Lets us consider

A row of lattice point a,b,c,d

Let the system process n- fold rotation symmetry , then rotation by angle of  theta other in clock wise and anti clockwise direction leads to some new lattice point p & s . let  ‘ t ‘  be translation vector  

 In    

       Cos  

       Cos  

        PQ = t cos

Bp = t   , cs = t  , p s = mt

Pq + qr + rs = mt (  “m “ is intiger )

T cos   + t + t cos = mt

2t cos  + t = mt

2t cos  = mt – t                     ( mt – t =  ( m- 1 ) t

2t cos   = ( m-1 ) t

2cos  = m-1

Cos  =       ( m-1   is also intiger )

       =

WHERE   N = m-1

Cos    =   [

Cos has value lying between   + 1  &  – 1

From eq –  1

-2     ( -2,-1, 0 , 1, 2 )

symmetry

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