Semi-Euclidean geometry

In geometry, Dehn introduced two examples of planes, a semi-Euclidean geometry and a non-Legendrian geometry, that have infinitely many lines parallel to a given one that pass through a given point, but where the sum of the angles of a triangle is at least π. A similar phenomenon occurs in hyperbolic geometry, except that the sum of the angles of a triangle is less than π. Dehn's examples use a non-Archimedean field, so that the Archimedean axiom is violated. They were introduced by Max Dehn (1900) and discussed by Hilbert (1902, p.127–130, or p. 42-43 in some later editions).

To construct his geometries, Dehn used a non-Archimedean ordered Pythagorean field Ω(t), a Pythagorean closure of the field of rational functions R(t), consisting of the smallest field of real-valued functions on the real line containing the real constants, the identity function t (taking any real number to itself) and closed under the operation ω → √(1+ω²). The field Ω(t) is ordered by putting x>y if the function x is larger than y for sufficiently large reals. An element x of Ω(t) is called finite if m<x<n for some integers m,n, and is called infinite otherwise.

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. It is discussed in Rucker (1982, page 98).

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