Fuzzy Geometry

Probabilistic version of geometry, labeled the “fuzzy” geometry, is introduced. The fifth postulate of Euclid (Playfair’s axiom) is adopted in the following probabilistic form: consider a line and a point not an line, there is exactly one line through the point with probability P, where 0≤P≤1. Playfair’s axiom is logically independent of the rest of the Hilbert system of axioms of the Euclidian geometry. Thus, the probabilistic version of the Playfair axiom may be combined with other Hilbert axioms. P=1 corresponds to the standard Euclidean geometry; P=0 corresponds to the elliptic- and hyperbolic-like geometries. 0<P<1 corresponds to the fuzzy geometry. Parallel constructions in this case are Bernoulli trials. Theorems of the fuzzy geometry are discussed. Given a triangle and a line drawn from a vertex parallel to the opposite side, the event that this line is actually parallel occurs with probability P. Otherwise, the line may intersect the side or diverge. Parallelism is not transitive in the fuzzy geometry. Fuzzy geometry occurs on the surface with a stochastically variable Gaussian curvature. Alternative fuzzy geometries adopting various versions of the probabilistic Playfair axiom are introduced. Probabilistic non-Archimedean fuzzy geometry is addressed. Applications of the fuzzy geometry are discussed.