  The paper deals with a particular type of a projective ring plane defined
over the ring of double numbers over Galois fields, R\_{\otimes}(q) \equiv
GF(q) \otimes GF(q) \cong GF(q)[x]/(x(x-1)). The plane is endowed with (q^2 + q
+ 1)^2 points/lines and there are (q + 1)^2 points/lines incident with any
line/point. As R\_{\otimes}(q) features two maximal ideals, the neighbour
relation is not an equivalence relation, i. e. the sets of neighbour points to
two distant points overlap. Given a point of the plane, there are 2q(q+1)
neighbour points to it. These form two disjoint, equally-populated families
under the reduction modulo either of the ideals. The points of the first family
merge with (the image of) the point in question, while the points of the other
family go in a one-to-one fashion to the remaining q(q + 1) points of the
associated ordinary (Galois) projective plane of order q. The families swap
their roles when switching from one ideal to the other, which can be regarded
as a remarkable, finite algebraic geometrical manifestation/representation of
the principle of complementarity. Possible domains of application of this
finding in (quantum) physics, physical chemistry and neurophysiology are
briefly mentioned.
