syerpinsky carpets and cellular automatas applied at Cross method

[hgjf2]

If you using cross method for calculus the structures statical undetermined or for approximated at plates calculus (applying the biharmonical equation
[(df^2/d^2x)+(df^2/d^2y)]^2=a, where a=1 if x=0 and y=0 and a=0 if x<>0 or y<>0. If approximating formulas and taking last digit of integer part. This grid look like a automata cellular if the structures have infinity grid.

[hgjf2]

If you using cross method for calculus the structures statical undetermined or for approximated at plates calculus (applying the biharmonical equation
[(df^2/d^2x)+(df^2/d^2y)]^2=a, where a=1 if x=0 and y=0 and a=0 if x<>0 or y<>0. If approximating formulas and taking last digit of integer part. This grid look like a automata cellular if the structures have infinity grid.

For to using Cross's method for the differential eqution of the biharmonicity must using: f(x-2;y)-8*f(x-1;y)+20*f(x;y)-8*f(x+1;y)+f(x+2;y)+2*f(x-1;y-1)-8*f(x;y-1)+2*f(x+1;y-1)+2*f(x-1;y+1)-8*f(x;y+1)+2*f(x+1;y+1)+f(x;y-2)+f(x;y+2)=P where P is the force applied on the plate at the point (x;y).
If adding new condition , as example like f(5;y)=0 with f(-5;y)=0 with f(x;7)=0 with f(x;-7)=0 then the infinity matrix of the values f(x;y) that x and y are integer numbers look like a automata cellular and like a fractal persan carpet for P=1 when x=0 and y=0 else P=0. Let's try this on MATHCAD.

[hgjf2]

An example of so infinity matrix whick using the biharmonicity equation but for the calculus of efforts for a board suppossed at vibration is:

    -1    ;0    ;0    ;0    ;0    ;0    ;0    ;...
     3    ;1    ;0    ;0    ;0    ;0    ;0    ;...
     24   ;-2   ;-1   ;0    ;0    ;0    ;0    ;...
     -64  ;-8   ;1    ;1    ;0    ;0    ;0    ;...
     0     ;0    ;0    ;0    ;-1   ;0    ;0    ;...
     64   ;8    ;0    ;0    ;-1   ;1    ;0    ;...
     424  ;0   ;8   ;0     ;8     ;2    ;-1  ;...
.............................................................
 this is only a part of this infinity matrix

[DarkBeam]

This looks interesting but I cannot understand you!
Please can you explain step by step how you mix matrix... differential equations ... and finally the automatas?
Those concepts are so distant that I just cannot imagine how to mix em

[hgjf2]

This looks interesting but I cannot understand you!
Please can you explain step by step how you mix matrix... differential equations ... and finally the automatas?
Those concepts are so distant that I just cannot imagine how to mix em

This part of the infinity "carpets" of number not need an differential equation. Just applying only the formula f(x-2;y)-8*f(x-1;y)+
20*f(x;y)-8*f(x+1;y)+f(x+2;y)+2*f(x-1;y-1)-8*f(x;y-1)+2*f(x+1;y-1)+2*f(x-1;y+1)-8*f(x;y+1)+2*f(x+1;y+1)+
f(x;y-2)+f(x;y+2)=0

The ordering is:   
                            1
                        2; -8; 2
                     1;-8;20;-8;1
                         2;-8; 2
                             1
     is just numerical interpolation for equation ((d2f/dx2)+(f2f/dy2))^2=0 the biharmonical equation.