Dehn Planes
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).
Read more about Dehn Planes: Dehn's Non-archimedean Field Ω(t), Dehn's Semi-Euclidean Geometry, Dehn's Non-Legendrian Geometry
Famous quotes containing the word planes:
“After the planes unloaded, we fell down
Buried together, unmarried men and women;”
—Robert Lowell (19171977)