ever have been written , by using permutation , to find it¡K ) - so , is there any possibility of ¡§urreal¡¨ moves in GOB ¡K yes and no: Yes, because if a machine computes all moves, there will be mistakes coming up. No ¡V because it would be an contradiction in itself . ¡§ Now, mate , I wanna know ¡V whats an ¡§contradiction¡§ ? ¡K Ah¡Kshut up !! ¡§ ) 2. Associative law I have struggled with that one - intuitively it should be what is called a " Zugumstellung " - if you change the sequence of moves e.g in the opening , but come to the same result at the end. Simple example : 1.e2-e4 e7-e5 2.d2-d4 d7-d5 gives the same arrangement as 1. d2-d4 d7-d5 2.e2-e4 e7-e5 More mathematically: The sum of the chess moves gives the same result , just the order / sequence is different. Question for the very intelligent: Try to work out an estimate, how many subgroups within GOB are " associative " - how many move constellations lead to the same result. How many subgroups exist altogether ?!? 3. Existence of an identity element "There is in the group an element e ( called the identity) which satisfies for every element g of the group : eg = ge = g " Wow ...now lets define e = <NothingHappens> h2-h4 * <NothingHappens>= h2-h4 . That¡¦s incredible ¡K incredibly simple. The * sign here is just another ¡§ operator :¡¨ sth like a thing , that does something to other things ¡§ . Really simple: in 2*2=4 , the *sign is a operator , that multiplies numbers . But the law of the identity element holds even deeper miracles . We can combine the identity element : <NothingHappens> * <NothingHappens> = <AbsolutelyNothingHappens> This element is part of the "uncertainty principle" . ( Be aware - during the last 4 lines I took you for a ride ... the following parts are sound again ) Question for the very , very intelligent: Are there situations in computer chess, where the identity element is really relevant ? 4. Existence of the Inverse ¡§ For each element g of the group there is an element g(-1) (called the inverse of g ) which satisfies g(-1)*g = e ¡§ . e is the identity in mathematics. On the chessboard it is simply the situation : ¡§ nothing has happened ¡§ . General Formula: I have defined before : e = <NothingHappens> Let¡¦s the Inverse call <TakeMoveBack> Then Move*<TakeMoveBack> = <NothingHappens> Special Case : e2-e4 *<TakeMoveBack>= <NothingHappens ¡§ Can you slacker not simply say, that <TakeMoveBack> here is simply e4-e2 ? ¡§Exactly. Smartass¡K and <Nothing happens> could be also called <OriginalPosition> .May be¡K You see, you simply have the power to define things as you like. It just has to be ¡¨ logically coherent¡¨ ¡K or in plain English: ¡§ sound¡¨. ¡§ OUTLOOK There are more advanced parts of Group Theory , were everything is not so easy¡K Topological groups ¡V ¡§set of elements not only equipped with a group operation, but also with a topology ¡§ . That means ¡§ two group elements are close to one another ¡§. The so called ¡§Lie Groups¡¨ ( found by the mathematician Sophus Lie , probably using a Lie ¡V detector) are ¡§ topological groups in which it is possible to label the group elements by a finite number of coordinates ( Chess ! ) , in such a way that the coordinates¡K¡¨ etc. Here I would have to check out communitative groups ( abelian groups after the Mathematician Niels Abel ) , with the formula : g*h = h*g ( or : 2*4 = 4*2 , and that makes eight. (. Do you start to see now how incredibly difficult even the basics of Group Theory are to understand ? ƒº Key words of the rest of the article: Geometric symmetries, Space-time symmetries, Gauge symmetries, Dynamic symmetries etc. Question: How can you explain Group Theory by using poetry, examples from everyday life, colours, graphics , sound, vision, calculations, MindMaps ¡K and Mind Tools !?! Source: Arthurs S.Wightman , Article on ¡§Group Theory¡¨ , page240 ¡V 241 , Encyclopedia of Science and Technology ,Publisher: McGraw-Hill O.K - thats it. I will take a creative break for a couple of weeks and " do the things I really love ( apologies to Dr. Robert Hyatt - had no clue that there are big names participating here ) - What I personally love - rock climbing and personal development of my emotional intelligence.
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