The Quadrant magic (feature)  

The following defined the possibe symmetry types of a pattern  
Quadrant  totally symmetric pattern of m cells  
Total  totally symmetric pattern (any other number of cells)  
Windmill  rotation symmetric pattern  
Lines  pattern symmetric in both central lines  
Diagonals  pattern symmetric in both diagonals  
Center  pattern symmetric in center  
Main diagonal  pattern symmetric in main diagonal  
sub diagonal  pattern symmetric in sub diagonal  
Horizontal  pattern symmetric in central horizontal line  
Vertical  pattern symmetric in central vertical line  
Not  pattern has no symmetry  
The following define the copying possibilities of a pattern The "Placement Policy" 

mirrored  pattern is mirrored  
copyed  pattern is copied  
rotated  pattern is rotated  
order dependent quadrant definitions  
orders  quadrants 
1st kind total patterns 
2nd kind total patterns 
4k1 
2k1 by 2k1 subsquare free central line from adjoining quadrants 
(^{(2k1)^2}_{4k1})  (^{(2k1)^2}_{2k1}) 
2k by 2k subsquare common central line to two adjoining quadrants 
(^{(2k)^2}_{4k1})  (^{(2k)^2}_{2k})  
4k 
2k by 2k subsquare adjoining quadrants 
(^{(2k)^2}_{4k})  (^{(2k)^2}_{2k}) 
4k+1 
2k+1 by 2k+1 subsquare common central line to two adjoining quadrants 
(^{(2k+1)^2}_{4k+1})  (^{(2k+1)^2}_{2k+1}) 
2(2k+1) 
2k+1 by 2k+1 subsquare adjoining quadrants 
(^{(2k+1)^2}_{2(2k+1)})  (^{(2k+1)^2}_{2k+1}) 
pattern counting formulae  
4k1  quadrant  none  
windmill  none  
lines/diags  _{j=1}∑^{k1} (^{(2k2)}_{j})(^{(k1)^2}_{kj})  
line/diag.  _{j=0}∑^{k}(^{2k1}_{2k12j})(^{(k1)(2k1)}_{k+j})  
center  (^{2k(k1)}_{k1})  
4k  quadrant 
_{j=0}∑^{k div 2} (^{k}_{k2j})(^{k(k1)/2}_{j}) 

windmill  (^{k^2}_{k})  
lines  (^{k^2}_{k})  
diagonals  _{j=0}∑^{k} (^{2k}_{2k2j})(^{k(k1)}_{j})  
line  (^{2k^2}_{2k})  
diagonal  _{j=0}∑^{k} (^{2k}_{2k2j})(^{k(2k1)}_{k+j})  
center  (^{2k^2}_{2k})  
4k+1  quadrant  _{j=0}∑^{k div 2}(^{k(k1) div 2}_{j})(^{2k}_{k2j})  
windmill  (^{k(k+1)}_{k})  
lines/diags  _{j=0}∑^{k} (^{2k}_{2j})(^{k*k}_{kj})  
line/diag.  _{j=0}∑^{k}(^{2k+1}_{2j+1})(^{k(2k+1)}_{2kj})  
center  (^{2k(k+1)}_{2k})  
2(2k+1)  quadrant  none  
windmill  none  
lines/diags  _{j=0}∑^{k} (^{2k}_{2j+2})(^{k*k}_{kj})  
line/diag.  _{j=0}∑^{k}(^{2k+1}_{2j})(^{k(2k+1)}_{2k+1j})  
center  (^{2k(k+1)}_{2k+1}) 