A correlation for a balanced design not based on a finite geometry is defined as an anti-isomorphism between the treatments and blocks of a given design. Clearly,the treatments and blocks must have the same cardinality. The product of two correlations is necessarily an automorphism. If the product of a correlation with itself is an isomorphism, then the correlation is termed a polarity.
BIBD with parameters v=19, k = 3, λ = 1 continuing the example given here, and re-numbering treatment 0 as 19. Developing this design, the second BIBD is isomorphic to the first, but differently labelled: in order to make sense of the geometric approach,it is convenient to renumber the second set treatments (12,5,17,10,3,15,8,1,13,6,18,11,4,16,9,2,14,7,19) i.e. so that treatment 1 becomes 12, 2 becomes 5, etc. The second set treatments occurring with each of the first treatments are given in the table below. This gives a design in which the initial blocks are {(1,1;7,7;11,11),(4,4;9,9;6,6),(16,16;17,17;5,5)} where the first set of treatments is cycled as usual, but the second set according to the permutation already given i.e. (12,5,17,10,3,15,8,1,13,6,18,11,4,16,9,2,14,7,19).
The following table works through this process for each of the points.
| Point | Plane | Point | Plane | Point | Plane | Point |
|---|---|---|---|---|---|---|
| 1 | 1'={1,3,5,6,7,8,11,14,15} | 11 | 11' | 7 | 7' | 1 |
| 2 | 2'={1,4,7,8,13,15,17,18,19} | 3 | 3' | 14 | 14' | 2 |
| 3 | 3'={1,6,8,10,11,12,13,16,19} | 14 | 14' | 2 | 2' | 3 | \\
| 4 | 4'={1,3,4,5,6,9,12,13,18} | 6 | 6' | 9 | 9' | 4 |
| 5 | 5'={2,5,6,11,13,15,16,17,18} | 17 | 17' | 16 | 16' | 5 |
| 6 | 6'={4,6,8,9,10,11,14,17,18} | 9 | 9' | 4 | 4' | 6 |
| 7 | 7'={1,2,3,4,7,10,11,16,18} | 1 | 1' | 11 | 11' | 7 |
| 8 | 8'={3,4,9,11,13,14,15,16,19} | 12 | 12' | 18 | 18' | 8 |
| 9 | 9'={2,4,6,7,8,9,12,15,16} | 4 | 4' | 6 | 6' | 9 |
| 10 | 10'={1,2,5,8,9,14,16,18,19} | 15 | 15' | 13 | 13' | 10 |
| 11 | 11'={1,2,7,9,11,12,13,14,17} | 7 | 7' | 1 | 1' | 11 |
| 12 | 12'={2,4,5,6,7,10,13,14,19} | 18 | 18' | 8 | 8' | 12 |
| 13 | 13'={3,6,7,12,14,16,17,18,19} | 10 | 10' | 15 | 15' | 13 |
| 14 | 14'={5,7,9,10,11,12,15,18,19} | 2 | 2' | 3 | 3' | 14 |
| 15 | 15'={2,3,4,5,8,11,12,17,19} | 13 | 13' | 10 | 10' | 15 |
| 16 | 16'={1,4,5,10,12,14,15,16,17} | 5 | 5' | 17 | 17' | 16 |
| 17 | 17'={3,5,7,8,9,10,13,16,17} | 16 | 16' | 5 | 5' | 17 |
| 18 | 18'={1,2,3,6,9,10,15,17,19} | 8 | 8' | 12 | 12' | 18 |
| 19 | 19'={2,3,8,10,12,13,14,15,18} | 19 | 19' | 19 | 19' | 19 |
These planes define a SBIBD, as they must, by the definition of a pergola. If these planes are numbered analogously to the treatments, so that plane 1' corresponds to treatment 1, etc, the points corresponding to the planes under this correlation are given by the intersections of the planes corresponding to the points in that plane. For example, plane 1', corresponding to point 1, is defined by the points {1,3,5,6,7,8,11,14,15}: so the point corresponding to plane 1' is defined by the planes {1',3',5',6',7',8',11',14',15'}, namely, 11. The points corresponding to the planes {1', 2',...,19'} are, respectively: {11, 3, 14, 6, 17, 9, 1, 12, 4, 15, 7, 18, 10, 2, 13, 5, 16, 8, 19}. So this is a correlation. The point orbits are: {(1,11,7)(2,3,14)(4,6,9)(5,17,16)(8,12,18)(10,15,13)(19)}