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Degenerate planes

Degenerate planes do not fulfill the third condition in the definition of a projective plane. They are not structurally complex enough to be interesting in their own right, but from time to time they arise as special cases in general arguments. There are seven kinds of degenerate plane according to (Albert & Sandler 1968). They are:

These seven cases are not independent, the fourth and fifth can be considered as special cases of the sixth, while the second and third are special cases of the fourth and fifth respectively. The special case of the seventh plane with no additional lines can be seen as an eighth plane. All the cases can therefore be organized into two families of degenerate planes as follows (this representation is for finite degenerate planes, but may be extended to infinite ones in a natural way):

1) For any number of points P,..., P, and lines L,..., L,

2) For any number of points P,..., P, and lines L,..., L, (same number of points as lines)