Dehn Planes - Dehn's Semi-Euclidean Geometry

Dehn's Semi-Euclidean Geometry

The set of all pairs (x, y), where x and y are any (possibly infinite) elements of the field Ω(t), and with the usual metric

which takes values in Ω(t), gives a model of Euclidean geometry. The parallel postulate is true in this model, but if the deviation from the perpendicular is infinitesimal (meaning smaller than any positive rational number), the intersecting lines intersect at a point that is not in the finite part of the plane. Hence, if the model is restricted to the finite part of the plane (points (x,y) with x and y finite), a geometry is obtained in which the parallel postulate fails but the sum of the angles of a triangle is π. This is Dehn's semi-Euclidean geometry.