In geometry, Dehn constructed 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).
Other articles related to "dehn planes, dehn, plane":
... In the same paper, Dehn also constructed an example of a non-Legendrian geometry where there are infinitely many lines through a point not meeting another line, but the sum of the angles in a triangle exceeds π ... Riemann's elliptic geometry over Ω(t) consists of the projective plane over Ω(t), which can be identified with the affine plane of points (xy1) together with the "line at infinity ... of the angles of a triangle is at most π, but assumes Archimedes's axiom, and Dehn's example shows that Legendre's theorem need not hold if Archimedes' axiom is dropped ...
Famous quotes containing the word planes:
“After the planes unloaded, we fell down
Buried together, unmarried men and women;”
—Robert Lowell (19171977)