Simpler foundations for the hyperbolic plane-Reference-Cited by-同舟云学术

Simpler foundations for the hyperbolic plane

Published:2023-03-03 Issue:5 Volume:35 Page:1301-1325
ISSN:0933-7741
Container-title:Forum Mathematicum
language:en
Short-container-title:

Author:

Bamberg John¹^ORCID,Penttila Tim²

Affiliation:

1. Department of Mathematics and Statistics , The University of Western Australia , Crawley , WA 6009 , Australia

2. School of Mathematical Sciences , The University of Adelaide , Adelaide , SA 5005 , Australia

Abstract

Abstract H. L. Skala (1992) gave the first elegant first-order axiom system for hyperbolic geometry by replacing Menger’s axiom involving projectivities with the theorems of Pappus and Desargues for the hyperbolic plane. In so doing, Skala showed that hyperbolic geometry is incidence geometry. We improve upon Skala’s formulation by doing away with Pappus and Desargues altogether, by substituting for them two simpler axioms.

Funder

Australian Research Council

Publisher

Walter de Gruyter GmbH

Subject

Applied Mathematics,General Mathematics

Link

https://www.degruyter.com/document/doi/10.1515/forum-2022-0268/pdf

Reference31 articles.

1. J. C. Abbott, The projective theory of non-Euclidean geometry. I, Rep. Math. Colloq. (2) 3 (1941), 13–27.

2. J. C. Abbott, The Projective Theory of Non-Euclidean Geometry, ProQuest LLC, Ann Arbor, 1942; Thesis (Ph.D.)–University of Notre Dame.

3. J. C. Abbott, The projective theory of non-Euclidean geometry. II, Rep. Math. Colloq. (2) 4 (1943), 22–30.

4. J. C. Abbott, The projective theory of non-Euclidean geometry, Rep. Math. Colloq. (2) 5(6) (1944), 43–52.

5. F. Bachmann, Aufbau der Geometrie aus dem Spiegelungsbegriff, Grundlehren Math. Wiss. 96, Springer, Berlin, 1973.