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Dehn plane
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 and discussed by .
==Dehn's non-archimedean field Ω(''t'')==
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.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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