In this paper the axiomatic basis will be a general absolute plane A = (P,L, α,≡) in the sense of [6], where P and L denote respectively the set of points and the set of lines, α the order structure and ≡ the congruence, and where furthermore the word “general” means that no claim is made on any kind of continuity assumptions. Starting from the classification of general absolute geometries introduced in [5] by means of the notion of congruence, singular or hyperbolic or elliptic, we get now a complete characterization of the different possibilities which can occur in a general absolute plane studying the value of the angle δ defined in any Lambert–Saccheri quadrangle or, equivalently, the sum of the angles of any triangle. This yelds, in particular, a Archimedes-free proof of a statement generalizing the classical “first Legendre theorem” for absolute planes.

Pianta, S., Marchi, M., Karzel, H., Legendre-like theorems in a general absolute geometry., <<RESULTS IN MATHEMATICS>>, 2007; (51): 61-71. [doi:10.1007/s00025-007-0258-0] [http://hdl.handle.net/10807/54951]

Legendre-like theorems in a general absolute geometry.

Pianta, Silvia;Marchi, Mario;Karzel, Helmut
2007

Abstract

In this paper the axiomatic basis will be a general absolute plane A = (P,L, α,≡) in the sense of [6], where P and L denote respectively the set of points and the set of lines, α the order structure and ≡ the congruence, and where furthermore the word “general” means that no claim is made on any kind of continuity assumptions. Starting from the classification of general absolute geometries introduced in [5] by means of the notion of congruence, singular or hyperbolic or elliptic, we get now a complete characterization of the different possibilities which can occur in a general absolute plane studying the value of the angle δ defined in any Lambert–Saccheri quadrangle or, equivalently, the sum of the angles of any triangle. This yelds, in particular, a Archimedes-free proof of a statement generalizing the classical “first Legendre theorem” for absolute planes.
2007
Inglese
Pianta, S., Marchi, M., Karzel, H., Legendre-like theorems in a general absolute geometry., <<RESULTS IN MATHEMATICS>>, 2007; (51): 61-71. [doi:10.1007/s00025-007-0258-0] [http://hdl.handle.net/10807/54951]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10807/54951
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