Catastrophic 8x8 Chess

Chess is a catastrophic game. A bad or blunder move can be termed as a Catastrophic Move. By that I simply mean a sudden change of situation which automatically leads to an unbalanced position. During a chess game there may be one or many catastrophic moves. A catastrophic move can lead to the loss of a game for one player unless he can reestablish the balance or the superiority by playing an excellent move. Sometimes, it is irreversible. The catastrophic move may occur at the opening, the middle or the endgame. At the opening it results from the control of the fundamental squares d4, e4, d4, e5.

For this case, I have designed the following 8x8 Catastrophic chessboard (see board(1)). It reflects the aspects of approach, compromise, behavior, action, cost, retreat of the fighters. The cusp of the fold (F) represents the catastrophic turning point.

Mathematically (without going into details) the catastrophic chessboard can be seen as a surface (S) with cubic equation

   (1)   X3= a + bX

where a,b are the control axes. The fold (F) is given by differentiating the equation (1). This gives

   (2)   3X2 = b

The bifurcation set (B) is the projection of the fold (F) onto the control space (C). It is given by eliminating X from the equations (1) and (2). We obtain

   (3)   27a2 = 4b3

The fold(F) separates the cubic surface (S) onto two pieces: the larger piece (G) is given by the inequality

   (4)   3X2 >= b

It is a single sheeted outside the cusp. The smaller piece (S-G) is given by the inequality

   (5)   3X2 < b

It is a single sheet over the inside of the cusp. In (C2) of the bifurcation set (B) there is the fear. Because the 8x8 chess is a dynamic game, a catastrophic chessboard may be appropriate because it reflects the evolution of changes during the game. To see this, let's consider the following simple chess game played between the french master KERMUR DE LEGALL and an unknown:

 
   1. e2-e4    e7-e5     2. Ng1-f3    d7-d6      
   3. Bf1-c4   Bc8-g4    4. Nb1-c3!   g7-g6   

(Here we have a bad move which white will explore. It is almost a catastrophic move and so the fold (F) can be located at the squares f6,g6)

   5.Nf3 x e5!!

(excellent move which will create the irreversible catastrophic situation if black falls into the trap)

   5. ...     Bg4 x Qd1  6. Bc4 x f7+   Ke8-e7 
   7. Nc3-d5  Mate.

In this game, we observe that one catastrophic move made by black has led to an irreversible unbalanced position which causes the loss of the game for black.